Multidimensional Stochastic Processes as Rough Paths: Theory and Applications (Cambridge Studies in Advanced Mathematics)

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 120 Editorial Board ´ W. FULTON, A. KATOK, F. KIRWAN, B. BOLLOBAS, P. SARNAK, B. SIMON, B. TOTARO

MULTIDIMENSIONAL STOCHASTIC PROCESSES AS ROUGH PATHS Rough path analysis provides a fresh perspective on Itˆo’s important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin–Wentzell theory, the Stroock–Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong–Zakai, McShane’s counterexample) receive “obvious” rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations, and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.

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Multidimensional Stochastic Processes as Rough Paths Theory and Applications PETER K. FRIZ NICOLAS B. VICTOIR

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Dubai, Tokyo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521876070 © P. K. Friz and N. B. Victoir 2010 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published in print format 2010 ISBN-13

978-0-511-68004-5

eBook (EBL)

ISBN-13

978-0-521-87607-0

Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

To Wendy and Laura

Contents

Preface Introduction

page xiii 1

1 2 3

The story in a nutshell From ordinary to rough differential equations Carnot–Caratheodory geometry Brownian motion and stochastic analysis

4 4 8 13

I

Basics

1 1.1 1.2 1.3 1.4 1.5

Continuous paths of bounded variation Continuous paths on metric spaces Continuous paths of bounded variation on metric spaces Continuous paths of bounded variation on Rd Sobolev spaces of continuous paths of bounded variation Comments

19 19 21 29 39 44

2 2.1 2.2 2.3

Riemann–Stieltjes integration Basic Riemann–Stieltjes integration Continuity properties Comments

45 45 49 52

3 3.1 3.2 3.3 3.4 3.5 3.6

Ordinary differential equations Preliminaries Existence Uniqueness A few consequences of uniqueness Continuity of the solution map Comments

53 53 55 59 60 62 67

4 4.1 4.2

ODEs: smoothness Smoothness of the solution map Comments

68 68 76

viii

Contents

5 5.1 5.2 5.3 5.4 5.5 5.6

Variation and H¨older spaces H¨older and p-variation paths on metric spaces Approximations in geodesic spaces H¨older and p-variation paths on Rd Generalized variation Higher-dimensional variation Comments

77 77 88 92 99 104 111

6 6.1 6.2 6.3 6.4 6.5

Young integration Young–L´oeve estimates Young integrals Continuity properties of Young integrals Young–L´oeve–Towghi estimates and 2D Young integrals Comments

112 112 115 118 119 122

II

Abstract theory of rough paths

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7

Free nilpotent groups Motivation: iterated integrals and higher-order Euler schemes Step-N signatures and truncated tensor algebras Lie algebra tN (Rd ) and Lie group 1 + tN (Rd ) Chow’s theorem Free nilpotent groups The lift of continuous bounded variation paths on Rd Comments

125 125 128 134 140 142 156 163

8 8.1 8.2 8.3 8.4 8.5 8.6 8.7

Variation and H¨older spaces on free groups p-Variation and 1/p-H¨older topology Geodesic approximations Completeness and non-separability The d0 /d∞ estimate Interpolation and compactness Closure of lifted smooth paths Comments

165 166 174 175 175 177 178 181

9 9.1 9.2 9.3 9.4 9.5

Geometric rough path spaces The Lyons-lift map x → SN (x) Spaces of geometric rough paths Invariance under Lipschitz maps Young pairing of weak geometric rough paths Comments

182 183 191 196 197 211

Contents

ix

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

Rough differential equations Preliminaries Davie’s estimate RDE solutions Full RDE solutions RDEs under minimal regularity of coefficients Integration along rough paths RDEs driven along linear vector fields Appendix: p-variation estimates via approximations Comments

212 212 215 221 241 248 253 262 268 279

11 11.1 11.2 11.3 11.4

RDEs: smoothness Smoothness of the Itˆo–Lyons map Flows of diffeomorphisms Application: a class of rough partial differential equations Comments

281 281 289 294 301

12 12.1 12.2 12.3

RDEs with drift and other topics RDEs with drift terms Application: perturbed driving signals and impact on RDEs Comments

302 302 316 324

III

Stochastic processes lifted to rough paths

13 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10

Brownian motion Brownian motion and L´evy’s area Enhanced Brownian motion Strong approximations Weak approximations Cameron–Martin theorem Large deviations Support theorem Support theorem in conditional form Appendix: infinite 2-variation of Brownian motion Comments

327 327 333 339 354 357 359 367 370 381 383

14 14.1 14.2 14.3 14.4

Continuous (semi-)martingales Enhanced continuous local martingales The Burkholder–Davis–Gundy inequality p-Variation rough path regularity of enhanced martingales Burkholder–Davis–Gundy with p-variation rough path norm

386 386 388 390 392

x

Contents

14.5 Convergence of piecewise linear approximations 14.6 Comments

395 401

15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10

Gaussian processes Motivation and outlook One-dimensional Gaussian processes Multidimensional Gaussian processes The Young–Wiener integral Strong approximations Weak approximations Large deviations Support theorem Appendix: some estimates in G3 (Rd ) Comments

402 402 404 416 433 436 442 445 448 451 452

16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10

Markov processes Motivation Uniformly subelliptic Dirichlet forms Heat-kernel estimates Markovian rough paths Strong approximations Weak approximations Large deviations Support theorem Appendix: analysis on free nilpotent groups Comments

454 454 457 463 464 467 480 483 484 493 499

IV

Applications to stochastic analysis

17 17.1 17.2 17.3

Stochastic differential equations and stochastic flows Working summary on rough paths Rough paths vs Stratonovich theory Stochastic differential equations driven by non-semi-martingales Limit theorems Stochastic flows of diffeomorphisms Anticipating stochastic differential equations A class of stochastic partial differential equations Comments

17.4 17.5 17.6 17.7 17.8

18 Stochastic Taylor expansions 18.1 Azencott-type estimates

503 503 506 515 517 521 523 525 526 528 528

Contents

xi

18.2 Weak remainder estimates 18.3 Comments

531 532

19 Support theorem and large deviations 19.1 Support theorem for SDEs driven by Brownian motion 19.2 Support theorem for SDEs driven by other stochastic processes 19.3 Large deviations for SDEs driven by Brownian motion 19.4 Large deviations for SDEs driven by other stochastic processes 19.5 Support theorem and large deviations for a class of SPDEs 19.6 Comments

533 533 536 538 541 542 544

20 20.1 20.2 20.3 20.4 20.5

545 545 549 550 553 566

Malliavin calculus for RDEs H-regularity of RDE solutions Non-degenerate Gaussian driving signals Densities for RDEs under ellipticity conditions Densities for RDEs under H¨ormander’s condition Comments

Appendices A A.1 A.2 A.3 A.4 A.5

Sample path regularity and related topics Continuous processes as random variables The Garsia–Rodemich–Rumsey estimate Kolmogorov-type corollaries Sample path regularity under Gaussian assumptions Comments

571 571 573 582 587 596

B B.1 B.2 B.3 B.4

Banach calculus Preliminaries Directional and Fr´echet derivatives Higher-order differentiability Comments

597 597 598 601 602

C C.1 C.2

Large deviations Definition and basic properties Contraction principles

603 603 604

D D.1 D.2 D.3 D.4

Gaussian analysis Preliminaries Isoperimetry and concentration of measure L2 -expansions Wiener–Itˆo chaos

606 606 608 610 610

xii

Contents

D.5 D.6

Malliavin calculus Comments

613 614

E E.1 E.2 E.3 E.4 E.5 E.6 E.7

Analysis on local Dirichlet spaces Quadratic forms Symmetric Markovian semi-groups and Dirichlet forms Doubling, Poincar´e and quasi-isometry Parabolic equations and heat-kernels Symmetric diffusions Stochastic analysis Comments

615 615 617 620 623 625 627 635

Frequently used notation References Index

636 638 652

Preface

This book is split into four parts. Part I is concerned with basic material about certain ordinary differential equations, paths of H¨older and variation regularity, and the rudiments of Riemann–Stieltjes and Young integration. Nothing here will be new to specialists, but the material seems rather spread out in the literature and we hope it will prove useful to have it collected in one place. Part II is about the deterministic core of rough path theory, a` la T. J. Lyons, but actually inspired by the direct approach of A. M. Davie. Although the theory can be formulated in a Banach setting, we have chosen to remain in a finitedimensional setting; our motivation for this decision comes from the fact that the bulk of classic texts on Brownian motion and stochastic analysis take place in a similar setting, and these are the grounds on which we sought applications. In essence, with rough paths one attempts to take out probability from the theory of stochastic differential equations – to the extent possible. Probability still matters, but the problems are shifted from the analysis of the actual SDEs to the analysis of elementary stochastic integrals, known as L´evy’s stochastic area. In Part III we start with a detailed discussion of how multidimensional Brownian motion can be turned into a (random) rough path; followed by a similar study for (continuous) semi-martingales and large classes of multidimensional Gaussian – and Markovian – processes. In Part IV we apply the theory of rough differential equations (RDEs), pathby-path, with the (rough) sample paths constructed in Part III. In the setting of Brownian motion or semi-martingales, the resulting (random) RDE solutions are identified as solutions to classical stochastic differential equations. We then give a selection of applications to stochastic analysis in which rough path techniques have proved useful. The prerequisites for Parts I and II are essentially a good command of undergraduate analysis. Some knowledge of ordinary differential equations (existence, uniqueness results) and basic geometry (vector fields, geodesics) would be helpful, although everything we need is discussed. In Part III, we assume a general background in measure theoretic probability theory and the basics of stochastic processes, such as Brownian motion. Stochastic area (for Brownian motion) is introduced via stochastic integration, with alternatives described in the text. In the respective chapters on semi-martingales,

xiv

Preface

Gaussian and Markovian processes, the reader is assumed to have the appropriate background, most of which we have tried to collect in the appendices. Part IV deals with applications to stochastic analysis, stochastic (partial) differential equations in particular. For a full appreciation of the results herein, the reader should be familiar with the relevant background; textbook references are thus given whenever possible at the end of chapters. Exercises are included throughout the text, often with complete (or sketched) solutions. It is our pleasure to thank our mentors, colleagues and friends. This book would not exist without the teachings of our PhD advisors, S. R. S. Varadhan and T. J. Lyons; both remained available for discussions at various stages throughout the writing process. Once we approached completion, a courageous few offered to do some detailed reading: C. Bayer, M. Caruana, T. Cass, A. Deya, M. Huesmann, H. Oberhauser, J. Teichmann and S. Tindel. Many others offered their time and support in various forms: G. Ben Arous, C. Borrell, F. Baudoin, R. Carmona, D. Chafa¨ı, T. Coulhon, L. Coutin, M. Davis, B. Davies, A. Davie, D. Elworthy, M. Gubinelli, M. Hairer, B. Hambly, A. Iserles, I. Karatzas, A. Lejay, D. L´epingle, P. Malliavin, P. Markowitch, J. Norris, Z. Qian, J. Ram´ırez, J. Robinson, C. Rogers, M. Sanz-Sole and D. Stroock. This is also a welcome opportunity to thank C. Obtresal, C. Schmeiser, R. Schnabl and W. Wertz for their early teachings. The first author expresses his deep gratitude to the Department of Pure Mathematics and Mathematical Statistics, Cambridge and King’s College, Cambridge, where work on this book was carried out under ideal circumstances; he would also like to thank the Radon Institute and his current affiliations, TU and WIAS Berlin, where this book was finalized. Partial support from the Leverhulme Trust and EPSRC grant EP/E048609/1 is gratefully acknowledged. The second author would like to thank the Mathematical Institute, Oxford and Magdalen College, Oxford, where work on the early drafts of this book was undertaken. Finally, it is our great joy to thank our loving families. Peter K. Friz (Cambridge, Berlin) Nicolas B. Victoir (Hong Kong) June 2009

Introduction One of the remarkable properties of Brownian motion is that we can use it to construct (stochastic) integrals of the type . . . dB. The reason this is remarkable is that almost every Brownian sample path (Bt (ω) : t ∈ [0, T ]) has inﬁnite variation and there is no help from the classical Stieltjes integration theory. Instead, Itˆ o’s theory of stochastic integration relies crucially on the fact that B is a martingale and stochastic integrals themselves are constructed as martingales. If one recalls the elementary interpretation of martingales as fair games one sees that Itˆ o integration is some sort of martingale transform in which the integrand has the meaning of a gambling strategy. Clearly then, the integrand must not anticipate the random movements of the driving Brownian motion and one is led to the class of so-called previsible processes which can be integrated against Brownian motion. When such integration is possible, it allows for a theory of stochastic diﬀerential equations (SDEs) of the form1 dY =

d

Vi (Y ) dB i + V0 (Y ) dt , Y (0) = y0 .

(∗)

i=1

Without going into too much detail, it is hard to overstate the importance of Itˆo’s theory: it has a profound impact on modern mathematics, both pure and applied, not to speak of applications in ﬁelds such as physics, engineering, biology and ﬁnance. It is natural to ask whether the meaning of (∗) can be extended to processes other than Brownian motion. For instance, there is motivation from mathematical ﬁnance to generalize the driving process to general (semi-)martingales and luckily Itˆ o’s approach can be carried out naturally in this context. We can also ask for a Gaussian generalization, for instance by considering a diﬀerential equation of the form (∗) in which the driving signal may be taken from a reasonably general class of Gaussian processes. Such equations have been proposed, often in the setting of fractional Brownian motion of Hurst parameter H > 1/2,2 as toy models to study the ergodic behaviour B = B 1 , . . . , B d is a d-dimensional Brownian motion. 2 Hurst parameter H = 1/2 corresponds to Brownian motion. For H > 1/2, one has enough sample path regularity to use Young integration. 1 Here

2

Introduction

of non-Markovian systems or to provide new examples of arbitrage-free markets under transactions costs. Or we can ask for a Markovian generalization. Indeed, it is not hard to think of motivating physical examples (such as heat ﬂow in rough media) a in which Brownian motion B may be replaced by a Markov ij X process 1 with uniformly elliptic generator in divergence form, say 2 i,j ∂i a ∂j · , without any regularity assumptions on the symmetric matrix aij . The Gaussian and Markovian examples have in common that the sample path behaviour can be arbitrarily close to Brownian motion (e.g. by taking H = 1/2 ± ε resp. a uniformly ε-close to the identity matrix I). And yet, Itˆ o’s theory has a complete breakdown! It has emerged over recent years, starting with the pioneering work of T. Lyons [116], that diﬀerential equations driven by such non-semimartingales can be solved in the rough path sense. Moreover, the soobtained solutions are not abstract nonsense but have ﬁrm probabilistic justiﬁcation. For instance, if the driving signal converges to Brownian motion (in some reasonable sense which covers ε → 0 in the aforementioned examples) the corresponding rough path solutions converge to the classical Stratonovich solution of (∗), as one would hope. While this alone seems to allow for ﬂexible and robust stochastic modelling, it is not all about dealing with new types of driving signals. Even in the classical case of Brownian motion, we get some remarkable insights. Namely, the (Stratonovich) solution to (∗) can be represented as a deterministic and continuous image of Brownian motion and L´evy’s stochastic area t t 1 B j dB k − B k dB j Ajt k (ω) = 2 0 0 alone. In fact, there is a “nice” deterministic map, the Itˆ o–Lyons map, (y0 ; x) → π (0, y0 ; x) which yields, upon setting x = B i , Aj,k : i, j, k ∈ {1, . . . , d} a very pleasing version of the solution of (∗). Indeed, subject to suﬃcient regularity of the coeﬃcients, we see that (∗) can be solved simultaneously for all starting points y0 , and even all coeﬃcients! Clearly then, one can allow the starting point and coeﬃcients to be random (even dependent on the entire future of the Brownian driving signals) without problems; in stark contrast to Itˆ o’s theory which struggles with the integration of non-previsible integrands. Also, construction of stochastic ﬂows becomes a trivial corollary of purely deterministic regularity properties of the Itˆ o–Lyons map. This brings us to the (deterministic) main result of the theory: continuity of the Itˆo–Lyons map x → π (0, y0 ; x) in “rough path” topology. When applied in a standard SDE context, it quickly gives an entire catalogue of limit theorems. It also allows us to

Introduction

3

reduce (highly non-trivial) results, such as the Stroock–Varadhan support theorem or the Freidlin–Wentzell estimates, to relatively simple statements about Brownian motion and L´evy’s area. Moreover, and at no extra price, all these results come at the level of stochastic ﬂows. The Itˆo–Lyons map is also seen to be regular in certain perturbations of x which include (but are not restricted to) the usual Cameron–Martin space, and so there is a natural interplay with Malliavin calculus. At last, there is increasing evidence that rough path techniques will play an important role in the theory of stochastic partial diﬀerential equations and we have included some ﬁrst results in this direction. All that said, let us emphasize that the rough path approach to (stochastic) diﬀerential equations is not set out to replace Itˆo’s point of view. Rather, it complements Itˆo’s theory in precisely those areas where the former runs into diﬃculties. We hope that the topics discussed in this book will prove useful to anyone who seeks new tools for robust and ﬂexible stochastic modelling.

The story in a nutshell 1 From ordinary to rough diﬀerential equations Rough path analysis can be viewed as a collection of smart estimates for diﬀerential equations of type dy = V (y) dx ⇐⇒ y˙ =

d

Vi (y) x˙ i .

i=1

Although a Banach formulation of the theory is possible, we shall remain in ﬁnite dimensions here. of simplicity, let us assume that the For the sake driving signal x ∈ C ∞ [0, T ] , Rd and that the coeﬃcients V1 , . . . , Vd ∈ C ∞,b (Re , Re ), that is bounded with bounded derivatives of all orders. We are dealing with a simple time-inhomogenous ordinary diﬀerential equation (ODE) and there is no question about existence and uniqueness of an Re valued solution from every starting point y0 ∈ Re . The usual ﬁrst-order Euler approximation, from a ﬁxed time-s starting point ys , is obviously t yt − ys ≈ Vi (ys ) dxi . s

(We now adopt the summation convention over repeated up–down indices.) A simple Taylor expansion leads to the following step-2 Euler approximation, t

yt − ys ≈ Vi (ys )

s

t

dxi + Vik ∂k Vj (ys )

s

r

s

dxi dxj

=E(y s ,x s , t )

with xs,t =

t

t

s

r

dx ⊗ dx

dx, s

∈ Rd ⊕ Rd×d .

(1)

s

Let us now make the following H¨ older-type assumption: there exists c1 and α ∈ (0, 1] such that, for all s < t in [0, T ] and all i, j ∈ {1, . . . , d}, t dxi ∨ (Hα ) : s

t s

s

r

1/2 α dxi dxj ≤ c1 |t − s| .

(2)

t r 2 Note that s s dxi dxj is readily estimated by 2 |t − s| , where = |x| ˙ ∞;[0,T ] is the Lipschitz norm of the driving signal, and so (Hα ) holds,

1 From ordinary to rough diﬀerential equations

5

somewhat trivially for now, with c1 = and α = 1. [We shall see later that (Hα ) also holds for d-dimensional Brownian motion for any α < 1/2 and a random variable c1 (ω) < ∞ a.s. provided the double integral is understood in the sense of stochastic integration. Nonetheless, let us keep x deterministic and smooth for now.] It is natural to ask exactly how good these approximations are. The answer is given by Davie’s lemma which says that, assuming (Hα ) for some α ∈ (1/3, 1/2], one has the “step-2 Euler estimate” θ

|yt − ys − E (ys , xs,t )| ≤ c2 |t − s|

where θ = 3α > 1. The catch here is uniformity : c2 = c2 (c1 ) depends on x only through the H¨ older bound c1 but not on its Lipschitz norm. Since it is easy to see that (Hα ) implies α

E (ys , xs,t ) ≤ c3 |t − s| , c3 = c3 (c1 ) , the triangle inequality leads to α

|yt − ys | ≤ c4 |t − s| ,

c4 = c4 (c1 ) .

(3)

As often in analysis, uniform bounds allow for passage to the limit. We therefore take xn ∈ C ∞ [0, T ] , Rd with uniform bounds t i dxn ∨ sup n s

t s

s

r

1/2

dxin dxjn

α

≤ c1 |t − s|

such that, uniformly in t ∈ [0, T ], t t r (1) (2) i i j ∈ Rd ⊕ Rd×d . dxn , dxn dxn → xt ≡ xt , xt 0

0

0

d d×d The and the class of d limiting object x is a path with values in R ⊕ R d×d R ⊕R -valued paths obtained in this way is precisely what we call the α-H¨ older rough paths.1 Two important remarks are in order.

(i) The condition α ∈ (1/3, 1/2] in Davie’s estimate is intimately tied to the fact that the condition (Hα ) involves the ﬁrst two iterated integrals. (ii) The space Rd ⊕Rd×d is not quite the correct state space for x. Indeed, the calculus product rule d xi xj = xi dxj + xj dxi implies that2 t t t r 1 Sym dx ⊗ dx = dx ⊗ dx . 2 0 0 0 0 1 To 2

be completely geometric α-H¨ older rough path. honest, we call this a weak Sym (A) := 12 A + A T , Anti (A) := 12 A − A T for A ∈ Rd ×d .

6

The story in a nutshell

Figure 1. We plot s → (xis , xjs ) and the chord which connects (xi0 , xj0 ), on the lower left side, say, with (xit , xjt ) on the right side. The (signed) enclosed area (2 ) (here positive) is precisely Anti(xt )i , j .

This remains valid in the limit so that x (t) must take values in 1 (1) (2) d d×d (2) (1) (1) ∈R ⊕R = x ⊗x . x = x ,x : Sym x 2 We can get rid of this algebraic redundancy by switching from x to3 x(1) , Anti(x(2) ) ∈ Rd ⊕ so (d) . At least for a smooth path x (·) , this has an appealing geometric interpretation. Let (xi· , xj· ) denote the projection to two distinct coordinates (i, j); basic multivariable calculus then tells us that t t i 1 (2) xs − xi0 dxjs − xjs − xj0 dxis Anti(xt )i,j = 2 0 0 is the area (with multiplicity and orientation taken into account) between the curve {(xis , xjs ) : s ∈ [0, t]} and the chord from (xit , xjt ) to (xi0 , xj0 ). See Figure 1. Example 1 Consider d = 2 and xn (t) = n1 cos 2n2 t , n1 sin 2n2 t ∈ R2 . Then (Hα ) holds with α = 1/2, as may be seen by considering sepa1/2 rately the cases where 1/n is less resp. greater than (t − s) . Moreover, the limiting rough path is 0 0 t , , (4) xt ≡ 0 −t 0 since we run around the origin essentially n2 t/π times, sweeping out area π/n2 at each round. 3 As will be discussed in Chapter 7, this is precisely switching from the step-2 free nilpotent Lie group (with d generators) to its Lie algebra.

1 From ordinary to rough diﬀerential equations

7

We are now ready for the passage to the limit on the level of ODEs. To this end, consider (y n ) ⊂ C ([0, T ] , Re ), obtained by solving, for each n, the ODE dy n = V (y n ) dxn , y n (0) = y0 . By Davie’s lemma the sequence (yn ) has a uniform α-H¨ older bound c4 and by Arzela–Ascoli we see that (yn ) has at least one limit point in C ([0, T ] , Re ). Each such limit point is called a solution to the rough differential equation (RDE) which we write as dy = V (y)dx, y (0) = y0 .

(5)

The present arguments apply immediately for V ∈ C 2,b , that is bounded with two bounded derivatives, and more precisely for V ∈ Lipγ −1 ,γ > 1/α, in the sense of Stein.4 As in classical ODE theory, one additional degree of regularity (e.g. V ∈ Lipγ , γ > 1/α) then gives uniqueness5 and we will write y = π (V ) (0, y0 ; x) for such a unique RDE solution. At last, it should not be surprising from our construction that the RDE solution map (a.k.a. Itˆ o–Lyons map) x → π (V ) (0, y0 ; x) is continuous in x (e.g. under uniform convergence with uniform H¨ older bounds).

t r t with smooth Example 2 Assume xt = 0 dxi , 0 0 dxj dxk i,j,k ∈{1,...,d}

x. Then y = π (V ) (0, y0 ; x) is the classical ODE solution to dy = V (y) dx, y (0) = y0 .

Example 3 Assume x is given by (4) and V = (V1 , V2 ). Then y = π (V ) (0, y0 ; x) can be identiﬁed as the classical ODE solution to dy = [V1 , V2 ] (y) dt where [V1 , V2 ] = V1i ∂i V2 − V2i ∂i V1 is the Lie bracket of V1 and V2 .

4 Writing γ = γ+{γ} with integer γ and {γ} ∈ (0, 1] this means that V is bounded and has up to γ bounded derivatives, the last of which is H¨ o lder with exponent {γ}. 5 With more eﬀort, uniqueness can be shown under Lip 1 / a -regularity.

The story in a nutshell

8

Example 4 Assume B = B 1 , . . . , B d is a d-dimensional Brownian motion. Deﬁne enhanced Brownian motion by t t i j k Bt = dB , B ◦ dB 0

i,j,k ∈{1,...,d}

0

(where ◦ indicates stochastic integration in the Stratonovich sense). We shall see that B is an α-H¨older rough path for α ∈ (1/3, 1/2) and identify Yt (ω) := π (V ) (0, y0 ; B) as a solution to the Stratonovich stochastic diﬀerential equation6 dY =

d

Vi (Y ) ◦ dB i .

i=1

2 Carnot–Caratheodory geometry We now try to gain a better understanding of the results discussed in the last section. To this end, it helps to understand the more general case of H¨ older-type regularity with exponent α = 1/p ∈ (0, 1]. As indicated in remark (i), this will require consideration of more iterated integrals and we need suitable notation: given x ∈ C ∞ [0, T ] , Rd we generalize (1) to7 t

xt := SN (x)0,t :=

1,

dx, 0

∆ 2[0 , t ]

dx ⊗ dx, . . . ,

∆N [0 , t ]

dx ⊗ · · · ⊗ dx , (6)

called the step-N signature of x over the interval [0, t] , with values in ⊗2 ⊗N T N Rd := R ⊕ Rd ⊕ Rd ⊕ · · · ⊕ Rd . Observe that we added a zeroth scalar component in our deﬁnition of xt which is always set to 1. This is pure convention but has some algebraic advantages. To go further, we note that T N Rd has the structure of a (truncated) tensor-algebra with tensor-multiplication ⊗. (Elements with scalar component equal to 1 are always invertible with respect to ⊗.) Computations are simply carried out by considering the standard basis (ei ) of Rd as non-commutative indeterminants; for instance, i j a ei ⊗ b ej = ai bj (ei ⊗ ej ) = ai bj (ej ⊗ ei ). 6 A drift term V (y) dt can be trivially included by considering the time-space process 0 (t, B). 7 ∆k denotes the k-dimensional simplex over [0, t]. [0 , t ]

2 Carnot–Caratheodory geometry

9

The reason we are interested in this sort of algebra is that the trivial t xs,t ≡ (−xs ) + xt = dx =: xs,t s

generalizes to xs,t ≡

x−1 s

t ⊗ xt = 1, dx, s

∆ 2[s , t ]

dx ⊗ dx, . . . ,

∆N [s , t ]

dx ⊗ · · · ⊗ dx .

As a consequence, we have Chen’s relation xs,u = xs,t ⊗ xt,u , which tells us precisely how to “patch together” iterated integrals over adjacent intervals [s, t] and [t, u]. Let us now take on remark (ii) of the previous section. One can see that the step-N lift of a smooth path x, as given in (6), takes values in the free step-N nilpotent (Lie) group with d generators, realized as restriction of T N Rd to GN Rd = exp Rd ⊕ Rd , Rd ⊕ Rd , Rd , Rd ⊕ . . . ≡ exp gN Rd where gN Rd is the free step-N nilpotent Lie algebra and exp is deﬁned by the usual power-series based on ⊗. Example 5 [N = 2] Note that Rd , Rd = so (d). Then exp Rd ⊕ Rd , Rd 1 d = 1, v, v ⊗ v + A : v ∈ R , A ∈ so (d) 2 which is precisely the algebraic relation we pointed out in remark (ii) of the previous section. If the discussion above tells us that T N Rd is too big astate space for lifted smooth paths, Chow’s theorem tells us that GN Rd is the correct state space. It asserts that for all g ∈ GN Rd there exists γ : [0, 1] → Rd , which may be taken to be piecewise linear such that SN (γ)0,1 = g. One can then deﬁne the Carnot–Caratheodory norm g = inf length γ|[0,1] : SN (γ)0,1 = g , where the inﬁmum is achieved for some Lipschitz continuous path γ ∗ : [0, 1] → Rd , some sort of geodesic path associated with g. The Carnot– Caratheodory distance is then simply deﬁned by d (g, h) := g −1 ⊗ h. A Carnot–Caratheodory unit ball is plotted in Figure 2. 0 0 a Example 6 Take g = , ∈ G2 R2 . Then γ ∗ is the 0 −a 0 shortest path which returns to its starting point and sweeps√out area a. From basic isoperimetry, γ ∗ must be a circle and g = 2 πa1/2 . See Figure 3.

10

The story in a nutshell

Figure 2. After identifying G2 Rd with the 3-dimensional Heisenberg group, x 0 a i.e. , ≡ (x, y, a), we plot the (apple-shaped) unit-ball with y −a 0 respect to the Carnot–Caratheodory distance. It contains (and is contained in) a Euclidean ball.

Figure 3. We plot the circle γ ∗ . The z-axis represents the wiped-out area and runs from 0 to a.

2 Carnot–Caratheodory geometry

11

In practice, we rarely need to compute precisely the CC norm of an element g = 1, g 1 , . . . , g N ∈ GN Rd . Instead we rely on the so-called equivalence of homogenous norms, which asserts that ∃κ > 0 : where

1 |||g||| ≤ 1, g 1 , . . . , g N ≤ κ |||g||| κ 1/i |||g||| := max g i (Rd ) ⊗i. i=1,...,N

Here,both “norms” · and |||·||| are homogenous with respect to dilation on GN Rd , δ λ : 1, g 1 , . . . , g N → 1, λ1 g 1 , . . . , λN g N , λ ∈ R. It is time to make the link to our previous discussion. Recall condition (Hα ) from equation (2), which expressed a H¨older-type assumption of the form t r 1/2 t α dx ∨ dx ⊗ dx ≤ c1 |t − s| . s

s

s

But this says exactly that, for all 0 ≤ s < t ≤ T , the corresponding “group” increment xs,t = S2 (x)s,t ∈ G2 Rd satisﬁes α

xs,t = d (xs , xt ) c1 |t − s| , where d is the Carnot–Caratheodory metric on G2 Rd , which is equivalent to d (xs , xt ) xα -H¨o l;[0,T ] ≡ sup α c1 . s,t∈[0,T ] |t − s| This regularity persists under passage to the limit and hence any (weak, geometric) α-H¨older rough path is a genuine α-H¨older path with values in G2 Rd . Conversely, given an abstract α-H¨older path in G2 Rd equipped with Carnot–Caratheodory distance, we can construct a path xn by con catenating geodesic paths associated with the increments xt i ,t i + 1 : i = 0, . . . , 2n } , (ti ) = (i2−n T ); the resulting sequence (xn ) then satisﬁes condition (Hα ) uniformly and converges uniformly, together with its iterated integrals, to the path x (·) with which we started. Nothing of all this is restricted to α ∈ (1/3, 1/2] ←→ N = 2: for any α = 1/p ∈ (0, 1] a weak, geometric 1/p-H¨ older rough path x is precisely a 1/p-H¨ older path in the metric space G[p] Rd , d where d denotes the Carnot–Caratheodory distance. Davie’s lemma extends to the step-[p] setting and we are led to a theory of (rough path) diﬀerential equations, formally written as dy = V (y) dx,

The story in a nutshell

12

where x is a (weak, geometric) 1/p-H¨older rough path. For V ∈ Lipγ −1 one has existence and V ∈ Lipγ with γ > p uniqueness.8 Once in possession of a unique solution y = π (V ) (0, y0 ; x) one may ask for regularity of the Itˆ o–Lyons map x → y. In fact, one can construct the RDE solution as a (weak, geometric) 1/pH¨ older rough path in its own right, say y = π (V ) (0, y0 ; x) with values in G[p] (Re ) and ask for regularity of the full Itˆo–Lyons map (y0 , V, x) → y. It turns out that this solution map is Lipschitz continuous on bounded sets, provided we measure the distance between two driving signals x, x ˜ with a (non-homogenous9 ) 1/p-H¨older distance given by ˜) := ρ1/p-H¨o l (x, x

max

sup

i=1,...,[p] s,t∈[0,T ]

i xs,t − x ˜is,t i/p

|t − s|

.

For most applications it is enough to have (uniform) continuity (on bounded sets), in which case one can work with the (homogenous10 ) 1/p-H¨older distance given by ˜) := d1/p-H¨o l (x, x

sup s,t∈[0,T ]

d (xs,t , x ˜s,t ) |t − s|

i/p

.

The latter often makes computations more transparent and can become indispensible in a probabilistic context (e.g. when studying “exponentially good” approximations in a large deviation context). But no matter which distance is more practical in a given context, both induce the lder rough path” topology on the rough path space same “1/p-H¨ o C 1/p-H¨o l [0, T ] , G[p] Rd . 8 With

more eﬀort, uniqueness can be shown under Lip p -regularity. respect to dilation since, in general,

9 . . . with

ρ1 / p -H ¨o l (δ λ x, δ λ x ˜ ) = |λ| ρ1 / p -H ¨o l (x, x ˜) . 1 0 . . . again

with respect to dilation, d1 / p -H ¨o l (δ λ x, δ λ x ˜ ) = |λ| d1 / p -H ¨o l (x, x ˜) .

3 Brownian motion and stochastic analysis

13

Figure 4. A typical 2-dimensional Brownian sample path. The (signed) area between the straight cord and the sample path corresponds to a typical L´evy area increment.

3 Brownian motion and stochastic analysis Let B be a d-dimensional Brownian motion. Almost every realization of enhanced Brownian motion (EBM) t → Bt (ω) =

t

Bs ⊗ ◦dBs

1, Bt ,

= exp (Bt + A0,t )

0

t with so (d)-valued L´evy area As,t (ω) = 12 s (Bs,r ⊗ dBr − dBr ⊗ Bs,r ) is a (weak) geometric rough path, namely B· (ω) ∈ C α -H¨o l [0, T ] , G2 Rd , d , α ∈ (1/3, 1/2). In Figure 4 we plot a Brownian path with an associated L´evy area increment. Granted the usual α-H¨older regularity of Brownian motion, this statement is equivalent to the question “Is it true that for α < 1/2 :

sup s,t∈[0,1]

|As,t | |t − s|

2α

< ∞ a.s. ?”

The reader is encouraged to think about this before reading on! Perhaps the most elegant way to establish this “rough path regularity” of L´evy area relies on scaling properties of enhanced Brownian motion. Namely, D

D

Bs,t = B0,t−s = δ (t−s) 1 / 2 B0,1 ,

The story in a nutshell

14

so that

2q 2q q = E Bs,t ≤ (const) × |t − s| E d (Bs , Bt )

for any q < ∞. Kolmogorov’s criterion applies without any trouble and so B is indeed a.s. α-H¨ older, α < 1/2, with respect to d. QED. Let us also mention a convergence result: we have dα -H¨o l;[0,T ] (B, S2 (B n )) → 0 in probability where B n denotes a piecewise linear approximation to B based on dissections Dn = {tni : i} with the mesh of Dn tending to 0. We then have two important conclusions: (i) Thanks to α-H¨ older regularity of B, the (random) RDE dY = V (Y ) dB can be solved for a.e. ﬁxed ω and yields a continuous stochastic process (7) Y· (ω) = π (V ) (0, y0 ; B (ω)) . (ii) By continuity of the Itˆo–Lyons map with respect to the rough path metric dα -H¨o l;[0,T ] it follows that π (V ) (0, y0 ; B n ) → π (V ) (0, y0 ; B (ω)) with respect to α-H¨older topology and in probability. Clearly, y n ≡ π (V ) (0, y0 ; B n ) is a solution to the (random) ODE dy n = V (y n ) dB n , y n (0) = y0 and the classical Wong–Zakai theorem11 allows us to identify (7) as the classical Stratonovich solution to dY = V (Y ) ◦ dB =

d

Vi (Y ) ◦ dB i .

i=1

But why is all this useful? The following list should give some idea ... • π (V ) (0, y0 ; B (ω)) is simultaneously deﬁned for all starting points y0 and coeﬃcient vector ﬁelds V of suitable regularity. In particular, the construction of stochastic ﬂows is a triviality and this itself can be the starting point for the robust treatment of certain stochastic partial diﬀerential equations. 1 1 For

example, the books of Ikeda–Watanabe [88] or Stroock [160].

3 Brownian motion and stochastic analysis

15

• Every approximation in rough path topology implies a limit theorem (even on the level of ﬂows). This includes classical piecewise linear approximations and non-standard variations a` la McShane, Sussmann. It also includes a variety of weak limit theorems such as a Donskertype invariance principle. • Various stochastic Taylor expansions (`a la Azencott, Platen, . . . ) can be obtained via deterministic rough path estimates. • Support descriptions a` la Stroock–Varadhan and large deviation estimates `a la Freidlin–Wentzell are reduced to the respective (relatively simple) statements about B in the rough path topology. • The Young integral allows us to perturb B simultaneously in all C q -var [0, 1] , Rd -directions with q < 2. Since Cameron–Martin ⊂ C 1-var ⊂ C q -var this implies in particular path space regularity of the SDE solution beyond Malliavin and there is a natural interplay with Malliavin calculus. • Starting points and vector ﬁelds can be fully anticipating. • At last, for the bulk of these results we can replace Brownian motion at little extra cost by martingales, Gaussian processes or Markov processes provided we can construct a suitable stochastic area and establish the correct rough path regularity!

Part I

Basics

1 Continuous paths of bounded variation We discuss continuous paths, deﬁned on a ﬁxed time horizon, with values in a metric space E. The emphasis is on paths with nice regularity properties and in particular on continuous paths of bounded variation.1 We then specialize to the case when E = Rd . Finally, we discuss simple Sobolev-type regularity of paths.

1.1 Continuous paths on metric spaces We start by deﬁning the supremum or inﬁnity distance. Deﬁnition 1.1 Let (E, d) be a metric space and [0, T ] ⊂ R. Then C ([0, T ], E) denotes the set of all continuous paths x : [0, T ] → E. The supremum or inﬁnity distance of x, y ∈ C ([0, T ] , E) is deﬁned by d∞;[0,T ] (x, y) := sup d (xt , yt ) . t∈[0,T ]

For a single path x ∈ C ([0, T ] , E) , we set |x|0;[0,T ] :=

sup u ,v ∈[0,T ]

d (xu , xv ) ,

and, given a ﬁxed element o ∈ E, identiﬁed with the constant path ≡ o, |x|∞;[0,T ] := d∞;[0,T ] (o, x) = sup d (o, xu ) . u ∈[0,T ]

If no confusion is possible we shall omit [0, T ] and simply write d∞ , |·|0 and |·|∞ . If E has a group structure such as Rd , + the neutral element is the usual choice for o. In the present generality, however, the deﬁnition of |·|∞ depends on the choice of o. Notation 1.2 Of course, [0, T ] can be replaced by any other interval [s, t] in which case one considers x : [s, t] → E. All notations adapt by replacing [0, T ] by [s, t]. Let us also agree that Co ([s, t] , E) denotes those paths in C ([s, t] , E) which start at o, i.e. Co ([s, t] , E) = {x ∈ C ([s, t] , E) : x (s) = o} . 1 Also

known as rectiﬁable paths.

Continuous paths of bounded variation

20

Many familiar properties of real-valued functions carry over. For instance, any continuous mapping from [0, T ] into E is uniformly continuous.2 It is also fairly easy to see that C ([0, T ] , E) is a metric space under d∞ (the induced topology will be called the uniform or supremum topology). Also, if (E, d) is complete then (C ([0, T ] , E) , d∞ ) is complete.

Deﬁnition 1.3 A set H ⊂ C ([0, T ] , E) is said to be equicontinuous if, for all ε > 0 there exists δ such that |t − s| < δ implies d (xs , xt ) < ε for all x ∈ H. It is said to be bounded if supx∈H |x|∞ < ∞.

Theorem 1.4 (Arzela–Ascoli) Let (E, d) be a complete metric space in which bounded sets have compact closure. Then a set H ⊂ C ([0, T ] , E) has compact closure if and only if H is bounded and equicontinuous. As a consequence, a bounded, equicontinuous sequence in C ([0, T ] , E) has a convergent subsequence and, conversely, any convergent sequence in C ([0, T ] , E) is bounded and equicontinuous.

Proof. Let us recall that a subset of a complete metric space has compact closure if and only if it is totally bounded, i.e. for all ε > 0, it can be covered by ﬁnitely many ε-balls. “⇐=”: We show that the assumption “H bounded and equicontinuous” implies total boundedness. We ﬁx ε > 0, and then δ > 0 such that for every f ∈ H, |t − s| < δ =⇒ d (fs , ft ) < ε/4.

(1.1)

Cover [0, T ] with a ﬁnite number of neighbourhoods ti − 2δ , ti + 2δ , i = 1, . . . , m, and deﬁne Ht i = {ft i , f∈ H}; as Ht i ⊂ E is bounded, its closure is compact, and so is its union 1≤i≤m Ht i ; let c1 , . . . , cn ∈ 1≤i≤m Ht i be such that 1≤i≤m Ht i is covered by the union of the ε/4-balls centred around some cj . Then, consider Φ, the set of functions from {1, . . . , m} into {1, . . . , n}. For each ϕ ∈ Φ, denote by Lϕ,ε the set of all functions f ∈ C ([0, T ] , E) such that maxi d ft i , cϕ (i) ≤ 4ε . Observe that from the deﬁnition of cj it follows that H is covered by the union of the (Lϕ,ε )ϕ∈Φ . To end the proof, we need only show that the diameter of each Lϕ,ε is ≤ ε. 2 For

example, Dieudonn´e [43], (3.6.15).

1.2 Continuous paths of bounded variation on metric spaces

21

If f, g are both in Lϕ,ε , then d∞ (f, g) = ≤

sup d (ft , gt ) t∈[0,T ]





max d (ft i , gt i ) +

1≤i≤m

d (fs , ft i ) + d (gs , gt i )

sup s∈(t i − δ2 ,t i + δ2

)

ε from (1.1) 2 ε ≤ max d ft i , cϕ (i) + max d gt i , cϕ(i) + i i 2 ≤ ε by deﬁnition of Lϕ,ε .

≤

max d (ft i , gt i ) +

1≤i≤m

“=⇒”: Since compact sets are bounded, only equicontinuity needs proof. By assumption H has compact closure and therefore is totally bounded. Fix ε > 0 and pick h1 , . . . , hn such that H ⊂ 1≤i≤n B hi , ε/3 where B (h, ε) denotes the open ε-ball centred at h. By continuity of each hi (·), there exists δ = δ (ε) such that |t − s| < δ =⇒ max d his , hit < ε/3. i=1,...,n

But then, for every h ∈ H, d (hs , ht ) ≤ ε/3+maxi=1,...,n d his , hit +ε/3 ≤ ε provided |t − s| < δ and so H is equicontinuous. The consequences for sequences are straightforward and left to the reader.

1.2 Continuous paths of bounded variation on metric spaces 1.2.1 Bounded variation paths and controls Let us write D ([s, t]) for the set of all dissections of some interval [s, t] ⊂ R, thus a typical element in D ([s, t]) is written D = {s = t0 < t1 < · · · < tn = t} and consists of #D = n adjacent intervals [ti−1 , ti ]. The mesh of D is deﬁned as |D| := maxi=1,...,n |ti − ti−1 | and we shall denote by Dδ ([s, t]) the set of all dissections of [s, t] with mesh less than or equal to δ. Deﬁnition 1.5 Let (E, d) be a metric space and x : [0, T ] → E. For 0 ≤ s ≤ t ≤ T , the 1-variation of x on [s, t] is deﬁned as3 |x|1-var;[s,t] = sup d xt i , xt i + 1 . (t i )∈D([s,t])

3 Let

i

us agree that |x|1 -va r;[s , s ] = 0 for 0 ≤ s ≤ T .

22

Continuous paths of bounded variation

If |x|1-var;[s,t] < ∞, we say that x is of bounded variation or of ﬁnite 1variation on [s, t]. The space of continuous paths of ﬁnite 1-variation on [0, T ] is denoted by C 1-var ([0, T ] , E), its subset of paths started at o ∈ E is denoted by Co1-var ([0, T ] , E). In the discussion of 1-variation regularity (and later p-variation regularity for p ≥ 1), the notion of control or control function, deﬁned on the simplex ∆ := ∆T = {(s, t) : 0 ≤ s ≤ t ≤ T } turns out to be extremely useful. Deﬁnition 1.6 A map ω : ∆T → [0, ∞) is called superadditive if for all s ≤ t ≤ u in [0, T ], ω(s, t) + ω(t, u) ≤ ω(s, u). If, in addition, ω is continuous and zero on the diagonal, i.e. ω (s, s) = 0 for 0 ≤ s ≤ T we call ω a control or, more precisely, a control function on [0, T ] . Deﬁnition 1.7 We say that the 1-variation of a map x : [0, T ] → E is dominated by the control ω, or controlled by ω, if there exists a constant C < ∞ such that for all s < t in [0, T ], d (xs , xt ) ≤ Cω (s, t) .

θ

Simple examples of controls are given by (s, t) → |t − s| for θ ≥ 1 or the integral of a non-negative function in L1 ([0, T ]) over the interval [s, t]. Trivially, a positive linear combination of controls yields another control. If ω is a control and x : [0, T ] → E a map controlled by ω then x is continuous. Exercise 1.8 Let φ ∈ C ([0, ∞), [0, ∞)) be increasing, convex with φ (0) = 0. Assuming that ω is a control, show that φ ◦ ω : (s, t) → φ (ω (s, t)) is also a control. Solution. Fix 0 < a < b and observe that by convexity φ (a) − φ (0) φ (a + b) − φ (b) ≥ a a so that φ (a + b) ≥ φ (a) + φ (b). Interchanging a, b if needed, this holds for all a, b ≥ 0 and we conclude that φ [ω (s, t)] + φ [ω(t, u)] ≤ φ [ω(s, t) + ω(t, u)] ≤ φ [ω(s, u)] . Exercise 1.9 Assume ω, ω ˜ are controls. (i) Show that ω.˜ ω is a control. (ii) Show that max (ω, ω ˜ ) need not be a control. ω β is a control. (iii) Given α, β > 0 with α + β ≥ 1, show that ω α .˜

1.2 Continuous paths of bounded variation on metric spaces

23

Solution. (iii) By Exercise 1.8, it is enough to consider the case α + β = 1. But this follows from H¨ older’s inequality, α 1 β 1 1 1 a ˜ β + ˜b β . ∀a, a ˜, b, ˜b ≥ 0 : a˜ a + b˜b ≤ a α + b α

Exercise 1.10 Let ω be a control on [0, T ] and consider s < u in [0, T ]. Show that there exists t ∈ [s, u] such that max {ω (s, t) , ω (t, u)} ≤ ω (s, u) /2. Solution. By continuity and monotonicity of controls, there exists t such that ω (s, t) = ω (t, u). By super-additivity, 2ω (s, t) = 2ω (t, u) = ω (s, t) + ω (t, u) ≤ ω (s, u)

and the proof is ﬁnished.

Proposition 1.11 Consider x : [0, T ] → E and ω = ω (s, t) super-additive, with s < t in [0, T ]. If d (xs , xt ) ≤ ω (s, t) for all s < t in [0, T ], then |x|1-var;[s,t] ≤ ω (s, t). Proof. Let D = (ti ) be a dissection of [s, t] . Then, by assumption, # D −1

≤ d xt i ,t i + 1

i=0

# D −1

ω (ti , ti+1 )

i=0

≤

ω (s, t) by super-additivity of ω.

Taking the supremum over all such dissections ﬁnishes the proof. Proposition 1.12 Let x ∈ C 1-var ([0, T ] , E). Then (s, t) → ω x (s, t) := |x|1-var;[s,t] deﬁnes a control on [0, T ] such that for all 0 ≤ s < t ≤ T d(xs , xt ) ≤ |x|1-var;[s,t] . This control is additive: for all 0 ≤ s ≤ t ≤ u ≤ T, |x|1-var;[s,u ] = |x|1-var;[s,t] + |x|1-var;[t,u ] . In particular, t ∈ [0, T ] → (t) := |x|1-var;[0,t] ∈ R is continuous, increasing and hence of ﬁnite 1-variation.

(1.2)

Continuous paths of bounded variation

24

Proof. Trivially, |x|1-var;[s,s] = 0 for all s ∈ [0, T ] . To see super-additivity it suﬃces to take dissections D1 , D2 of [s, t] and [t, u] respectively; noting that the union of D1 and D2 is a dissection of [s, u] we have d xt i , xt i + 1 + d xt j , xx j + 1 ≤ |x|1-var;[s,u ] t i ∈D 1

t j ∈D 2

and ω x (s, t) + ω x (t, u) ≤ ω x (s, u) follows from taking the supremum over all dissections D1 and D2 . For additivity of ω x we establish the reverse inequality. Let D = (vi ) be a dissection of [s, u] so that t ∈ [vj , vj +1 ] for some j. We then have # D −1

j −1 d xv i ,v i + 1 = d xv i , xv i + 1 +

i=0

i=0

+

# D −1

d xv j , xv j + 1

≤ d (x v j ,x t )+d (x t ,x v j + 1 )

d xv i , xv i + 1 .

i=j +1

But, as j −1 d xv i , xv i + 1 + d xv j , xt

≤

|x|1-var;[s,t] ,

≤

|x|1-var;[t,u ] ,

i=0 D −1 # d xv i , xv i + 1 d xt , xv j + 1 + i= j +1

we have

# D −1

d xv i ,v i + 1 ≤ |x|1-var;[s,t] + |x|1-var;[t,u ] .

i=0

Taking the supremum over all dissections shows additivity of ω x . It only remains to prove its continuity. To this end, ﬁx s < t in [0, T ]. From monotonocity of ω x , we see that the limits |x|1-var;[s + ,t − ] := :=

lim

|x|1-var;[s+h 1 ,t−h 2 ] , |x|1-var;[s − ,t + ]

lim

|x|1-var;[s−h 1 ,t+ h 2 ]

h 1 ,h 2 0 h 1 ,h 2 0

exist and that |x|1-var;[s + ,t − ] ≤ |x|1-var;[s,t] ≤ |x|1-var;[s − ,t + ] .

(1.3)

We aim to show the inequalities in (1.3) are actually equalities. To establish “continuity from inside”, i.e. |x|1-var;[s + ,t − ] = |x|1-var;[s,t] , we deﬁne

1.2 Continuous paths of bounded variation on metric spaces

25

ω (s, t) = |x|1-var;[s + ,t − ] , and pick s < t < u in [0, T ] , and h1 , h2 , h3 , h4 four (small) positive numbers. If (ai ) is a dissection of [s + h1 , t − h2 ] , and (bj ) a dissection of [t + h3 , u − h4 ] , then by deﬁnition of the 1-variation of x, d xa i , xa i + 1 + d xb j , xb j + 1 ≤ |x|1-var;[s+ h 1 ,u −h 4 ] . i

j

Taking the supremum over all possible dissections (ai ) and (bi ) , we obtain |x|1-var;[s+ h 1 ,t−h 2 ] + |x|1-var;[t+ h 3 ,u −h 4 ] ≤ |x|1-var;[s+h 1 ,u −h 4 ] . Letting h1 , h2 , h3 , h4 go to 0 and using the continuity of x we obtain that (s, t) → |x|1-var;[s + ,t − ] is super-additive. We also easily see that for all s, t ∈ [0, T ] , d (xs , xt ) ≤ |x|1-var;[s + ,t − ] . Hence, using Proposition 1.11, we obtain |x|1-var;[s + ,t − ] ≥ |x|1-var;[s,t] , and hence we have proved that |x|1-var;[s + ,t − ] = |x|1-var;[s,t] for all s < t in [0, T ]. The remaining part of the proof is “continuity from outside”, i.e. |x|1-var;[s − ,t + ] = |x|1-var;[s,t] .Using additivity of |x|1-var;[.,.] it is easy to see that |x|1-var;[s − ,t + ]

=

|x|1-var;[0,T ] − |x|1-var;[0,s − ] − |x|1-var;[t + ,T ]

=

|x|1-var;[0,T ] − |x|1-var;[0,s] − |x|1-var;[t,T ] = |x|1-var;[s,t]

and this ﬁnishes the proof. Exercise 1.13 Assume ω is a control on [0, T ]. Assume f ∈ C (∆, [0, ∞)) where ∆ = {(s, t) : 0 ≤ s ≤ t ≤ T } , non-decreasing in the sense that [s, t] ⊂ [u, v] implies f (s, t) ≤ f (u, v) . Show that (s, t) → f (s, t) ω (s, t) is a control. As application, given x ∈ C 1-var ([0, T ] , E) and y ∈ C ([0, T ] , E), show that (s, t) → |y|∞;[s,t] |x|1-var;[s,t] is a control where |·|∞;[s,t] is deﬁned with respect to some ﬁxed o ∈ E. Proposition 1.14 Let x ∈ C ([0, T ] , E). Then for all δ > 0 and 0 ≤ s ≤ t ≤ T, |x|1-var;[s,t] = sup d xt i , xt i + 1 ∈ [0, ∞] . (t i )∈Dδ ([s,t])

i

Proof. Clearly, ω x,δ (s, t) :=

sup (t i )∈Dδ ([s,t])

d xt i , xt i + 1 ≤ ω x (s, t) = |x|1-var;[s,t] . i

Continuous paths of bounded variation

26

Super-addivitity of ω x,δ follows from the same argument as for ω x . Take any D = (ui ) ∈ Dδ ([s, t]) so that s = u0 < u1 < · · · < un = t with ui+1 − ui < δ. It follows that d (xs , xt ) ≤ d (xs , xu 1 ) + · · · + d xu n −1 , xt ≤ ω x,δ (s, t) . From Proposition 1.11, we conclude that |x|1-var;[s,t] ≤ ω x,δ (s, t) , which concludes the proof. We now observe lower semi-continuity of the function x → |x|1-var in the following sense. Lemma 1.15 Assume (xn ) is a sequence of paths from [0, T ] → E of ﬁnite 1-variation. Assume xn → x pointwise on [0, T ]. Then, for all s < t in [0, T ], |x|1-var;[0,T ] ≤ lim inf |xn |1-var;[0,T ] . n →∞

Proof. Let D = {0 = t0 < t1 < · · · < tK = T } be a dissection of [0, T ] . By assumption, xn → x pointwise and so K −1

d xt i , xt i + 1

i=0

= lim inf n →∞

d xnti , xnti + 1 i

≤ lim inf |xn |1-var;[0,T ] . n →∞

Taking the supremum over all the dissections of [s, t] ﬁnishes the 1-variation estimate. In general, the inequality in Lemma 1.15 can be strict. The reader is invited to construct an example in the following exercise. Exercise 1.16 Construct (xn ) ∈ C 1-var ([0, 1] , R) such that |xn |∞;[0,1] ≤ 1/n but so that |xn |1-var = 1 for all n. Conclude that the inequality in Lemma 1.15 can be strict.

1.2.2 Absolute continuity Deﬁnition 1.17 Let (E, d) be a metric space. A path x : [0, T ] → E is absolutely continuous if for all ε > 0, there existsδ > 0, such that for all s 1 < t1 ≤ s2 < t2 ≤ · · · < sn < tn in [0, T ] with i |ti − si | < δ, we have d (x , x ) < ε. s t i i i Proposition 1.18 Any absolutely continuous path is a continuous path of bounded variation. Proof. If x : [0, T ] → E is absolutely continuous it is obviously continuous. Furthermore, by deﬁnition there exists δ > 0, such that for all s1 < t1 ≤ s2 < t2 ≤ · · · < sn < tn ∈ [0, T ] with i |ti − si | ≤ δ, we have

1.2 Continuous paths of bounded variation on metric spaces

27

d (xs i , xt i ) ≤ 1. Pick D = (ti )1≤i≤n a dissection of [0, T ]. Then, deﬁne j0 = 1 and jk = max i, ti − tj k −1 ≤ δ , and observe that j[T /δ ]+1 = jk for all k ≥ [T /δ] + 1. i

n −1

[T /δ ]+ 1 j k + 1 −1

d xt i , xt i + 1 ≤

i=1

k =0

i=j k

d xt i , xt i + 1 .

j k + 1 −1 |ti+1 − ti | = tj k + 1 − tj k ≤ δ, hence By deﬁnition of the jk s, i= j k j k + 1 −1 n −1 d xt i , xt i + 1 ≤ 1, which implies that i=1 d xt i , xt i + 1 ≤ [T /δ]+ i= j k 1. Taking the supremum over all dissections ﬁnishes the proof. In general, the converse of the above is not true, as seen in the following. Example 1.19 (Cantor function) Each x ∈ [0, 1] has a base-3 decimal expansion x = j ≥1 aj 3−j where aj ∈ {0, 1, 2}. This expansion is unique unless x is of the form p3−k for some p, k ∈ N (we may assume p is not divisible by 3) and in this case x has two expansions: one with aj = 0 for j > k and one with aj = 2 for j > k. One of them has ak = 1, the other will have ak ∈ {0, 2}. If we agree always to use the latter, we see that a1 a1

= =

1 iﬀ x ∈ (1/3, 2/3) 1, a2 = 1 iﬀ x ∈ (1/9, 2/9) ∪ (7/9, 8/9)

and so forth. The Cantor set C is then deﬁned as the set of all x ∈ [0, 1] that have a base-3 expansion x = aj 3−j with aj = 1 for all j. Thus C is obtained from K0,1 = [0, 1] by removing the open middle third, leaving us with the union of K1,1 = [0, 1/3] , K1,2 = [2/3, 1]; followed by removing all open middle thirds, leaving us with the union of K2,1 = [0, 1/9] , K2,1 = [2/9, 3/9] , K2,1 = [6/9, 7/9] , K2,1 = [8/9, 1] 2 and so forth, so that in the end C = ∩∞ n =1 ∪i=1 Kn ,i . Let us now deﬁne the Cantor function f on C by aj 2−j , x ∈ C. f (x) = 2 n

j ≥1

This series is the base-2 expansion of a number in [0, 1] and since any number in [0, 1] can be obtained this way we see that f (C) = [0, 1]. One readily sees that if x, y ∈ C and x < y, then f (x) < f (y) unless x and y are the endpoints of one of the open intervals removed from [0, 1] to obtain C. In this case, f (x) = p2−k for some p, k ∈ N and f (x) = f (y), given by the two base-2 expansions of this number. We can therefore extend f to a map from [0, 1] to itself by declaring it to be constant on the intervals missing from C. This extended f is still increasing, and since its range is all of [0, 1] it cannot have any jump discontinuities, hence it is continuous.

28

Continuous paths of bounded variation

Being increasing, f is obviously of bounded variation on [0, 1]. We now show that f is not absolutely continuous. Given any δ > 0 we can take si , ti as the boundary points of the intervals (Kn ,i )i=1,...,2 n with n chosen large 2 n enough so that i=1 (ti − si ) < δ. Then, since f is constant on [ti , si+1 ] for i = 1, . . . , 2n − 1, we have n

2

|f (ti ) − f (si )| = f (1) − f (0) = 1.

i=1

1.2.3 Lipschitz or 1-H¨ older continuity Deﬁnition 1.20 Let (E, d) be a metric space. A path x : [0, T ] → E is Lipschitz or 1-H¨older continuous4 if |x|1-H¨o l;[0,T ] :=

sup s,t∈[0,T ]

d (xs , xt ) < ∞. |t − s|

The space of all such paths is denoted by C 1-H¨o l ([0, T ] , E), the subset of paths started at o ∈ E is denoted by Co1-H¨o l ([0, T ] , E). We observe that every Lipschitz path is absolutely continuous. In particular, it is of bounded variation and we note |x|1-var;[s,t] ≤ |x|1-H¨o l;[s,t] × |t − s| . older if and only if it is conFurthermore, x ∈ C 1-var ([0, T ] , E) is 1-H¨ trolled by (s, t) → |t − s|. It is easy to construct examples which are of bounded variation but not Lipschitz (e.g. t → t1/2 ). On the other hand, every continuous bounded variation path is a continuous time-change (or reparametrization) of a Lipschitz path. Proposition 1.21 A path x ∈ C ([0, T ], E) is of ﬁnite 1-variation if and only if there exists a continuous non-decreasing function φ from [0, T ] onto [0, 1] and a path y ∈ C 1-H¨o l ([0, 1] , E) such that x = y ◦ φ. Proof. We may assume |x|1-var;[0,T ] = 0 (otherwise, x|[0,T ] is constant and there is nothing to show). By Propostion 1.12, φ(t) =

|x|1-var;[0,t] |x|1-var;[0,T ]

deﬁnes a continuous increasing function from [0, T ] onto [0, 1] . Then, there exists a function y such that (y ◦ φ) (t) = x (t) , as φ (t1 ) = φ (t2 ) =⇒ 4 . . . in view of the later deﬁnition of H¨ o lder continuity and in order to avoid redundant notation . . .

1.3 Continuous paths of bounded variation on Rd

29

x (t1 ) = x (t2 ) . Now, sup

0≤u < v ≤1

d (y (u) , y (v)) |u − v|

=

sup

0≤u < v ≤T

d (y (φ (u)) , x (y (v))) |φ (u) − φ (v)|

≤

|x|1-var;[u ,v ] |x|1-var;[0,T ] |x|1-var;[0,u ] − |x|1-var;[0,v ]

=

|x|1-var;[0,T ] .

This shows that y is in C 1-H¨o l ([0, 1] , E). The converse direction is an obvious consequence of the invariance of variation norms under reparametrization. Remark 1.22 The 1-variation (i.e. length) of a path is obviously invariant under reparametrization and so it is clear that |y|1-var;[0,1] = |x|1-var;[0,T ] . On the other hand, for the particular parametrization φ (·) used in the previous proof (essentially the arc-length parametrization) we saw that |y|1-H¨o l;[0,1] ≤ |x|1-var;[0,T ] . With the trivial |y|1-var;[0,1] ≤ |y|1-H¨o l;[0,1] we then see that |y|1-H¨o l;[0,1] = |x|1-var;[0,T ] . Lemma 1.23 Assume (xn ) is a sequence of paths from [0, T ] → E of ﬁnite 1-variation. Assume xn → x pointwise on [0, T ]. Then, for all s < t in [0, T ], |x|1-H¨o l; [s,t] ≤ lim inf |xn |1-H¨o l;[s,t] . n →∞

Proof. The H¨ older statement is a genuine corollary of Lemma 1.15: it suﬃces to note that for any u, v ∈ [s, t] , d (xu , xv )

≤

|x|1-var;[u ,v ]

≤

lim inf |xn |1-var;[u ,v ] .

≤

|v − u| lim inf |xn |1-H¨o l;[s,t] .

n →∞

n →∞

1.3 Continuous paths of bounded variation on Rd Unless otherwise stated, Rd shall be equipped with Euclidean structure. In particular, if a ∈ Rd has coordinates a1 , . . . , ad its norm is given by |a| =

2

2

|a1 | + · · · + |ad | .

30

Continuous paths of bounded variation

Given a map x : [0, T ] → Rd the group structure of Rd , + allows us to speak of the increments of x (·) and we write5 xs,t := xt − xs .

1.3.1 Continuously diﬀerentiable paths

diﬀerWe deﬁne inductively the set C k [0, T ] , Rd of k-times continuously 0 d d [0, T ] , R to be C [0, T ] , R , and then entiable paths by ﬁrst deﬁning C k d in C [0, T ] , R . C k +1 [0, T ] , Rd to be the set of paths with a derivative ∞ d [0, T ] , R to be the interFinally, we deﬁne the set of smooth paths C section of all C k [0, T ] , Rd , for k ≥ 0. For continuously diﬀerentiable paths, the computation of 1-variation is a simple matter. Proposition 1.24 Let x ∈ C 1 [0, T ] , Rd . Then t ∈ [0, T ] → (t) := |x|1-var;[0,t] ∈ R is continuously diﬀerentiable and ˙ (t) = |x˙ (t)| for t ∈ (0, T ). In particular, |x|1-var;[s,t] =

t

|x˙ u | du s

for all s < t in [0, T ]. Proof. We ﬁrst note that |xt − xs | ≤ obtain that

t s

|x˙ u | du; using Proposition 1.11, we

(t) − (s) = |x|1-var;[s,t] ≤

t

|x˙ u | du. s

Equality in the above estimate will follow immediately from ˙ (t) = |x˙ (t)| and this is what we now show. Take t ∈ [0, T ) and h small enough (so that t + h ≤ T ). Clearly, | (t + h) − (t)| 1 |xt,t+h | ≤ ≤ h h h

t+ h

|x˙ u | du t

and upon sending h ↓ 0 we see that is diﬀerentiable at t from the right with derivative equal to |x˙ t |. The same argument applies “from the left” and so is indeed diﬀerentiable with derivative |x|. ˙ By assumption on x, this derivative is continuous and the proof is ﬁnished. 5 Later on, we shall replace Rd by a Lie group (G, ·) and increments will be deﬁned as (xs )−1 · xt .

1.3 Continuous paths of bounded variation on Rd

31

1.3.2 Bounded variation The results of Section 1.2 applied to Rd equipped with Euclidean distance allow us in particular to consider the space C 1-var [0, T ] , Rd . Theorem 1.25 C 1-var [0, T ] , Rd is Banach with norm x → |x (0)| + |x|1-var;[0,T ] . The closed subspace of paths in C 1-var [0, T ] , Rd started at 0, denoted by C01-var [0, T ] , Rd , is also Banach under x → |x|1-var;[0,T ] . These Banach spaces are not separable. Proof. It is easy to see that C 1-var [0, T ] , Rd , C01-var [0, T ] , Rd are normed linear spaces under the given norms. We thus focus on completeness. Noting that sup |x (t)| ≤ |x (0)| + |x|1-var;[0,T ] ,

t∈[0,T ]

a Cauchy sequence (xn ) with respect to x → |x (0)| + |x|1-var;[0,T ] is also Cauchy in uniform topology and thus (uniformly) convergent to some continuous path x (·). By Lemma 1.15 it is clear that x has ﬁnite 1-variation and it only remains to see that xn → x in 1-variation norm. To this end, let D = {0 = t0 < · · · < tK = t} be an arbitrary dissection of [0, T ]. For every ε > 0 there exists N = N (ε) large enough so that for all n, m ≥ N (ε) sup D

K −1

< ε/2. d xnti ,t i + 1 , xm t i ,t i + 1

i=0

On the other hand, we can ﬁx D and ﬁnd m large enough so that K −1

< ε/2 d xm t i ,t i + 1 , xt i ,t i + 1

i=0

which implies that for n ≥ N (ε) large enough K −1

d xnti ,t i + 1 , xt i ,t i + 1 ≤ ε,

i=0

uniformly over all D. But this precisely says that xn → x in 1-variation. Non-separability follows from the example below. Example 1.26 (non-separability) We give an example of an uncountable family of functions (fα ) in C 1-var ([0, 1] , R) for which |fα − fα |1-var ≥ 1 if α = α . To this end, take α = (αn )n ≥1 to be a {0, 1}-sequence, and 1) as the write [0, union of the disjoint interval In , n ≥ 1, where In ≡ 1 − 2 n1−1 , 1 − 21n . If αn = 0 then deﬁne fα to be zero on In . Otherwise, deﬁne fα on In by s 1 fα (tn + s) = sin nπ −n 2n 2

Continuous paths of bounded variation

32

so that, using Proposition 1.24, |fα |1-var;I n = 1. By construction fα (tn ) = 0 for all n and hence fα is continuous on [0, 1). (Left) continuity at 1 is also clear: thanks to the decay factor 1/n we see that fα (t) → 0 as t 1. A simple approximation of a path x on Rd is given by its piecewise linear approximation.6 Deﬁnition 1.27 Let x : [0, T ] → Rd , and D = (ti )i a dissection of [0, T ] . We deﬁne the piecewise linear approximation to x by t − ti xt ,t if ti ≤ t ≤ ti+1 . xD t = xt i + ti+1 − ti i i + 1 Proposition 1.28 Let x ∈ C 1-var [0, T ] , Rd . Then, for any dissection D of [0, T ] and any s < t in [0, T ] , D x ≤ |x|1-var;[s,t] . (1.4) 1-var;[s,t] If (Dn ) is an arbitrary sequence of dissections with mesh |Dn | → 0, then xD n converges uniformly to x. (We can write this more concisely as xD → x uniformly on [0, T ] as |D| → 0.) Proof. The estimate (1.4) boils down to the fact that the shortest way to connect two points in Rd is via a straight line. The convergence result requires the remark that x (·) is uniformly continuous on [0, T ]. The easy details are left to the reader. The question arises if (or when) xD → x in 1-variation as |D| → 0. Since piecewise linear approximations are absolutely continuous, the following result tells us that there is no hope unless x is absolutely continuous. (We shall see later that xD → x in 1-variation as |D| → 0 indeed holds true provided x is absolutely continuous.) Proposition 1.29 The set of absolutely continuous functions from [0, T ] → Rd is closed in 1-variation and a Banach space under 1-variation norm. Proof. We prove that if xn is absolutely continuous and converges to x in 1-variation norm, then x is absolutely continuous. Fix ε > 0, and n ∈ N such that ε |x − xn |1-var + |x0 − xn0 | < . 2 Then, as xn is absolutely continuous, there existsδ > 0, such that for all t2 ≤ · · · < sn < tn in [0, T ] with i |ti − si | < δ, we have s 1 < tn1 ≤ s2 < ε i xs i ,t i < 2 . This implies that xns ,t + xs ,t − xns ,t sup |xs ,t | ≤ i

i

i

i

i

≤

i

i

D =(t i ) of [0,T ]

i

i

xns ,t + |x − xn | 1-var ≤ ε i i i

and the proof is ﬁnished. 6A

powerful generalization of this will be discussed in Section 5.2.

i

i

1.3 Continuous paths of bounded variation on Rd

33

Exercise 1.30 By Proposition 1.29 it is clear that piecewise linear approximations cannot converge (in 1-variation) to the Cantor function f : [0, 1] → [0, 1] given in Example 1.19. (By Proposition 1.29 any 1-variation limit point is absolutely continuous; but the Cantor function is not absolutely continuous as was seen in Exercise 1.19). Verify this by an explicit computation. More precisely, set Dn = {j3−n ; j = 0, . . . , 3n } and show that f − f D n = |f − I|1-var;[0,1] 1-var;[0,1] where I (x) = x and conclude that f D f in 1-variation as |D| → 0. Solution. f is self-similar, in the sense that for all n ≥ 1, k ∈ {0, . . . , 3n } , f k3−n + 3n x − f k3−n = 2n f (x) . Using self-similarity, we see that, if I denotes the identity function on [0, 1], f − f D n = 2n |f − I|1-var;[0,1] . 1-var; [ 3jn , j3+n 1 ] Hence, f − f D n 1-var;[0,1]

=

n 2 −1

f − f D n 1-var; [

j =0

=

j 3n

, j3+n 1 ]

|f − I|1-var;[0,1] > 0.

1.3.3 Closure of smooth paths in variation norm

Let us deﬁne C 0,1-var [0, T ] , Rd as the closure of smooth paths from [0, T ] → Rd in 1-variation norm. Obviously, C 0,1-var is a closed, linear subspace of C 1-var [0, T ] , Rd and thus a Banach space. Restricting to paths with x (0) = 0 yields a further subspace (also Banach) denoted by C00,1-var [0, T ] , Rd . By Proposition 1.29 any element of C 0,1-var must be absolutely continuous (a.c.) and so C 0,1-var [0, T ] , Rd ⊂ x : [0, T ] → Rd a.c. C 1-var [0, T ] , Rd . We shall show that the ﬁrst inclusion is in fact an equality. Proposition 1.31 The map y →

· 0

yt dt is a Banach space isomorph from

L1 [0, T ] , Rd → C00,1-var [0, T ] , Rd .

Continuous paths of bounded variation

34

[0, T ] ,Rd if and only if there exists a As a consequence, x ∈ C 0,1-var (uniquely determined) x˙ ∈ L1 [0, T ] , Rd , see Remark 1.33, such that · x ≡ x0 + x˙ t dt 0

˙ L 1 holds. and in this case the Banach isometry |x|1-var = |x| Proof. Without lossof generality started at x0 = 0. we consider paths For · any smooth y ∈ C ∞ [0, T ] , Rd we have x = 0 yt dt ∈ C ∞ [0, T ] , Rd and so, by Proposition 1.24, |x|1-var = |y|L 1 . Obviously, this allows us to extend the map · yt dt ∈ C ∞ [0, T ] , Rd ι : y ∈ C ∞ [0, T ] , Rd → x = 0

to the respective closures. From the very deﬁnition of the space C00,1-var and by density of smooth paths in L1 it follows that i extends to (a Banach space isomorphism) ˆι : L1 [0, T ] , Rd → C00,1-var [0, T ] , Rd . To see that ˆι still has the simple representation as an indeﬁnite integral, let y ∈ L1 [0, T ] , Rd , take smooth approximations y n in L1 and pass to the limit in t

ι (y n )t =

T

ysn ds = 0

ysn 1[0,t] (s) ds, 0

n 1 using the simple fact that y n → y in L1 implies y 1[0,t] → y1[0,t] in L for 1 d every ﬁxed t ∈ [0, T ]. At last, given x ∈ ˆι(L [0, T ] , R we write x˙ rather than y for the uniquely determined ˆι−1 (x) ∈ L1 [0, T ] , Rd . The next proposition requires some background in basic measure theory (Lebesgue–Stieltjes measures, Radon–Nikodym theorem, . . . ).7 d Proposition 1.32 Let x : [0, T ] → Then it

R be absolutely continuous. can be written in the form x0 + 0 x˙ t dt with x˙ ∈ L1 [0, T ] , Rd . As a consequence, C 0,1-var [0, T ] , Rd = x : [0, T ] → Rd absolutely continuous .

Proof. It suﬃces to consider d = 1. The function x determines a signed Borel measure on R via µ ((−∞, t]) = x0,t ≡ xt − x0 for t ∈ [0, T ] 7 See

Folland and Stein’s book [55] for instance.

1.3 Continuous paths of bounded variation on Rd

35

and putting zero mass on R\ [0, T ]. The assumption of absolute continuity of x implies that µ is absolutely continuous (in the sense of measures) with respect to Lebesgue measure λ. By the Radon–Nikodym theorem, there exists an integrable density function y = dµ/dλ, an integrable function from [0, T ] to R, uniquely deﬁned up to Lebesgue null sets, such that

t

xt = µ ((0, t]) =

ys ds. 0

Hence, using Proposition 1.31, x ∈ C 0,1-var [0, T ] , Rd . The converse inclusion follows directly from Proposition 1.29. Remark 1.33 Our notation for x˙ for the unique L1 -function with the property t x˙ t dt xt = x0 + 0

for absolutely continuous x is consistent with the fundamental theorem of calculus for Lebesgue integrals (e.g. [55], p. 106). It states that a real-valued function x on [0, T ] is absolutely continuous if and only if its derivative xt+ h − xt lim h→0 h exists for almost every t ∈ [0, T ] and gives an L1 -function whose indeﬁnite integral is xt − x0 . We have not shown (and will not use) the fact that x˙ is the almost-sure limit of the above diﬀerence quotient. Corollary 1.34 Let x ∈ C 1-var [0, T ] , Rd . Then piecewise linear approximations converge in 1-variation, x − xD

1-var;[0,T ]

→ 0 as |D| → 0

if and only if x ∈ C 0,1-var [0, T ] , Rd . Proof. “=⇒” : Any 1-variation limit of piecewise linear approximation is . absolutely continuous and hence in C 0,1-var “ ⇐= ” : Fix ε > 0, and x ∈ C 0,1-var [0, T ] , Rd . From the very deﬁnition of this space there exists a smooth path y such that |x − y|1-var;[0,T ] ≤

ε . 3

We claim that for all dissections D with small enough mesh (depending on y and ε), ε |y − y D |1-var;[0,T ] < . 3

36

Continuous paths of bounded variation

Indeed, this follows from Proposition 1.24 and the computation t i + 1 y˙ − y˙ D 1 y˙ s − yt i ,t i + 1 ds for D = {ti } ⊂ [0, T ] = L [0,1] ti+1 − ti ti i ti + 1 = |y˙ (s) − y˙ (ξ i )| ds with ξ i ∈ (ti , ti+1 ) ti

i

≤

|¨ y |∞

ti + 1

ti

i

|s − ξ i | ds ≤ |¨ y |∞ |D| T. D

By the triangle inequality and the contraction property of (·) as linear map from C 1-var [0, T ] , Rd into itself, see (1.4), we have x − xD ≤ |x − y|1-var;[0,T ] + y − y D 1-var;[0,T ] 1-var;[0,T ] + xD − y D 1-var;[0,T ] ≤ 2 |x − y| + y − y D 1-var;[0,T ]

≤

1-var;[0,T ]

ε

and this ﬁnishes the proof. Corollary 1.35 The space C 0,1-var [0, T ] , Rd is a separable Banach space (and hence Polish). Proof. Let Dn be the dyadic dissection {T k/2n : i = k, . . . , 2n } and deﬁne Ωn to be the set of paths from [0, T ] to Rd , linear on the dyadic times in Qd . Then, Ω := n Ωn intervals of Dn with values at dyadic is a countable set. If x ∈ C 0,1-var [0, T ] , Rd and ε > 0, there exists n such that x − xD n 1-var < ε/2. It is then easy to ﬁnd y ∈ Ωn such that D x n − y < ε/2, which proves that Ω is dense in C 0,1-var [0, T ] , Rd . 1-var This shows that C 0,1-var [0, T ] , Rd is separable.

1.3.4 Lipschitz continuity

older paths. We now turn to C 1-H¨o l [0, T] , Rd , the set of Lipschitz or 1-H¨ It includes, for instance, C 1 [0, T ] , Rd and elementary examples (e.g. t → |t|) show that this inclusion is strict. Proposition 1.36 C 1-H¨o l [0, T ] , Rd is Banach with norm x → |x (0)| + |x|1-H¨o l;[0,T ] . The closed subspace of paths in C 1-H¨o l [0, T ] , Rd started at 0, is also Banach under x → |x|1-H¨o l;[0,T ] . These Banach spaces are not separable. Proof. Non-separability follows from Example 1.26 together with |x|1-var;[0,T ] ≤ |x|1-H¨o l;[0,T ] or using the (well-known) non-separability of

1.3 Continuous paths of bounded variation on Rd

37

L∞ [0, T ] , Rd in conjunction with Proposition 1.37 below. All other parts of the proof are straightforward and left to the reader.

· Proposition 1.37 The map y → 0 yt dt is a Banach space isomorph from L∞ [0, T ] , Rd → C01-H¨o l [0, T ] , Rd . As a consequence, x ∈ C 1-H¨ol [0, T ] , Rd if and only if there exists a (uniquely determined) x˙ ∈ L∞ [0, T ] , Rd such that · x ≡ x0 + x˙ t dt 0

˙ L ∞ holds. and in this case the Banach isometry |x|1-H¨o l = |x| Proof. Similar to Proposition 1.31 and left to the reader. From general principles, any continuous path of ﬁnite 1-variation can be reparametrized to a 1-H¨older path. In the present context of Rd -valued paths this can be done so that the reparametrized path has constant speed. We have Proposition 1.38 Let x ∈ C 1-var [0, T ] , Rd , not constant. Deﬁne y (·) by y ◦ φ = x where φ(t) = |x|1-var;[0,t] / |x|1-var;[0,T ] . speed. More precisely, y is the Then y ∈ C 1-H¨o l [0, 1] , Rd has constant indeﬁnite integral of some y˙ ∈ L∞ [0, 1] , Rd and |y˙ (t)| ≡ |x|1-var;[0,T ] = |y|1-H¨o l;[0,1] for a.e. t ∈ [0, 1]. Proof. By the precise argument of the proof of Proposition 1.21, y is well-deﬁned and in C 1-H¨o l [0, 1] , Rd . From the very deﬁnition of y and invariance of 1-variation under reparametrization we have |y|1-var;[0,φ(t)] = |x|1-var;[0,t] = cφ(t) where c = |x|1-var;[0,T ] . On the other hand, by Propositions 1.37 and 1.31, y is the indeﬁnite integral of some y˙ ∈ L∞ [0, 1] , Rd and |y|1-var;[0,φ(t)] =

φ(t)

|y˙ (s)| ds. 0

It follows that |y| ˙ ≡ c almost surely. At last, the equality c = |y|1-H¨o l;[0,1] was noted in Remark 1.22.

38

Continuous paths of bounded variation

Remark 1.39 More generally, the proof shows that x can be reparame˙ ≡ 1 almost trized to y ∈ C 1-H¨o l [0, c] , Rd with unit speed, i.e. |y| surely. The reader will notice that the continuous embedding C 1-H¨o l [0, T ] , Rd → C 1-var [0, T ] , Rd is a consequence of the trivial estimate |x|1-var;[0,T ] ≤ |x|1-H¨o l;[0,T ] T. ol [0, T ] , Rd , As in the previous section it is natural to consider C 0,1-H¨ deﬁned as the closure of smooth paths in C 1-H¨o l [0, T ] , Rd . The resulting closure is a space we have already encountered. Proposition 1.40 The closure of smooth paths in C 1-H¨o l [0, T ] , Rd equals C 1 [0, T ] , Rd . Proof. Let us ﬁrst observe that the norm x → |x0 | + supt∈[0,T ] |x˙ t | on C 1 [0, T ] , Rd makes C 1 [0, T ] , Rd a Banach space. To avoid trivialities (norms vs semi-norms), let us assume that all paths are null at 0. Using C 1 [0, T ] , Rd ∼ = ⊕di=1 C 1 ([0, T ] , R) and similar for C 1-H¨o l it suﬃces to consider d = 1. Given a smooth path x : [0, T ] → R with x (0) = 0 we ﬁrst show that |x|1-H¨o l ≡

sup s,t∈[0,T ]

|x (t) − x (s)| |t − s|

equals

sup |x˙ t | . t∈[0,T ]

Indeed, from |x (t + h) − x (h)| ≤ |x|1-H¨o l h we see that |x˙ t | ≤ |x|1-H¨o l for all t ∈ [0, T ] while the converse estimate follows from the intermediate value theorem, |x (t) − x (s)| t−s

=

|x˙ (ξ)|

for ξ ∈ (s, t)

≤

|x| ˙ ∞;[0,T ] .

Any sequence (xn ) of smooth paths which converges (in 1-H¨ older norm) to some path x is also Cauchy in 1-H¨older. By the previous argument, it is ˜ ∈ C 1 ([0, T ] , R). also Cauchy in C 1 ([0, T ] , R) and so converges to some x 1 Since both 1-H¨ older and C -norm imply pointwise convergence we must have x = x ˜ ∈ C 1 ([0, T ] , R) and the proof is ﬁnished.

1.4 Sobolev spaces of continuous paths of bounded variation

39

1.4 Sobolev spaces of continuous paths of bounded variation 1.4.1

Paths of Sobolev regularity on Rd

We saw in Proposition 1.31 that a path x is in C 0,1-var [0, T ] , Rd if and only if · x0 + x˙ t dt

0

0

1

d

∞

d

˙ L 1 . We then saw, with x˙ ∈ L [0, T ] , R and in this case |x|1-var = |x| Proposition 1.37, that a path x is Lipschitz, in symbols x ∈ C 1-H¨o l ([0, T ] , Rd , if and only if · x0 + x˙ t dt ˙ L ∞ . This suggests with x˙ ∈ L [0, T ] , R and in this case |x|1-H¨o l = |x| considering the following path spaces. Deﬁnition 1.41 For p ∈ [1, ∞] , we deﬁne W 1,p [0, T ] , Rd to be the space of Rd -valued functions on [0, T ] of the form · x (·) = x0 + ydt (1.5) 0

with y ∈ Lp [0, T ] , Rd . Writing x˙ instead of y we further deﬁne ˙ L p ;[0,T ] = |x|W 1 , p ;[0,T ] := |x|

T

1/p p

|x| ˙ du

.

0

The set of such paths with x0 = o ∈ Rd is denoted by Wo1,p [0, T ] , Rd . As always, [0, T ] may be replaced by any other interval [s, t] ⊂ R. It is clear from the deﬁnition that W 1,1 = C 0,1-var and hence (Proposition 1.32) precisely the set of absolutely continuous paths, while W 1,∞ is precisely the set of Lipschitz or 1-H¨older paths. It is also clear from the usual inclusions of Lp -spaces that W 1,∞ ⊂ W 1,p ⊂ W 1,1 . In particular, any path in W 1,p is absolutely continuous (and then of course of bounded variation). Proposition 1.42 The space W 1,p [0, T ] , Rd is a Banach space under the norm x → |x0 | + |x|W 1 , p ;[0,T ] . The closed subspace of paths in W 1,p [0, T ] , Rd started at 0, is also Banach under x → |x|W 1 , p ;[0,T ] . These Banach spaces are separable if and only if p ∈ [1, ∞).

40

Continuous paths of bounded variation

Proof. Since Lp ⊂ L1 , we can use Proposition 1.31 to see that the map ˙ L p [0,T ] . The closed x → x˙ is well-deﬁned, as is its norm x → |x0 | + |x| 1,p d subspace of paths in W [0, T ] , R started at 0 is isomorphic (as normed space) to Lp [0, T ] , Rd and hence Banach. The separability statement now follows from well-known facts about Lp -spaces. Exercise 1.43 Let p ∈ [1, ∞] and recall that we equipped W 1,p [0, T ] , Rd with Banach norm |x (0)| + |x| ˙ L p ;[0,T ] . ˙ L p ;[0,T ] , for all Show that an equivalent norm is given by |x|L q ;[0,T ] + |x| q ∈ [1, ∞]. Solution. Lp -control of x˙ gives a modulus for x and in particular |x0,t | ≤ |x|W 1 , p ;[0,T ] t1−1/p where 1/p = 0 for p = ∞.Using |xt | ≤ |x0 | + |x0,t | one controls the supremum of x over t ∈ [0, T ] and then any Lq -norm. Every path in W 1,p ⊂ W 1,1 is continuous and of ﬁnite 1-variation. For p = ∞, such paths are Lipschitz or 1-H¨ older continuous; more precisely |xs,t | ≤ |x|W 1 , ∞ ;[s,t] |t − s|. Observe that the right-hand side is a control so that |xs,t | in the above estimate can be replaced by |x|1-var;[s,t] . In the following theorem we see that a similar statement holds true for all p > 1. Theorem 1.44 Let p ∈ (1, ∞). Given x ∈ W 1,p [0, T ] , Rd , ω (s, t) = |x|W 1 , p ;[s,t] (t − s)

1−1/p

deﬁnes a control function on [0, T ] and we have |x|1-var;[s,t] ≤ ω (s, t) for all s < t in [0, T ]. In particular, we have the continuous embedding W 1,p [0, T ] , Rd → C 1-var [0, T ] , Rd . Proof. Without loss of generality x0 = 0. By Proposition 1.31, x is the older’s indeﬁnite integral of some x˙ ∈ L1 . Deﬁne α = 1 − 1/p. Using H¨ inequality with conjugate exponents p and 1/α |xs,t | ≤

t

α

|x˙ r | dr ≤ (t − s) s

=

|x|W 1 , p ;[s,t] (t − s)

=

ω (s, t) .

t

1/p p

|x˙ r | dr s

α

We show that ω is a control. Continuity of ω is obvious from the fact that p p ˙ , over [s, t]. |x|W 1 , p ;[s,t] is the integral of an integrable function, namely |x| Only super-additivity, ω (s, t) + ω (t, u) ≤ ω (s, u) with s ≤ t ≤ u, remains

1.4 Sobolev spaces of continuous paths of bounded variation

41

to be shown. From H¨older’s inequality with conjugate exponents p and p/ (p − 1) = 1/α we obtain α

α

≤

|x|W 1 , p ;[s,t] (t − s) + |x|W 1 , p ;[t,u ] (u − t) 1/p ! "(p−1)/p p p α p α p (t − s) p −1 + (u − t) p −1 |x|W 1 , p ;[s,t] + |x|W 1 , p ;[t,u ]

=

|x|W 1 , p ;[s,u ] (u − t) .

α

By Proposition 1.11, we conclude that |x|1-var;[s,t] ≤ ω (s, t). In particular, |x|1-var;[0,T ] ≤ ω (0, T ) = |x|W 1 , p ;[0,T ] T 1−1/p which gives the continuous embedding. d Proposition 1.45 Let p ∈ (1, ∞). A function x : [0, T ] → R is in 1,p d [0, T ] , R if and only if Mp (x) < ∞ where W

Mp (x)

:

=

sup (t i )∈D([0,T ])

=

lim

xt

i ,t i + 1

p p−1

i

sup

|ti+1 − ti | xt i ,t i + 1 p

δ →0 (t i )∈Dδ ([0,T ])

|ti+1 − ti |

i

p−1

and in this case p

|x|W 1 , p ;[0,T ] = Mp (x) . Proof. Without loss of generality x0 = 0 and we assume x ∈ W 1,p is the older’s inequality gives indeﬁnite integral of some x˙ ∈ Lp . Then, H¨ xt

≤ |ti+1 − ti |1/p i ,t i + 1

ti + 1

1/p p

|x˙ u | du

ti

where 1/p + 1/p = 1. It immediately follows that

T

Mp (x) ≤ 0

p

p

|x| ˙ du = |x|W 1 , p ;[0,T ] .

(1.6)

Conversely, suppose that Mp (x) < ∞; given s1 < t1 ≤ s2 < t2 ≤ · · · < older’s inequality yields sn < tn in [0, T ] , H¨ n

|xt i − xs i |

=

i=1

n |xt i − xs i | 1/p

i=1

≤

|ti+1 − ti | 1/p

(Mp (x))

i

|ti+1 − ti |

1/p

1/p |ti+1 − ti |

Continuous paths of bounded variation

42

which shows that x is absolutely continuous, hence precisely in C00,1-var , and (Proposition 1.32) the indeﬁnite integral of some x˙ ∈ L1 [0, T ]. We

T p show that x˙ ∈ Lp [0, T ], with 0 |x˙ u | du bounded by Mp (x). Let Dn = i Dn → x in 1-variation norm, and n T : i = 0, . . . , n . By Corollary 1.34, x therefore we have the convergence Dn

x˙

n n = x ( i −1 ) T n T i=1

, inT

By passing to a subsequence

1[ ( i −1 ) T n

, inT

1 ) → x˙ ∈ L [0, T ] .

˜ k = (Dn ) we can achieve that D k

˜

x˙ tD k →k →∞ x˙ t for almost every t ∈ [0, T ] with respect to Lebesgue measure. By Fatou’s lemma we then see that T T D˜ k p p |x˙ u | du ≤ lim inf x˙ t du k →∞ 0 0 xt ,t p i i+ 1 = lim inf p−1 k →∞ ˜ k |ti+1 − ti | i:t i ∈D xt ,t p i i+ 1 ˜ ≤ lim sup p−1 =: Mp (x) . δ →0 |D |≤δ |t − ti | i:t i ∈D i+1 p ˜ p (x) and with the trivRecalling (1.6) we get Mp (x) ≤ |x|W 1 , p ;[0,T ] ≤ M ˜ p (x) ≤ Mp (x) we must have equality throughout. This ﬁnishes the ial M proof.

1.4.2

Paths of Sobolev regularity on metric spaces

We already remarked that W 1,1 (resp. W 1,∞ ) coincides with the set of absolutely continuous (resp. 1-H¨older) paths and this kind of regularity only requires paths with values in an abstract metric space (E, d). Proposition 1.45 suggests how to deﬁne W 1,p -regularity in a metric setting. Although we shall only need p = 2 in later chapters (in particular, in our discussions of large deviations), the case p ∈ (1, ∞) is covered without extra eﬀort and has applications in large deviation-type results for diﬀusions on fractals (see comments below). Deﬁnition 1.46 For p ∈ (1, ∞) we deﬁne W 1,p ([0, T ] , E) as those paths x : [0, T ] → (E, d) for which 1/p d xt i , xt i + 1 p sup < ∞. |x|W 1 , p ;[0,T ] := p−1 |ti+1 − ti | (t i )∈D([0,T ]) i The subset of paths started at o ∈ E is denoted by Wo1,p ([0, T ] , E). As always, [0, T ] may be replaced by any other interval [s, t].

1.4 Sobolev spaces of continuous paths of bounded variation

43

We now give a generalization of Theorem 1.44. Theorem 1.47 For any x ∈ W 1,p ([0, T ] , E) we have for all s, t ∈ [0, T ] , 1−1/p

d (xs , xt ) ≤ |x|1-var;[s,t] ≤ |x|W 1 , p ;[s,t] (t − s)

.

(1.7)

In particular, W 1,p ([0, T ] , E) ⊂ C 1-var ([0, T ] , E) . Proof. From the very deﬁnition of |x|W 1 , p ;[0,T ] we have p

p

d (xs , xt ) ≤ |x|W 1 , p ;[s,t] |t − s|

p−1

and the estimate on d (xs , xt ) follows. We then show, exactly as in the proof of Theorem 1.44, that the map 1−1/p

(s, t) → |x|W 1 , p ;[s,t] (t − s)

(1.8)

is super-additive and the estimate on |x|1-var;[s,t] follows by Proposition 1.11. Remark 1.48 To see that (1.8) is actually a control function, one would have to undergo a similar continuity consideration as in Propo sition 1.12. As in the case of 1-variation (cf. Proposition 1.14) it is enough in the deﬁnition of |x|W 1 , p ;[0,T ] to look at dissections with small mesh. Proposition 1.49 For every x ∈ C ([0, T ] , E), d xt i , xt i + 1 p p |x|W 1 , p ;[0,T ] = lim sup p−1 ∈ [0, ∞] . δ →0 (t i )∈Dδ ([0,T ]) |ti+1 − ti | i:t i ∈D Proof. We assume |x|W 1 , p ;[0,T ] < ∞, leaving the case |x|W 1 , p ;[0,T ] = ∞ to the reader. It suﬃces to show that, for any s < t < u in [0, T ], p

|d (xs , xu )| |u − s|

p−1

p

≤

|d (xs , xt )| p−1

|t − s|

p

+

|d (xt , xu )| |u − t|

p−1

(1.9)

˜ as this will allow us to replace a given dissection D with a reﬁnement D with ˜ D < δ. (We used a similar argument in the proof of Proposition 1.14.) To p

this end, recall the elementary inequality (θa + (1 − θ) b) ≤ θap +(1 − θ) bp for a, b > 0 and θ ∈ (0, 1). Replacing θa by a and (1 − θ) b by b gives p

(a + b) ≤

ap θp−1

+

bp (1 − θ)

p−1

and this implies (1.9) with θ = (t − s) / (u − s) and d (xs , xu ) ≤ d (xs , xt ) + d (xt , xu ) ≡ a + b.

44

Continuous paths of bounded variation

Exercise 1.50 As usual, let C ([0, T ] , E) be equipped with the uniform topology. Let p ∈ (1, ∞). (i) Show that p

x ∈ C ([0, T ] , E) → Mp (x) := |x|W 1 , p ;[0,T ] ∈ [0, ∞] is lower semi-continuous. (ii) Assume that E has the Heine–Borel property, i.e. bounded sets have compact closure. Show that the level sets {x ∈ Co ([0, T ] , E) : Mp (x) ≤ Λ} with Λ ∈ [0, ∞) and o ∈ E are compact. (Hint: Arzela–Ascoli.) Solution. (i) Assume xn → x uniformly (or even pointwise) on [0, T ] and ﬁx a dissection D ⊂ [0, T ]. Then p d xnti , xnti + 1 d xt i , xt i + 1 p ≤ lim inf Mp (xn ) p−1 = lim ninf p−1 →∞ n →∞ |ti+1 − ti | |ti+1 − ti | i:t i ∈D i:t i ∈D and taking the sup over all dissections ﬁnishes the proof. (ii) By (i) it is clear that level sets are closed. Thanks to Theorem 1.47 we know that Mp (x) ≤ Λ implies 1−1/p

d (xs , xt ) ≤ Λ1/p (t − s)

with equicontinuity and boundedness of {x ∈ Co ([0, T ] , E) : Mp (x) ≤ Λ}. Conclude with Arzela–Ascoli.

1.5 Comments Continuous paths of ﬁnite variation, also known as rectiﬁable paths, arise in many areas of analysis and geometry. Ultimately the focus of this book is on non-rectiﬁable paths and so we avoid the notion of rectiﬁability altogether. Topics such as absolute continuity of real-valued functions on R, the fundamental theorem of calculus for Lebesgue integrals or the Radon– Nikodym theorem are found in many textbooks on real analysis such as Rudin [149], Folland [55] or Driver [45]. The interplay between variation, H¨ older and W 1,p -spaces was studied in Musielak and Semadeni [132]; in particular, the authors of [132] attribute Proposition 1.45 to Riesz. A nice martingale proof of this can be found in Revuz and Yor [143]. The extension of W 1,p -regularity to paths in metric spaces is not for the sake of generality but arises, for instance, in the context of sample path large deviation(-type) estimates for symmetric diﬀusions; see Bass and Kumagai [6] and more speciﬁcally our later discussion of large deviation for Markov processes lifted to rough paths, Section 16.7.

2 Riemann–Stieltjes integration In this chapter we give a brief exposition of the Riemann–Stieltjes integral and its basic properties.

2.1 Basic Riemann–Stieltjes integration We will use the notation L Rd , Re for the space of linear maps from Rd into Re . We will always equip this space with its operator norm, that is if f ∈ L Rd , Re , then |f | = sup |f x|Re . x∈Rd |x|Rd =1

Deﬁnition y be two functions from [0, T ] into Rd and d e 2.1 Let x and n L R , R . Let Dn = (ti : i) be a sequence of dissections of [0, T ] with # D −1 |Dn | → 0, and ξ ni some points in tni , tni+1 . Assume i=0 n y (ξ ni ) xt ni ,t ni+ 1 converges when n tends to ∞ to a limit I independent of the choice of ξ ni and the sequence (Dn ). Then we say that the Riemann–Stieltjes integral of y against x (on [0, T ]) exists and write T T ydx := yu dxu := I. 0

0

We call y the integrand and x the integrator. Of course, [0, T ] may be replaced by any other interval [s, t]. Proposition 2.2 Let x ∈ C 1-var [0, T ] , Rd and y : [0, T ] → L Rd , Re

T piecewise continuous.1 Then the Riemann–Stieltjes integral 0 ydx exists, is linear in y and x, and we have the estimate T ydx ≤ |y|∞;[0,T ] |x|1-var;[0,T ] . 0 Moreover,2 t yu dxu − 0 1 This

0

s

t

yu dxu for all 0 ≤ s < t ≤ T .

yu dxu =

(2.1)

s

will cover all our applications. integrals in (2.1) are understood in the sense of Deﬁnition 2.1 with [0, T ] replaced by the intervals [0, t] , [0, s] , [s, t] respectively. 2 All

Riemann–Stieltjes integration

46

Proof. Let us say that a real-valued function y is dx-integrable (on the

T ﬁxed time interval [0, T ]) if the Riemann–Stieltjes integral 0 ydx exists. Step 1: Step-functions, i.e. functions of the form g (t) = a0 1[0,t 1 ] +

n −1

ai 1(t i ,t i + 1 ] (t) ,

i=1

with 0 < t1 < · · · < tn = T and ai ∈ L Rd , Re , are dx-integrable and

T

gdx = 0

n −1

ai xt i ,t i + 1 .

i=0

Step 2: The set of dx-integrable functions is a linear space, i.e. if g and h are dx-integrable, then so is αg + βh, with α, β ∈ R which readily implies that T

T

(αg + βh) dx = α 0

T

gdx + β 0

hdx. 0

Step 3: If y is dx-integrable then T ydx ≤ |y|∞;[0,T ] |x|1-var;[0,T ] . 0 Step 4: The space of dx-integrable functions is closed in supremum topology on [0, T ]. Indeed, assume (y n ) is a sequence of dx-integrable functions such that |y − y n |∞;[0,T ] → 0 as n → ∞. By steps 2 and 3, T T y n dx − y m dx ≤ |yn − ym |∞;[0,T ] |x|1-var;[0,T ] 0 0

T and so In = 0 y n dx deﬁnes a Cauchy sequence whose limit we denote by of [0, T ] with mesh |Dn | → 0 I. Let Dn = (tni )i be a sequence of dissections and ξ ni an arbitrary point in tni , tni+1 for all i, n. Then, # D −1 #D m −1 m m m m n y (ξ i ) xt i ,t i + 1 ≤ |I − In | + (y (ξ i ) − y (ξ i )) xt i ,t i + 1 I − i=0 i=0 #D m −1 y n (ξ m ) x + In − t i ,t i + 1 i i=0

≤ |I − In | + |y − y n |∞;[0,T ] |x|1-var;[0,T ] #D m −1 + In − y n (ξ m i ) xt i ,t i + 1 . i=0

(2.2)

2.1 Basic Riemann–Stieltjes integration

47

Fixing ε > 0 we can pick n large enough so that |I − In | + |y − y n |∞;[0,T ] |x|1-var;[0,T ] < ε/2. Then, since y n is dx-integrable, there exists M > 0 such that m > M implies that (2.2) < ε/2 and hence #D m −1 y (ξ m I − i ) xt i ,t i + 1 < ε. i=0

T But this shows precisely that the Riemann–Stieltjes integral 0 ydx exists and so y is dx-integrable. Step 5: Any y ∈ C [0, T ] , L Rd , Re is dx-integrable. Indeed, take ξ ni ∈ n n ti , ti+1 where Dn = (tni )i is as in the previous step and set

(# D n )−1

y n (t) := y (ξ n0 ) 1[0,t 1 ] (t) +

y (ξ ni ) 1(t i ,t i + 1 ] (t) .

i=1

It then suﬃces to observe that limn →∞ |y − y n |∞;[0,T ] = 0 because y is uniformly continuous on [0, T ] and we conclude with step 4. If y is only piecewise continuous (i.e. bounded with ﬁnitely many points of discontinuthat it contains all points of discontinuity. ity) it suﬃces to choose Dn such ˜ ∈ C 1-var [0, T ] , Rd , the last step shows that any y ∈ Step 6: Given x, x C [0, T ] , L Rd , Re is d (αx + β x ˜)-integrable, for any α, β ∈ R. This easily implies linearity of T ydx x ∈ C 1-var [0, T ] , Rd → 0

T and in conjunction with step 2 we obtain bilinearity of (x, y) → 0 ydx. Step 7: Fix s, t with 0 ≤ s < t ≤ T . If y is piecewise continuous then so is 1[0,t] yu and t T 1[0,t] (u) yu dx = yu dx. 0

0

Relation (2.1) then follows from 1[0,t] yu = 1[0,s] (u) yu + 1(s,t] (u) yu . The details are left to the reader. Exercise 2.3 Assume x ∈ C 1 [0, T ] , Rd and y ∈ C [0, T ] , L Rd , Re . Show that T T yu dxu = yu x˙ u du. 0

0

We then have the classical integration-by-parts formula. It can be obtained by a simple passage to the limit in an elementary partial summation formula for ﬁnite sums. The details are left to the reader.

Riemann–Stieltjes integration

48

Proposition 2.4 (integration by parts) Let x ∈ C 1-var [0, T ] , Rd and y ∈ C 1-var [0, T ] , L Rd , Re . Then

T

T

(dyu ) dxu = yT xT − y0 x0 .

yu dxu + 0

0

Exercise 2.5 Take (x, y) ∈ C 1-var [0, T1 ] , Rd × C [0, T1 ] , L Rd , Re and assume φ a continuous non-decreasing function φ from [0, T2 ] onto [0, T1 ]. Show that

t

yφ(·) d (x ◦ φ) = 0

φ(t)

ydx for all t ∈ [0, T2 ] . 0

Exercise 2.6 Let x ∈ C 1-var [0, T ] , Rd , φ a C ∞ function from R into

∞ R+ , compactly supported on [−1, 1] with −∞ φ (u) du = 1. Deﬁne Φt =

t φ (u) du and extend x to a continuous function from R into Rd by −∞ setting x ≡ x0 on (−∞, 0) and x ≡ xT on [T, ∞). Deﬁne for all ε > 0 the molliﬁer approximation to x by Φ(t−s)/ε dxs . xε : t ∈ [0, T ] → x0 + R

Show that (i) for all ε > 0, xε is inﬁnitely diﬀerentiable; (ii) for all ε > 0, |xε |1-var;[0,T ] ≤ |x|1-var;[0,T ] , and also |xε |1-H¨o l;[0,T ] ≤ |x|1-H¨o l;[0,T ] ; (iii) xε converges to x in supremum topology when ε tends to 0. Solution. (i) One can easily see that, for n ≥ 1, the nth derivative of xε

−d dxs , where Φ(n ) is the nth derivative of Φ. (ii) the is t → R ε Φ(n ) t−s ε ε 1-variation of x is given by T 1 (1) t − s ε dxs dt |x |1-var = εΦ ε 0 R 1 ( 1 ) t − s ≤ Φ . |dxs | dt ε R R ε 1 (1) t − s Φ dt . |dxs | ≤ ε R ε R ≤ |dxs | = |x|1-var;[0,T ]. R

The 1-H¨ older bound follows from integration by parts, 1 t−s 1 ds = xs φ xt+ εs φ (−s) ds. xεt = ε ε −1 R

(2.3)

2.2 Continuity properties

49

(iii) As x is continuous (and hence uniformly continuous), sup

lim

ε→0 s,t∈[0,1]×[0,T ]

|xt+ εs − xt | = 0,

and (2.3) implies that limε→0 supt∈[0,T ] |xεt − xt | = 0.

2.2 Continuity properties

Proposition 2.2 obviously implies that (x, y) → ydx, viewed as a map from C 1-var [0, T ] , Rd × C [0, T ] , L Rd , Re → C 1-var ([0, T ] , Re ) , is a bounded, bilinear map and hence continuous (and even Fr´echet smooth) in the respective norms.3 In particular, |y n − y|∞;[0,T ] → 0, |xn − x|1-var;[0,T ] → 0 implies that

·

y dx → n

0

n

·

ydx 0

in 1-variation. However, this is not the last word on continuity. For instance, the seemingly harmless assumption that all xn are piecewise smooth would already force us to restrict attention to x absolutely continuous (cf. Proposition 1.29). We thus formulate continuity statements that are applicable under the weaker assumption of uniform convergence with uniform 1-variation bounds. functions Proposition 2.7 Let y n , y : [0, T ] → L Rd , Re be continuous n n 1-var [0, T ] , Rd and and assume y → y uniformly. Assume x , x ∈ C xn → x uniformly with sup |xn |1-var;[0,T ] < ∞. n

Then

t

y dx → n

0

t

ydx uniformly for t ∈ [0, T ] .

n

0

1 -va r [0, T ] , Rd but a that what we call |·|1d-va r is a only a semi-norm on C 1 -va r genuine norm on C 0 [0, T ] , R and we can obviously assume x (0) = 0 as only dx is of interest. Alternatively, deﬁne an equivalence relation on C 1 -va r [0, T ] , Rd by setting 1 -va r , is Banach x ∼ y iﬀ t → xt − y t is constant; the resulting quotient space, say C

1 -va r. 1 -va r under |·|1 -va r and viewing (f, x) → f dx as a map C × C →C 3 Observe

Riemann–Stieltjes integration

50

Proof. Set c = supn |xn |1-var;[0,T ] . Then t t n n y dx − ydx ≤ 0

t t t n n n (y − y) dx + ydx − ydx 0 0 0 t t n n ydx − ydx ≤ c |y − y|∞;[0,T ] +

0

0

0

and so it is enough to show t t n ydx − ydx → 0 0

(2.4)

0

uniformly in t ∈ [0, T ] as n → ∞. Fix ε > 0 and pick m = m (ε) such that supn |xn |1-var;[0,T ] m

< ε/2.

Then, from uniform continuity of y on [0, T ], we can ﬁnd a dissection D = (ti ) such that the step function

(# D ) D

y (t) :=

y (ti−1 ) 1[t i −1 ,t i ) (t)

i=1

satisiﬁes y − y D ∞ ≤ 1/m. Observe that (using Lemma 1.15) |x|1-var;[0,T ] / m < ε/2. We estimate the

left-hand side of (2.4) by adding/subtracting the integrals y D dxn and y D dx. This leaves us with three terms, of which the ﬁrst two are dealt with by t t D D n y − y dx + sup y − y dx ≤ ε. sup sup n t∈[0,T ]

t∈[0,T ]

0

0

On the other hand, y D is constant over the (ﬁnitely many) intervals [ti , ti+1 ). Fix t ∈ [0, T ] and let tD ∈ D be the largest point in D for which tD ≤ t. Then t D n n + yt D xt D ,t − xntD ,t y d (x − x ) = yt i −1 xt i −1 ,t i − xt i −1 ,t i 0

i

where the sum that

i

runs over all integers i ≥ 1 for which ti−1 ≤ t. It follows

t D n y d (x − x ) ≤ (#D) × 2 |x − xn |∞;[0,T ] sup

t∈[0,T ]

0

(2.5)

2.2 Continuity properties

51

where #D denotes the number of points in D, dependent on m and hence on ε. It follows that t t n ydx − ydx ≤ ε + (#D) × 2 |x − xn |∞;[0,T ] . 0

0

Using |x − xn |∞;[0,T ] → 0 as n → ∞ it follows that t t lim sup ydxn − ydx ≤ ε n →∞ 0

0

and we conclude by sending ε ↓ 0. Another useful property of Riemann–Stieltjes integration is uniform continuity on bounded sets. and x, x ∈ C 1-var 2.8 Let y, y ∈ C [0, T ] , L Rd , Re Proposition [0, T ] , Rd . Then, · · ydx − y dx 0

≤

0

1-var;[0,T ]

|x|1-var;[0,T ] . |y − y |∞;[0,T ] + |y |∞;[0,T ] . |x − x |1-var;[0,T ] .

In particular, the map (x, y) ∈ C 1-var [0, T ] , Rd ×C [0, T ] , L Rd , Re →

· ydx ∈ C 1-var ([0, T ] , Re ) is locally Lipschitz. 0 Proof. It suﬃces to insert and subtract inequality.

· 0

y dx, followed by the triangle

applications, integrands frequently come in the form ϕ (xt ) ∈ L In Rd , Re for ϕ : Rd → L Rd , Re or V (yt ) for an Re -valued path y and V : Re → L Rd , Re . With focus on the latter, we state the following uniform continuity property; the simple proof is left to the reader. Corollary 2.9 Let x, x ∈ C 1-var [0, T ] , Rd , y, y ∈ C ([0, T ] , Re ) and V : Re → L Rd , Re continuous. Assume |x|1-var;[0,T ] , |x |1-var;[0,T ] , |y|∞;[0,T ] , |y |∞;[0,T ] < R and let ε > 0. Then there exists δ = δ (ε, R, V ) so that |x − x |1-var;[0,T ] + |y − y |∞;[0,T ] < δ

Riemann–Stieltjes integration

52

implies

· · V (y) dx − V (y ) dx 0

0

< ε. 1-var;[0,T ]

2.3 Comments Riemann–Stieltjes integration is discussed in many elementary analysis texts, for example Rudin [148] or Protter and Morrey [139].

3 Ordinary diﬀerential equations We develop the basic theory of ordinary diﬀerential equations of the form dy dxi = Vi (y) dt dt i=1 d

on a ﬁxed time horizon [0, T ]. Here, x and y are paths with values in Rd , Re respectively and we have coeﬃcients Vi : Re → Re , often viewed as “driving” vector ﬁelds on Re . When the driving signal x is continuously diﬀerentiable we are dealing with an example of a (time-inhomogenous) ordinary diﬀerential equation. We give a direct existence proof, via Euler approximations, that applies to continuous, ﬁnite-variation driving signals and continuous vector ﬁelds; uniqueness holds for Lipschitz continuous vector ﬁelds.

3.1 Preliminaries Given a collection of (continuous) vector ﬁelds V = (V1 , . . . , Vd ) on Re and continuous, ﬁnite-variation paths x, y with values in Rd , Re we set

t

V (y) dx := 0

d i=1

t

Vi (y) dxi .

0

From the point of view of vector-valued Riemann–Stieltjes integration, this amounts precisely to viewing V as a map d Vi (y) ai } ∈ L Rd , Re , y ∈ Re → {a = a1 , . . . , ad → i=1

where L Rd , Re is equipped with the operator norm, so that d sup Vi (y) ai . |V (y)| := |V (y)|op := a∈Rd :|a|=1

(3.1)

i=1

Deﬁnition 3.1 A collection of vector ﬁelds V = (V1 , . . . , Vd ) on Re , viewed e d e as V : R → L R , R , is called bounded if |V |∞ := sup |V (y)| < ∞. y ∈Re

Ordinary diﬀerential equations

54

For any U ⊂ Re we deﬁne the 1-Lipschitz norm (in the sense of E. M. Stein) by # $ |V (y) − V (z)| , sup |V (y)| . sup |V |Lip 1 (U ) := max |y − z| y ∈U y ,z ∈U :y = z We say that V ∈ Lip1 (Re ) if |V |Lip 1 ≡ |V |Lip 1 (Re ) < ∞ and locally 1Lipschitz if |V |Lip 1 (U ) < ∞ for all bounded subsets U ⊂ Re . (The concept of Lip1 regularity will later be generalized to Lipγ in the sense of E. M. Stein.) Observe that 1-Lipschitz paths are Lipschitz continuous paths that are bounded. We now state a classical analysis lemma. Lemma 3.2 (Gronwall’s lemma) Let x ∈ C 1-var [0, T ], Rd , and φ : [0, T ] → R+ a bounded measurable function. Assume that for all t ∈ [0, T ]

t

φ (t) ≤ K + L

φs |dxs |,

(3.2)

0

for some K, L ≥ 0. Then, for all t ∈ [0, T ] φt ≤ K exp L|x|1-var;[0,t] . If t → Kt is a non-negative, non-decreasing function, K may be replaced by Kt . Proof. After n iterated uses of (3.2) φ (t)

≤

t t1 t t n −1 K + KL |dxs | + · · · + KLn ··· |dxt n | · · · |dxt 1 | 0 0 0 0 t t1 t n −1 t n ··· φ (tn +1 ) |dxt n + 1 | · · · |dxt 1 |. +Ln +1 0

0

0

0

Since x is continuous, t 0

t1

···

0

t n −1

|dxt n | · · · |dxt 1 | =

0

|x|n1-var;[0,t] n!

.

Then,

φ (t) ≤ K exp L|x|1-var;[0,t] + |φ|∞;[0,T ]

n +1 L|x|1-var;[0,t] (n + 1)!

and sending n → ∞ gives the required estimate. The last statement, replacing K by some non-decreasing Kt , comes from the obvious remark that the previous estimate can be applied on the interval [0, t] with K = Kt .

3.2 Existence

55

3.2 Existence Let us ﬁrst deﬁne what we mean by solution of a (controlled, ordinary) diﬀerential equation: Deﬁnition 3.3 Given a collection of continuous vector ﬁelds V = (V1 , . . . , Vd ) on Re , a driving signal x ∈ C 1-var [0, T ] , Rd and an initial condition y0 ∈ Re , we write π (V ) (0, y0 ; x) for the set of all solutions to the ODE1 dyt =

d

Vi (yt ) dxit ≡ V (yt ) dxt

(3.3)

i=1

for t ∈ [0, T ] started at y0 . The above ODE is understood as a Riemann– Stieltjes integral equation, i.e. t V (ys ) dxs . y0,t := yt − y0 = 0

In case of uniqueness y = π (V ) (0, y0 ; x) denotes the solution. If necessary, π (0, y0 ; x) is only considered up to some explosion time. Similarly, π (V ) (s, ys ; x) stands for solutions of (3.3) started at time s from a point ys ∈ Re . We shall frequently describe π (V ) (0, y0 ; x) as an “ODE solution, driven by x along the vector ﬁelds V and started from y0 ”. Existence of a solution holds under minimal regularity conditions on the vector ﬁelds. Theorem 3.4 (existence) Assume that (i) V = (V1 , . . . , Vd ) is a collection of continuous, bounded vector ﬁelds on Re ; (ii) y0 ∈ Re is an initial condition; (iii) x is a path in C 1-var [0, T ] , Rd . Then there exists a (not necessarily unique) solution to the ODE (3.3). Moreover, for all 0 ≤ s < t ≤ T π (V ) (0, y0 ; x) ≤ |V |∞ |x|1-var;[s,t] . (3.4) 1-var;[s,t] Remark 3.5 In case of non-uniqueness, we abuse notation in the above estimate in the sense that π (V ) (0, y0 ; x) stands for an arbitrary solution to (3.3) started at y0 . Proof. Let D = (ti )i be a dissection of the interval [0, T ] , and deﬁne the Euler approximation y (D ) : [0, T ] → Re by # (D ) y 0 = y0 , (D )

yt 1 The

(D )

= yt i

(D )

+ V yt i

xt i ,t for t ∈ [ti , ti+1 ] .

ODE (3.3) is time-inhomogenous unless xt is proportional to t.

Ordinary diﬀerential equations

56

Then, it is easy to see that for all 0 ≤ s < t ≤ T (D ) (D ) ≤ |V |∞ |x|1-var;[s,t] . ys,t ≤ y 1-var;[s,t]

(3.5)

In particular, (D ) y

∞;[0,T ]

≤ |y0 | + |V |∞ |x|1-var;[0,T ] .

Moreover, if rD denotes the greatest real number in D less than r then t t! " (D ) ) (D ) V yr(D − V y dxr . V yr(D ) dxr = (3.6) y0,t − r D 0

0

of dissections, with mesh |Dn | → 0 as n tends to Now let (Dn) be a sequence ∞. Clearly y (D n ) is equicontinuous and bounded. From Arzela–Ascoli’s theorem we see that y (D n ) has at least one limit point y. After relabelling our sequence, we can assume that y (D n ) converges to y uniformly on [0, T ] . (D ) (D ) Fix r ∈ [0, T ]. From (3.5), limn →∞ yr n − yr D nn = 0. On the other hand, (D n )

yr

(D )

→ yr hence yr D nn → yr and by continuity of V, n) lim V yr(D n ) − V yr(D = 0. Dn n →∞

(3.7)

By dominated convergence,2 we can pass to the limit in (3.6) to see that t V (yr ) dxr = 0. y0,t − 0

Finally, for s, t ∈ [0, T ], for any solution y ∈ π (V ) (0, y0 ; x) , t |ys,t | = V (yu ) dxu

≤ s

=

s t

|V |∞ . |dxu |

|V |∞ |x|1-var;[s,t] .

The right-hand side being a control, we obtain inequality (3.4). If we only assume continuity of the vector ﬁelds (without imposing growth conditions) existence holds up to an explosion time: Theorem 3.6 Assume that (i) V = (V1 , . . . , Vd ) is a collection of continuous vector ﬁelds on Re ; 2 Which requires us to know that Riemann–Stieltjes integrals with continuous integrands coincide with Lebesgue–Stieltjes integrals.

3.2 Existence

57

(ii) y0 ∈ Re is an initial condition; (iii) x is a path in C 1-var [0, T ] , Rd . Then either there exists a (global) solution y : [0, T ] → Re to ODE (3.3) started at y0 or there exists τ ∈ [0, T ] and a (local) solution y : [0, τ ) → Re such that y is a solution on [0, t] for any t ∈ (0, τ ) and lim |y (t)| = +∞.

tτ

Proof. Without loss of generality take y0 = 0. Replace V by compactly supported vector ﬁelds V n which coincide with V on the ball {y : |y| ≤ n}. From the preceding existence theorem, there exists a (not necessarily unique) ODE solution y 1 := π (V 1 ) (0, y0 ; x) which we consider only up to time τ 1 = inf t ∈ [0, T ] : yt1 ≥ 1 ∧ T > 0. If τ 1 = T then y = y 1 is a solution on [0, T ] and we are done. Set τ 0 = 0. We now deﬁne τ n , yn inductively and assume τ n ∈ [0, T ] , y n ∈ C ([τ n −1 , τ n ] , Re ) have been deﬁned. We then deﬁne y n +1 := π (V n + 1 ) τ n , yτnn ; x as an (again, not necessarily unique) ODE solution started from y n +1 (τ n ) = yτnn driven by x along the vector ﬁelds V n +1 up to time τ n +1 = inf t ∈ [τ n , T ] : ytn +1 ≥ n + 1 ∧ T > 0. If at any step in this induction, τ n = T , then (0, T ] = ∪nk=1 (τ k −1 , τ k ] and y (t) = ytk for t ∈ (τ k −1 , τ k ] deﬁnes a solution on [0, T ] and we ﬁnd ourselves in case (i) of the statement of the theorem. Otherwise, we obtain an increasing sequence (τ n ) with τ = limn τ n →∞ ∈ (0, T ]. Any interval (0, t] ⊂ (0, τ ) can be covered by intervals (τ k −1 , τ k ] and a solution on (0, t] is constructed as above by setting y (t) = ytk for t ∈ (τ k −1 , τ k ]. Moreover, be deﬁnition of τ n we see that |y (τ n )| = n → ∞ as n → ∞ and the proof is ﬁnished. Theorem 3.7 Assume that (i) V = (V1 , . . . , Vd ) is a collection of continuous vector ﬁelds on Re of linear growth, i.e. ∃A ≥ 0 : |Vi (y)| ≤ A (1 + |y|) for all y ∈ Re ; (ii) y0 ∈ Re is an initial condition; (iii) x is a path in C 1-var [0, T ] , Rd , and ≥ A |x|1-var;[0,T ] . Then explosion cannot happen. Moreover, any solution y to (3.3) satisﬁes the estimates (3.8) |y|∞;[0,T ] ≤ (|y0 | + ) exp () ,

Ordinary diﬀerential equations

58

and, for all 0 ≤ s < t ≤ T , |y|1-var;[s,t] ≤ (1 + |y0 |) exp (2) A Proof. For s ∈ [0, min (τ (y) , T )), s V (yu ) dxu |ys | ≤ |y0 | + 0 s |dxu | + A ≤ |y0 | + A 0

t

|dxu | . s

s

|yu | . |dxu | . 0

Hence, by Gronwall’s inequality, for all s ∈ [0, min (τ (y) , T )), s s |dxu | exp A |dxu | . |ys | ≤ |y0 | + A 0

(3.9)

0

This implies in particular that explosion cannot happen in ﬁnite time, and that s s |dxu | exp A |dxu | . |y|∞;[0,s] ≤ |y0 | + A 0

0

Let us now take s, t ∈ [0, T ]. Clearly, for all u, v ∈ [s, t] , v v yu ,v = V (yr ) dxr = V (yu + yu ,r ) dxr u

u

so that

v

|yu ,v | ≤ A (1 + |yu |)

v

|dxu | + A u

|yu ,r | |dxr | . u

By Gronwall’s inequality, we obtain v |dxu | exp A |yu ,v | ≤ A (1 + |yu |) u

v

|dxr | .

u

v

v

u Now, using inequality (3.9) and 0 |dxr | + u |dxr | = 0 |dxr |, v v u |dxr | ≤ |y0 | + 1 + A |dxr | exp A |dxr | (1 + |yu |) exp A u 0 0 t t |dxr | + exp 2A |dxr | ≤ |y0 | exp A 0 0 t |dxr | , ≤ (1 + |y0 |) exp 2A 0

which gives

t

|yu ,v | ≤ A (1 + |y0 |) exp 2A 0

|dxr |

v

|dxu | , u

3.3 Uniqueness

59

and hence |y|1-var;[s,t] ≤ A (1 + |y0 |) exp 2A

t

t |dxr | |dxu | .

0

s

3.3 Uniqueness We now show uniqueness for ODEs driven along Lipschitz vector ﬁelds by establishing Lipschitz continuity of the ﬂow. Theorem 3.8 Assume that (i) V = (V1 , . . . , Vd ) is a collection of Lipschitz continuous vector ﬁelds on Re such that, for some υ ≥ 0, υ ≥ sup

y ,z ∈Rd

|V (y) − V (z)| ; |y − z|

(ii) x ∈ C 1-var [0, T ] , Rd with, for some ≥ 0, υ |x|1-var;[0,T ] ≤ . Then, for every initial condition there exists a unique ODE solution to dy = V (y) dx on [0, T ]. Moreover, the associated ﬂow is Lipschitz continuous in the following sense that, for any initial conditions y01 , y02 ∈ Re , π (V ) 0, y01 ; x − π (V ) 0, y02 ; x (3.10) ≤ y01 − y02 exp () . ∞;[0,T ] Moreover, for all s < t in [0, T ] we have π (V ) 0, y01 ; x − π (V ) 0, y02 ; x ≤ y01 − y02 exp (2) .υ |x|1-var;[s,t] . 1-var;[s,t] Proof. Lipschitz continuous vector ﬁelds are of linear growth and existence of solutions on [0, T ] is guaranteed by Theorem 3.7. Let us write y i ∈ π (V ) 0, y0i ; x , i = 1, 2 for an arbitrary solution started from y01 , y02 respectively and set y¯ = y 1 − y 2 . Then t 1 V ys − V ys2 dxs , y¯t = y¯0 + 0

and hence

|¯ yt | ≤ |¯ y0 | + υ

t

|¯ ys | . |dxs | . 0

60

Ordinary diﬀerential equations

Gronwall’s inequality then leads to the ﬁrst stated estimate. Moreover, taking y01 = y02 shows that y 1 ≡ y 2 and there is indeed a unique solution. For the second estimate, we have t 1 2 V yr − V yr dxs |¯ ys,t | = s t |¯ yr | |dxr | ≤ υ s t t |dxr | + υ |¯ ys,r | . |dxr | . ≤ |¯ ys | .υ s

s

Applying Gronwall gives

t

|¯ ys,t | ≤ |¯ ys | .υ

|dxr | . exp () . s

Using the estimate (3.10), we obtain |¯ ys,t | ≤ |¯ y0 | .υ

t

|dxr | . exp (2) . s

Noting that the right-hand side is a control, we obtain our estimate. Since uniqueness is a local property we immediately have Corollary 3.9 Given x ∈ C 1-var [0, T ] , Rd , there is a unique solution to dy = V (y) dx started at y0 along locally 1-Lipschitz vector ﬁelds V = (V1 , . . . , Vd ) up to its possible explosion time. If explosion can be ruled out (e.g. under an additional linear growth condition, cf. Theorem 3.7) then there exists a unique solution on [0, T ].

3.4 A few consequences of uniqueness We ﬁrst show that time-change commutes with solving diﬀerential equations. Proposition 3.10 Let x ∈ C 1-var [0, T1 ] , Rd and V = (V1 , . . . , Vd ) a collection of locally Lipschitz continuous vector ﬁelds on Re of linear growth. Assume φ is a continuous non-decreasing function from [0, T2 ] onto [0, T1 ] so that x ◦ φ ∈ C 1-var [0, T2 ] , Rd . Then π (V ) (0, y0 ; x)φ(·) ≡ π (V ) (0, y0 ; x ◦ φ) on [0, T2 ] .

3.4 A few consequences of uniqueness

61

Proof. Let y = π (V ) (0, y0 ; x) denote the (unique) ODE solution. For all t ∈ [0, T2 ] d φ(t) yφ(t) = y0 + Vi (yr ) dxir . 0

i=1

By a change of variable r = φ (s) for Riemann–Stieltjes integrals, we obtain yφ(t) = y0 +

d i=1

t

Vi yφ(s) dxiφ(s)

0

which says precisely that t → yφ(t) is an ODE solution driven x ◦ φ along vector ﬁelds V1 , . . . , Vd started at y0 . By uniqueness, we therefore have π (V ) (0, y0 ; x)φ(·) = π (V ) (0, y0 ; x ◦ φ) .

Deﬁnition 3.11 (concatenation, time-reversal) (i) Given x ∈ C [0, T ] , Rd and x ˜ ∈ C [T, U ] , Rd we deﬁne the concatenation x x ˜ as a d 3 path in C [0, U ] , R deﬁned by (x x ˜) (t) = xt if t ∈ [0, T ] (x x ˜) (t) = (xS − x ˜S ) + x ˜t if t ∈ [T, U ] . (ii) Next, the time-inverse of a path x ∈ C [0, T ] , Rd is deﬁned as the path x run backwards on [0, T ], i.e. ← −T : t ∈ [0, T ] → x d x T −t ∈ R . − for When [0, T ] is ﬁxed and no confusion is possible, we simply write ← x the time-inverse of x. As a simple consequence of uniqueness we have the following two propositions. ˜ ∈ C 1-var [S, T ] , Rd and Proposition 3.12 Let x ∈ C 1-var [0, S] , Rd , x V = (V1 , . . . , Vd ) a collection of locally Lipschitz continuous vector ﬁelds on Re of linear growth. Then π (V ) (0, y0 ; x) ≡ π (V ) (0, y0 ; x x ˜) on [0, S] and π (V ) S, π (V ) (0, y0 ; x)S ; x ˜ ≡ π (V ) (0, y0 ; x x ˜) on [S, T ] . 3 Of course, x, x ˜ need not be deﬁned on adjacent intervals but a simple reparametrization will bring things back to the above deﬁnition. Formally speaking, concatenation is an operation on paths modulo their parametrization.

62

Ordinary diﬀerential equations

Proof. Obvious. Proposition 3.13 Let x ∈ C 1-var [0, T ] , Rd and V = (V1 , . . . , Vd ) a collection of locally Lipschitz continuous vector ﬁelds on Re of linear growth so that there is a (unique) ODE solution y = π (V ) (0, y0 ; x). Then for all 0 ≤ t ≤ T, −T x = yt . π (V ) 0, yT ; ← T −t Proof. Same as for Proposition 3.10, just use φ (t) = T − t. We record a simple corollary. (As a preview to an application discussed later on: when applied to the left-invariant vector ﬁelds U1 , . . . , Ud on the −T over [0, T ] step-N nilpotent group it implies that the signature of x ← x is trivial.) Corollary 3.14 Let x ∈ C 1-var [0, T ] , Rd and V = (V1 , . . . , Vd ) a collection of locally Lipschitz continuous vector ﬁelds on Re of linear growth so − x that there is a (unique) ODE solution y = π (V ) (0, y0 ; x). Reparametrize ← ← − as a path on [T, 2T ], i.e. x (t) = x2T −t . Then, −T = y . x π (V ) 0, y0 ; x ← 0 2T

3.5 Continuity of the solution map We now investigate continuity properties of the solution map, i.e. the map (y0 , x) → y, the ODE solution to dy = V (y) dx =

d

Vi (y) dxi

i=1

started at time 0 at y0 ∈ Re . In fact, it will not complicate things to consider the map (y0 , V, x) → y.

3.5.1 Limit theorem for 1-variation signals Let us recall that our notion of 1-Lipschitz regularity includes the assumption of boundedness, cf. Deﬁnition 3.1. We start with our ﬁrst continuity statement for solutions of ordinary diﬀerential equations. Theorem 3.15 We consider (i) V 1 = V11 , . . . , Vd1 and V 2 = V12 , . . . , Vd2 are two collections of Lip1 vector ﬁelds on Re with, for some υ ≥ 0, max V i Lip 1 ≤ υ; i=1,2

3.5 Continuity of the solution map

63

(ii) y01 , y02 ∈ Re are initial conditions; (iii) x1 and x2 are two paths in C 1-var [0, T ] , Rd with, for some ≥ 0, max xi 1-var;[0,T ] ≤ . i=1,2

Then, if y i = π (V i ) 0, y0i ; xi for i = 1, 2, we have 1 2 1 y −y y01 −y02 +υ x1 −x2 V −V 2 exp (2υ) . ≤ + ∞;[0,T ] 0;[0,T ] ∞

Proof. Without loss of generality, x10 = x20 = 0 so that 12 x1 − x2 0;[0,T ] ≤ 1 x − x2 ≤ x1 − x2 0,[0,T ] . First note that for i = 1, 2 we have ∞;[0;T ] i y ≤ υ. Now, write for t ∈ [0, T ] , 1-var;[0,T ] 1 yt − yt2

1 t 1 1 2 1 2 1 V yr − V yr dxr ≤ y 0 − y0 + 0 t V 1 yr2 d x1r − x2r + 0 t 2 2 V yr − V 1 yr2 dx2r + 0

t 1 1 2 yr − yr2 . dx1r (3.11) ≤ y 0 − y0 + υ 0 t V 1 yr2 d x1r − x2r + V 1 − V 2 ∞ . + 0

We deduce from the integration-by-parts formula t t 2 1 1 1 2 xr − x2r dV 1 yr2 + V 1 yt2 . x1t − x2t V yr d xr − xr = 0

0

the bound t 2 1 1 2 y d x V − x r r r 0

≤ x1 − x2 ∞;[0,t] V 1 y·2 1-var;[0,t] + V 1 yt2 . x1 − x2 ∞;[0,t] ≤ υ x1 − x2 ∞;[0,t] 1 + y 2 1-var;[0,t] ≤ υ x1 − x2 ∞;[0,t] (1 + υ) .

This last inequality and inequality (3.11) give, for t ∈ [0, T ] , 1 yt − yt2 ≤ y01 − y02 + υ (1 + υ) x1 − x2 ∞;[0,t] t 1 yr − yr2 . dx1r + V 1 − V 2 +υ ∞ 0

64

Ordinary diﬀerential equations

which implies, using Gronwall’s lemma, that 1 yt − yt2 ≤ y01 − y02 + υ x1 − x2 (1 + υ) ∞;[0,t] t 1 dxr + V 1 − V 2 ∞ exp |V |Lip 1 0 1 V − V 2 exp (2υ) . y01 − y02 + υ x1 − x2 + ≤ ∞;[0,t] ∞

From the above theorem we see in particular that the solution map (y0 , x) → π (V ) (0, y0 ; x) is uniformly continuous in the sense of “uniform convergence with uniform 1-variation bounds”. By a localization argument we now weaken the boundedness assumption inherent to Lip1 regularity: Corollary 3.16 We consider (i) V 1 = V11 , . . . , Vd1 and V 2 = V12 , . . . , Vd2 are two collections of locally Lip1 vector ﬁelds on Re , with linear growth; with y0i ≤ R for some R ≥ 0; (ii) y01 , y02 ∈ Re are initial conditions, (iii) x1 and x2 are two paths in C 1-var [0, T ] ,Rd , with maxi=1,2 xi 1-var;[0,T ] ≤ for some ≥ 0. Then, if y i = π (V i ) 0, y0i ; xi for i = 1, 2, there exist constants C, M depending only on R, and the vector ﬁelds, such that for all s, t ∈ [0, T ] 1 1 y − y 2 y01 − y02 + x1 − x2 V − V 2 . ≤ C + ∞;[0,T ] 0;[0,T ] ∞;B (0,M ) Proof. We saw in Corollary 3.9 that under locally Lipschitz and linear growth assumptions on the vector ﬁelds, there is indeed a unique, nonexploding solution. In fact, thanks to the explicit estimate (3.8) there exists M = M (, R) so that maxi=1,2 y i ∞;[0,T ] ≤ M . We now modify the vector ﬁelds V i outside a ball of radius R such as to make them Lip1 -vector ﬁelds, say V˜ i , and note that y i = π (V i ) 0, y0i ; xi = π (V˜ i ) 0, y0i ; xi . This allows us to use Theorem 3.15 to ﬁnish the proof. Exercise 3.17 (change-of-variable formula) Assume f is C 1 (Re ) and y = π (V ) (0, y0 ; x) the unique solution to dy = V (y) dx, y (0) = y0 ∈ Re along locally 1-Lipschitz V = (V1 , . . . , Vd ) on Re , with linear vector ﬁelds growth, and x ∈ C 1-var [0, T ] , Rd . Show that f (yT ) − f (y0 ) =

T

(V f ) (ys ) dxs 0

3.5 Continuity of the solution map

65

where V f = (V1 f, . . . , Vd f ) and each Vi is identiﬁed with a ﬁrst-order differential operator Vi f =

e

Vik ∂k f.

k =1

T ] , Rd , this is just the fundamental theorem of Solution. For x ∈ C 1 [0, calculus. For x ∈ C 1-var [0, T ] , Rd we approximate (uniformly, with uniform 1-variation bounds) and use the limit theorem. One can also appeal to the direct change-of-variable formulae for Riemann–Stieltjes integrals . . .

3.5.2 Continuity under 1-variation distance Given a collection V of Lip1 -vector ﬁelds we ﬁrst show that (y0 , x) ∈ Re × C 1-var [0, T ] , Rd → π (V ) (0, y0 ; x) ∈ C 1-var ([0, T ] , Re ) is Lipschitz continuous on bounded sets. Again, it will not complicate things to include V in the following continuity result.

Theorem 3.18 We consider (i) V 1 = V11 , . . . , Vd1 and V 2 = V12 , . . . , Vd2 are two collections of Lip1 vector ﬁelds on Re , such that, for some υ ≥ 0, max V i Lip 1 ≤ υ;

i=1,2

(ii) y01 , y02 ∈ Re are viewed as two time-0 initial conditions; (iii) x1 and x2 are two paths in C 1-var [0, T ] , Rd such that, for some ≥ 0, max xi 1-var;[0,T ] ≤ .

i=1,2

Then, if y i = π (V i ) 0, y0i ; xi for i = 1, 2, we have 1 2 y −y

1-var;[0,T ]

≤ 2( y01 − y02 υ+υ x1 −x2 1-var;[0,T ] + V 1 − V 2 ∞ )e3υ . (3.12)

66

Ordinary diﬀerential equations

Proof. Take s < t in [0, T ] and observe that t t 1 1 1 2 2 2 1 2 ys,t − ys,t = V yr dxr − V yr dxr s s t t 1 1 V yr − V 1 yr2 dx2r + V 1 yr1 d x1r − x2r ≤ s s t 1 2 V yr − V 2 yr2 dx2r + s ≤ υ y 1 − y 2 ∞;[0,T ] + V 1 − V 2 ∞ x2 1-var;[s,t] +υ x1 − x2 1-var;[s,t] . As the right-hand side is a control, it follows that 1 2 1 x y − y 2 y 1 − y 2 V − V 2 ≤ υ + 1-var;[s,t] ∞;[0,T ] ∞ 1-var;[s,t] 1 +υ x − x2 . 1-var;[s,t]

Using Theorem 3.15, and replacing s, t by 0, T, we then obain (3.12), as claimed. Remark 3.19 The interval [0, T ] in the above theorem is of course arbitrary. In particular, this means that we also have for all s, t ∈ [0, T ] , 1 y − y 2 ≤ 2( ys1 − ys2 υ+υ x1 − x2 1-var;[s,t] + V 1 − V 2 ∞ )e3υ , 1-var;[s,t] (3.13) where is a bound on max x2 1-var;[s,t] , x1 1-var;[s,t] . As before, we can relax the assumption on the vector ﬁelds and still keep a uniform Lipschitz bound on bounded sets. Corollary 3.20 We consider (i) V 1 = V11 , . . . , Vd1 and V 2 = V12 , . . . , Vd2 are two collections of locally Lip1 vector ﬁelds on Re , with linear growth; i (ii) y01 , y02 ∈ Re are initial conditions, 0; with y0d ≤ R for some R ≥ 1 2 1-var [0, T ] , R ,with maxi=1,2 xi 1-var;[0,T ] (iii) x and x are two paths in C ≤ for some ≥ 0. Then, if y i = π (V i ) 0, y0i ; xi for i = 1, 2, there exist constants C, M depending only on R, and the vector ﬁelds, such that 1 2 1 y −y y01 − y02 + x1 − x2 V − V 2 . ≤ C + 1-var;[0,T ] 0;[0,T ] ∞;B (0,M ) The 1-variation estimates imply 1-H¨ older estimates: Exercise 3.21 Under the same assumptions as the one in Corollary 3.20, and assuming x1 and x2 to be 1-H¨ older, prove the existence of constants

3.6 Comments

67

C, M depending on maxi y0i and maxi=1,2 xi 1-H¨o l;[0,T ] and the vector ﬁelds, such that 1 2 1 y −y y01 − y02 + x1 −x2 V −V 2 . ≤ C + 1-H¨o l;[0,T ] 1-H¨o l;[0,T ] ∞;B (0,M ) Solution. We will use |V |Lip 1 |x|1-var;[s,t] ≤ |V |Lip 1 |x|1-H¨o l;[s,t] |t − s|. We may take the vector ﬁelds to be 1-Lipschitz, as the result then follows by a localization argument. Deﬁne = maxi xi 1-H¨o l;[0,T ] . From (3.13) we obtain 1 2 ys,t − ys,t ≤ 2 ys1 −ys2 .υ + υ x1 −x2 1-H¨o l;[0,T ] + V 1 − V 2 ∞ e3υ T . t−s Replacing ys1 − ys2 on the right-hand side by y 1 − y 2 ∞ , followed by taking the supremum over all s < t in [0, T ], leads to an estimate of the form 1 y − y 2 ≤ 2 y 1 − y 2 ∞ υ + · · · exp (3υT ) . 1-H¨o l;[0,T ] We then conclude with Theorem 3.15.

3.6 Comments There are many books on ODE theory, such as the authorative Hartman [84]; for a concise treatment, see the relevant chapters of Driver [45]. The class of ODEs studied here, where the time-inhomogeneity factorizes in the form of a multidimensional driving signal, is particularly important in (non-linear) control theory, see the relevant contributions in Agrachev [1] for instance. Continuity in the starting point (the “ﬂow”) is well known, and its further regularity is discussed in Chapter 4. Continuity in the driving signal is harder to ﬁnd in the literature but also well known, see Lyons and coworkers [120, 123] and the references cited therein.

4 ODEs: smoothness We remain in the ODE setting of the previous chapter; that is, we consider diﬀerential equations of the form dy = V (y) dx, y (0) = y0 , where x = x (t) is a Rd -valued continuous path of bounded variation. In the present chapter we investigate various smoothness properties of the solution, in particular as a function of y0 and x.

4.1 Smoothness of the solution map We saw in the last chapter (cf. Theorem 3.18) that Lip1 -regularity of the vector ﬁelds leads to (local Lipschitz) continuity of the solution map π (V ) (0, y0 ; x) as a function of the initial condition y0 , the driving signal x and the vector ﬁelds V = (V1 , . . . , Vd ). Under the slightly stronger regularity assumption of C1 -boundedness we now show that π (V ) (0, y0 ; x) is diﬀerentiable in y0 and x. (For simplicity, we do not discuss diﬀerentiability in V .) In fact, we shall see that Ck -boundedness allows for k derivatives of π (V ) (0, y0 ; x) in y0 and x. As earlier, in the following deﬁnition V = (V1 , . . . , Vd ) is regarded as a map from Re to L Rd , Re , equipped with operator norm. if (i) it Deﬁnition 4.1 We say that V : Re → L Rd , Re is Ck -bounded is k-times Fr´echet diﬀerentiable and (ii) V, DV, . . . , Dk V is a bounded function on Re . We then set |V |C k := max Di V ∞ . i=0,...,k

If only (i) holds, we write V ∈ Cklo c .

4.1.1 Directional derivatives Lemma 4.2 Let V = (V1 , . . . , Vd ) be a collection of continuously diﬀeren tiable vector ﬁelds, that is V ∈ C1 Re , L Rd , Re . Then, for all ε > 0 and for all bounded sets U ⊂ Re , there exists δ such that for all b, a ∈ U , |b − a| < δ =⇒ |V (b) − V (a) − DV (a) · (b − a)| ≤ ε |b − a| .

4.1 Smoothness of the solution map

69

Proof. By the fundamental theorem of calculus and the chain rule, |V (b) − V (a) − DV (a) · (b − a)| 1 = [DV (a + t (b − a)) − DV (a)] dt . (b − a) 0 1 |DV (a + t (b − a)) − DV (a)| dt. ≤ |b − a| 0

We conclude using the fact that DV is uniformly continuous on bounded sets in Re . Condition 4.3 (non-explosion) We say that a collection of vector ﬁelds V = (V1 , . . . , Vd ) on Re satisﬁes the non-explosion condition if for all R > 0, there exists M > 0 such that if (y0 , x) ∈ Re × C 1-var [0, T ] , Rd with |x|1-var + |y0 | ≤ R, then π (V ) (0, y0 ; x) < M. ∞;[0,T ]

Following our usual convention, we agree that, in the case of nonuniqueness, π (V ) (0, y0 ; x) stands for any ODE solutions driven by x along vector ﬁelds V started at y0 . For example, a collection of continuous vector ﬁelds of linear growth satisﬁes the non-explosion condition. Theorem 4.4 (directional derivatives in starting point and driving signal) We ﬁx a collection of C1lo c -vector ﬁelds on Re , V = (V1 , . . . , Vd ), satisfying the non-explosion condition. Then, (i) the map1 (y0 , x) ∈ Re × C 1-var [0, T ] , Rd → y ≡ π (0, y0 ; x) ∈ C 1-var ([0, T ] , Re ) has directional derivatives2 d π (V ) (0, y0 + εv; x + εh) D(v ,h) π (V )(0, y0 ; x) := ∈ C 1-var ([0, T ] , Re ) dε ε=0 in all directions (v, h) ∈ Re × C 1-var [0, T ] , Rd ; (ii) deﬁne the bounded variations paths t → Mt := Mty 0 ,x :=

d i=1

t

DVi (yr ) dxir ∈ Me (R)

(4.1)

0

(where Me (R) denotes real (e × e)-matrices) and also t → Ht :=

Hty 0 ,x;h

:=

d i=1

1 Since 2 The

t

Vi (yr ) dhir ∈ Re

(4.2)

0

V remains ﬁxed we write π instead of π (V ) . derivate exists as a (strong) limit in the Banach space C 1 -va r ([0, T ] , Re ).

ODEs: smoothness

70

then z = D(v ,h) π (V ) (0, y0 ; x) is the (unique) solution of the linear ODE

dzt = dMty 0 ,x · zt + dHty 0 ,x;h , z0 = v.

(4.3)

Remark 4.5 Observe that (y, z) = π (V ) (0, y0 ; x) , D(v ,h) π (V ) (0, y0 ; x) solves the ODE driven by (x, h) given by formal diﬀerentiation, namely dy = V (y) dx, dz = (DV (y) dx) · z + V (y) dh or, in more detail, #

d dyt = i=1 Vi (yt ) dxit d d dzt = i=1 (DVi (yt ) · zt ) dxit + i=1 Vi (yt ) dhit

started at (y0 , z0 ) = (y0 , v). With V ∈ C1lo c , DV is continuous and so the vector ﬁelds of the ODE for (y, z) are continuous but in general not C1lo c . Nonetheless, it has a unique solution (thanks to the speciﬁc structure: ﬁrst solve for y, then M, H, then z) which satisﬁes the non-explosion condition. Indeed, this is a straightforward application of the estimates for ODE solutions and Riemann–Stieltjes integrals: estimate y in terms of (y0 , x· ), then M, H in terms of (x· , h· , y· ) and ﬁnally z in terms of (v, M· , H· ). Proof. We ﬁrst notice that by a localization argument, we can assume that V is compactly supported. With (y0 , x) , (v, h) ∈ X ≡ Re ×C 1-var [0, T ] , Rd ﬁxed write ytε = π (V ) (0, y0 + εv; x + εh)t , y ≡ y 0 and also z ε = (y ε − y) /ε for ε > 0. Deﬁne z ∈ C 1-var ([0, T ] , Re ) as the (unique) ODE solution to (4.3). Step 1: We ﬁrst establish that lim z ε = z in Y∞

(4.4)

ε→0

with Y∞ := C [0, T ] , Rd , a Banach space under the ∞-norm. From the respective ODEs for y, y ε and z, ztε − zt

=

& d t% 1 (Vi (ysε ) − Vi (ys )) − DVi (ys ) · zs dxis ε i=1 0 d t + (Vi (ysε ) − Vi (ys )) dhis i=1

=

d

0

∆i1 (0, t) + ∆i2 (0, t) + ∆i3 (0, t)

i=1

4.1 Smoothness of the solution map

71

with ∆i1 (s, t) ∆i2 (s, t) ∆i3 (s, t)

t

DVi (yr ) · (zrε − zr ) dxir ,

=

s

s

t

=

1 [Vi (yrε ) − Vi (yr ) − DVi (yr ) · (yrε − yr )] dxir , ε

t

(Vi (yrε ) − Vi (yr )) dhir .

= s

First observe that Theorems 3.4 and 3.18 apply (as V ∈ C1c ⊂ Lip1 ) and we have R := sup |ytε | < ∞, t∈[0,T ] ε∈[0,1]

|z ε |1-var;[0,T ] ≤ c1 |v| + |h|1-var;[0,T ] =: c2 . Fix η > 0. From R < ∞ and Lemma 4.2, we see that there exists δ > 0 such that |yrε − yr | < δ implies

1 |Vi (yrε ) − Vi (yr ) − DVi (yr ) · (yrε − yr )| ≤ η |zrε | . ε

Using |yrε − yr | ≤ εc2 , this means that there exists ε0 > 0 such that ε < ε0 implies sup r ∈[0,T ]

1 |Vi (yrε ) − Vi (yr ) − DVi (yr ) · (yrε − yr )| ≤ η |zrε | ≤ c2 η. ε

In particular, we obtain that d i ∆2 (s, t) ≤ c3 η |x|

1-var;[s,t]

.

i=1

Bounding ∆i3 (s, t) is even easier; indeed, d i ∆3 (s, t) ≤

d

i=1

i=1

≤

|Vi |Lip 1 sup |yrε − yr | . |h|1-var;[s,t] r ∈[s,t]

c4 ε |h|1-var;[s,t] .

Finally, as the vector ﬁelds are Lipschitz, we have t d i ∆1 (s, t) = c5 |zrε − zr | . |dxr | . i=1

s

(4.5)

ODEs: smoothness

72

Putting things together, we obtain that for ε < ε0 , t ε |zsε − zs | . |dxs | + c4 ε |h|1-var;[0,t] + c3 η |x|1-var;[0,t] . |zt − zt | ≤ c5 0

By Gronwall’s lemma, we obtain that sup |ztε − zt | ≤ c4 ε |h|1-var;[0,T ] + c3 η |x|1-var;[0,T ] exp c5 |x|1-var;[0,T ] , t∈[0,T ]

(4.6) so that limε→0 |z ε − z|∞;[0,T ] ≤ c6 (η + ε) and since η > 0 was arbitrary it follows that limε→0 |z ε − z|∞;[0,T ] = 0. Step 2: Deﬁne zˆε to be the solution of dˆ ztε = dMty 0 + ε,x+ εh · zˆtε + dHty 0 +ε,x+εh;h , zˆ0ε = v. As there was nothing special about ε = 0 in the ﬁrst step, we have actually just showed that ε ∈ [0, 1] → π (V ) (0, y0 + εv; x + εh) ∈ Y∞ := C ([0, T ] , Re ) is diﬀerentiable with derivative zˆε . Now ε → Mty 0 + ε,x+εh , Hty 0 + ε,x+εh;h → zˆε ∈ Y1 = C 1-var ([0, T ] , Re ) is continuous (from the continuity properties of the solution map and Riemann–Stieltjes integration respectively). Therefore, from Proposition B.1 in Appendix B, ε ∈ [0, 1] → π (V ) (0, y0 + εv; x + εh) ∈ Y1 is diﬀerentiable; that is, the limit when ε → 0 of ε−1 π (V ) (0, y0 + εv; x + εh) − π (V ) (0, y0 + v; x + h) exists in Y1 . The proof is now ﬁnished. Proposition 4.6 (higher-order directional derivatives) Let k ∈ {1, 2, . . . }. Assume V = (V1 , . . . , Vd ) is a collection of Cklo c -vector ﬁelds on Re satisfying the non-explosion condition. Then (y0 , x) → π (V ) (0, y0 ; x) has (up to) kth-order directional derivatives in the sense that, for all (vi , ×k hi )1≤i≤k ∈ Re × C 1-var [0, T ] , Rd ,    k k   k ∂ k   π (0, y ; x) := π 0, y + ε v , x+ ε h D(v 0 0 j j j j (V ) i ,h i ) 1 ≤i ≤k ∂ε1 . . . ∂εk  j =1 j =1

ε=0

4.1 Smoothness of the solution map

73

exists as a strong limit in the Banach space C 1-var ([0, T ] , Re ). Furthermore, the directional derivatives satisfy the control ODEs obtained by formal differentiation. Proof. This follows by simple induction: for j ≥ 1, a solution of an ODE driven by Cjlo c -vector ﬁelds satisfying the non-explosion condition admits a derivative in any arbitrary direction in its starting point and driving signal, and the derivative in such directions, together with the driving signal, satisﬁes an ODE driven along Cjlo−1 c vector ﬁelds that satisﬁes the non-explosion condition.

4.1.2 Fr´echet diﬀerentiability We now show that the solution map to dy = V (y) dx is continuously Fr´echet diﬀerentiable in the starting point and driving signal. Theorem 4.7 Let V = (V1 , . . . , Vd ) be a collection of C1lo c-vector ﬁelds on Re satisfying the non-explosion condition. Then the map (y0 , x) ∈ Re × C 1-var ([0, T ] , Re ) → y ≡ π (V ) (0, y0 ; x) ∈ C 1-var ([0, T ] , Re ) is C 1 in the Fr´echet sense. Proof. From Corollary B.5, we only need to show that the map (y0, x), (v, h) ×2 into C 1-var ([0, T ] , → D(v ,h) π (V ) (0, y0 ; x) from Re × C 1-var [0, T ] , Rd e R ) is uniformly continuous on bounded sets. This follows from the uniform continuity on bounded sets of the maps   v φ1 φ2 φ (y0 , x) , (v, h) →  M y 0 ,x  →3 D(v ,h) π (V ) (0, y0 ; x) ; (y0 , x) , (v, h) → π (V ) (0, y0 ; x) H y 0 ,x φ1 and φ3 because of Theorem 3.20, and φ2 because of Corollary 2.8. We now discuss C k -Fr´echet diﬀerentiability of the map (y0 , x) → π (V ) (0, y0 ; x) . Proposition 4.8 (higher-order Fr´echet) Let k ≥ 1, and V = (V1 , . . . , Vd ) a collection of Cklo c -vector ﬁelds on Re satisfying the non-explosion condition. Then the map (y0 , x) ∈ Re × C 1-var ([0, T ] , Re ) → y ≡ π (V ) (0, y0 ; x) ∈ C 1-var ([0, T ] , Re ) is C k in the Fr´echet sense. k Proof. The map (y0 , x) , (vi , hi )1≤i≤k → D(v π (V ) (0, y0 ; x) is unii ,h i ) 1 ≤i ≤k formly continuous on bounded sets because of the uniform continuity on bounded sets of the solution map and the integral. This is enough to conclude the proof using Corollary B.11 in Appendix B.

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74

It can be convenient in applications to view π (V ) (0, ·; x) as a ﬂow of C k diﬀeomorphisms, that is, an element in the space of all φ : [0, T ] × Re → Re : (t, y) → φt (y) such that ∀t ∈ [0, T ] : φt is a C k -diﬀeomorphism of Re , ∀α : |α| ≤ k : ∂α φt (y) , ∂α φ−1 t (y) are continuous in (t, y) . Corollary 4.9 Under the assumptions of Proposition 4.8, the map (t, y0 ) → π (V ) (0, y0 ; x) is a ﬂow of C k -diﬀeomorphisms. Proof. It is clear from Proposition 4.8 that y0 ∈ Re → π (V ) (0, y0 ; x)t is in C k (Re , Re ). Moreover, it follows from Proposition 3.13 that −1 −) x π (V ) (0, ·; x)t = π (V ) (0, ·; ← t ← − where x (·) = x (t − ·) ∈ C 1-var [0, t] , Rd ; we see that π (V ) (0, ·; x)t is a bijection whose inverse is also in C k (Re , Re ) and conclude that each π (V ) (0, ·; x)t is indeed a C k-diﬀeomorphism of Re. At last, each ∂α -derivative −1 of π (V ) (0, ·; x)t resp. π (V ) (0, ·; x)t can be represented as a (non-explosive) ODE solution which plainly implies joint continuity in t and y0 .

Exercise 4.10 Prove Proposition 4.8 with C 1-var replaced throughout by (i) C 1-H¨o l and (ii) W 1,2 . We ﬁnish this section with a representation formula for directional derivatives. principle) Consider (y0 ,x) ∈ Re ×C 1-var Proposition 4.11 (Duhamel’s d 1 [0, T ] , R , a collection of Clo c -vector ﬁelds on Re , V = (V1 , . . . , Vd ) satisfying the non-explosion condition and write y ≡ π (V ) (0, y0 ; x) ∈ C 1-var([0, T ] , Re ) for the unique ODE solution. Deﬁne Mt =

d i=1

t

DVi (yr ) dxir ∈ Me (R)

0

and J· as the Me (R)-valued (unique) solution to the linear ODE, dJt = dMt · Jt , J0 = I

(4.7)

(where · denotes matrix multiplication and I the identity matrix). More generally, given 0 ≤ s ≤ t ≤ T write Jt←s for the solution of this ODE started at I at time s. Then Jt←s is the Jacobian of π (V ) (s, ·; x)t : Re → y s ,x to indicate this. Moreover, the Re at ys and we may write Jt←s =: Jt←s following representation formula holds: D(v ,h) π (V ) (0, y0 ; x)t = D(v ,h) yt =

y 0 ,x Jt←0 ·v+

d i=1

0

t

y s ,x Jt←s ·Vi (ys ) dhis . (4.8)

4.1 Smoothness of the solution map

75

Proof. By Theorem 4.4, for 0 ≤ t ≤ T , the ﬂow map y0 → π (V ) (0, y0 ; x)t from Re → Re admits partial derivatives in all directions. These are easily seen to be continuous (much more will be shown soon) and so π (V ) (0, ·; x)t ∈ C 1 (Re , Re ). Its diﬀerential (the “Jacobian”) at some point y0 , viewed as an Re×e -matrix, is of the form J˜t = (z1 | . . . |ze ) where zi = zi (t) = D(bi ,0) π (V ) (0, y0 ; x)t and (bi ) denotes the canonical basis of Re . From Theorem 4.4, zi = zi (t) is the solution of a linear ODE of the form dzi (t) = dMt · zi (t) with zi (0) = bi . Equivalently, J˜t is the solution of (4.7) started at I at time 0 and by ODE uniqueness, J˜t = Jt . The matrix Jt remains invertible for all t ∈ [0, T ]. Indeed, its inverse is constructed explictly as the (unique) ODE solution to dKt = −Kt · dMt with K0 = I. To see this, we just observe that d (Kt Jt ) = −Kt dMt Jt + Kt dMt Jt = 0. Of course, there is nothing special about time 0 and the same reasoning shows that, for 0 ≤ s ≤ t ≤ T , the ﬂow map π (V ) (s, ·; x)t is in C 1 (Re , Re ) with Jacobian given by (the invertible matrix) Jt←s . The chain rule in conjunction with π (V ) (s, ys ; x)u = π (V ) (t, π (s, ys , x)t ; x)u , 0 ≤ s ≤ t ≤ u ≤ T implies3 Ju ←s = Ju ←t · Jt←s , 0 ≤ s ≤ t ≤ u ≤ T −1

and by deﬁning Js←t := (Jt←s ) , 0 ≤ s ≤ t, this remains valid for all s, t, u ∈ [0, T ]. The validity of (4.8) is nothing more than a variationof-constants ODE argument (also known as Duhamel’s principle) which represents the solution to the inhomogenous equation dzt = dMt · zt + dHt , z0 = v, which is precisely D(v ,h) π (V ) (0, y0 ; x), in terms of the solution of the homogenous equation, i.e. the ODE satisﬁed by the Jacobian. More precisely, it suﬃces to observe that J0←t zt − v =

t

d (J0←s zs ) = 0

t

J0←s dHs = 0

d i=1

t

J0←s Vi (ys ) dhis .

0

−1

x x x ) · J0←s = Jt←s the representation formula (4.8) now follows Using (J0←t from simple algebra.

Remark 4.12 The underlying geometry helps to “read” these equations. The ﬂow π (V ) (s, ·; x)t maps ys → yt (where y is the solution of the ODE 3 The notation J t ←s (rather than J s →t ) has the advantage of suggesting the right order of matrix multiplication.

ODEs: smoothness

76

driven by x) and its matrix-valued Jacobian should be viewed as a linear map between the respective tangent spaces, i.e. y s ,x ∈ L (Ty s Re , Ty t Re ) . Jt←s = Jt←s

From the very nature of vector ﬁelds, Vi (ys ) = Vi |y s ∈ Ty s Re and Jt←s · Vi (ys ) ∈ Ty t Re . In particular, we should think of (4.8) as equality between elements in Ty t Re rather than just Re .

4.2 Comments Although (or maybe because) the results are unsurprising we are unaware of good references to the smoothness topics discussed here. In a more general Young context, related smoothness properties have been discussed by Li and Lyons [109]. Diﬀerential equations driven by W 1,2 -paths (a special case of Exercise 4.10) was Bismut’s starting point in [15]; the resulting Hilbert structure of the input signal is convenient in discussing non-degeneracy properties of the solution map. Diﬀerential equations driven by W 1,2 -paths also arise naturally in support and large-deviation statements for stochastic diﬀerential equations, which we shall encounter in Part IV.

5 Variation and H¨ older spaces We return to the abstract setting of Section 1.1, where we introduced C ([0, T ] , E), the space continuous paths deﬁned on [0, T ] with values in a metric space (E, d), followed by a detailed discussion of continuous paths of ﬁnite 1-variation (“bounded variation”). The purpose of the present chapter is to carry out a similar discussion for p-variation and 1/p-H¨ older regularity, p ∈ [1, ∞). In the later applications to rough paths, E will be a Lie group whose dimension depends on [p], the integer part of p.

5.1 H¨older and p-variation paths on metric spaces 5.1.1 Deﬁnition and ﬁrst properties We start by deﬁning α-H¨older and p-variation distances. Deﬁnition 5.1 Let (E, d) be a metric space. A path x : [0, T ] → E is said to be (i) H¨ older continuous with exponent α ≥ 0, or simply α-H¨ older, if |x|α -H¨o l;[0,T ] :=

sup 0≤s< t≤T

d (xs , xt ) α < ∞; |t − s|

(5.1)

(ii) of ﬁnite p-variation for some p > 0 if |x|p-var;[0,T ] :=

sup (t i )∈D([0,T ])

p d xt i , xt i + 1

1/p < ∞.

(5.2)

i

We will use the notations C α -H¨o l ([0, T ], E) for the set of α-H¨older paths x and C p-var ([0, T ], E) for the set of continuous paths x : [0, T ] → E of ﬁnite p-variation. It is obvious from these deﬁnitions that a path x : [0, T ] → E is constant, i.e. xt ≡ o for some o ∈ E, if and only if |x|α -H¨o l;[0,T ] = 0 and if and only if |x|p-var;[0,T ] = 0. (In particular, if E = Rd our quantities (5.1), (5.2) are only semi-norms.) Observe that C 0-H¨o l ([0, T ], E) is nothing but the set of continuous paths from [0, T ] into E and |x|0-H¨o l;[0,T ] = |x|0;[0,T ] , where the latter was deﬁned in Section 1.1. Any α > 0 can be written as α = 1/p and it is obvious that any (1/p)-H¨ older path is a continuous path of ﬁnite p-variation. Although

78

Variation and H¨ older spaces

a path of ﬁnite p-variation need not be continuous (e.g. a step-function), our focus is on continuous paths. The following simple proposition then explains why our main interest lies in α ∈ [0, 1] and p ≥ 1. Proposition 5.2 Assume x : [0, T ] → E is α-H¨ older continuous, with α ∈ (1, ∞), or continuous of ﬁnite p-variation with p ∈ (0, 1). Then x is constant, i.e. x (·) ≡ x0 . Proof. Since α-H¨older paths have ﬁnite p-variation with p = 1/α it suﬃces to consider the case when x is continuous of ﬁnite p-variation with p < 1. Consider a dissection D = (ti ) ∈ D ([0, T ]) with mesh |D|. Then d (x0 , xT ) ≤ d xt i , xt i + 1 i

≤

1−p p M max d xt i , xt i + 1 i

where M = |x|p-var;[0,T ] < ∞. Using uniform continuity of x on [0, T ], we can make maxi d xt i , xt i + 1 arbitrarily small by taking a dissection with small enough mesh |D| = maxi |ti+1 − ti |. The case p = 1 resp. α = 1 was already discussed in detail in Section 1.2 and heavily used in our discussion of ODEs driven by continuous paths of bounded variation. We now begin a systematic study of p-variation, generalizing much of the familiar p = 1 case. Proposition 5.3 Let x ∈ C ([0, T ] , E) . Then, if 1 ≤ p ≤ p < ∞, |x|p -var;[0,T ] ≤ |x|p-var;[0,T ] . In particular, C p-var ([0, T ], E) ⊂ C p

-var

([0, T ] , E) .

Proof. This follows from the elementary inequality 1/p 1/p p p |ai | ≤ |ai | .

Exercise 5.4 Formulate and prove the H¨ older version of Proposition 5.3. Proposition 5.5 (interpolation) Let x ∈ C ([0, T ] , E). (i) For 1 ≤ p < p < ∞, we have p/p 1−p/p |x|0;[0,T ] . |x|p -var;[0,T ] ≤ |x|p-var;[0,T ] (ii) For 1 ≥ α > α ≥ 0, we have α /α 1−α /α |x|0;[0,T ] . |x|α -H¨o l;[0,T ] ≤ |x|α -H¨o l;[0,T ]

5.1 H¨ older and p-variation paths on metric spaces

Proof. (i) Observe p d xt i , xt i + 1

=

p p −p d xt i , xt i + 1 d xt i , xt i + 1

i

i

≤

79

p −p

|x|0

p d xt i , xt i + 1 , i

then pass to the respective suprema over all dissections (ti ), and raise to the power 1/p . (ii) Follows from α /α α /α d (xs , xt ) d (xs , xt ) d (xs , xt ) 1−α /α 1−α /α ≤ d (xs , xt ) ≤ |x|0 α α α |t − s| |t − s| |t − s| and passing to the respective suprema. Proposition 5.6 Let p ≥ 1 and x ∈ C ([0, T ] , E). (i) x ∈ C p-var ([0, T ], E) is equivalent to p limδ →0 sup d xt i , xt i + 1 < ∞. (t i )∈Dδ ([0,T ])

(ii) If 1 ≤ q < p < ∞ and x ∈ C q -var ([0, T ], E) then p limδ →0 sup d xt i , xt i + 1 = 0. (t i )∈Dδ ([0,T ])

(5.3)

i

(5.4)

i

Remark 5.7 Proposition 5.9 implies that one can replace p The forthcoming p d xt i , xt i + 1 by |x|p-var;[t i ,t i + 1 ] in both (5.3) and (5.4). Proof. (i) If x is of ﬁnite p-variation then, trivially, (5.3) holds. Conversely, let us write ϕ (x) = xp ; it follows from (5.3) that we can ﬁnd δ > 0 small enough and c < ∞ so that ϕ d xt i , xt i + 1 < c t i ∈D

for any dissection D = (ti ) of [0, T ] with |D| < δ. Then, for an arbitrary dissection D of [0, T ], the number of intervals of length atleast δ cannot be more than T δ −1 and each of these contributes at most ϕ |x|0;[0,T ] where, by continuity of x, |x|0;[0,T ] ≡

sup d (xs , xt ) < ∞. s,t∈[0,T ]

Hence, for any dissection D of [0, T ], ϕ d xt i , xt i + 1 < c + T δ −1 ϕ |x|0;[0,T ] < ∞ t i ∈D

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80

which implies that x ∈ C p-var ([0, T ], E). (ii) Introduce the modulus of continuity, osc (x, δ) = sup {d (xs , xt ) : s, t ∈ [0, T ] , |t − s| ≤ δ} . By uniform continuity of x : [0, T ] → E we have osc (x, δ) → 0 as δ 0. The estimate p q p−q osc (x, |D|) d xt i , xt i + 1 ≤ d xt i , xt i + 1 t i ∈D

t i ∈D

then implies sup (t i )∈Dδ ([0,T ])

p q p−q d xt i , xt i + 1 ≤ |x|q -var;[0,T ] osc (x, δ) i

which converges to 0 with δ 0 as required. As in the discussion of 1-variation regularity, the notion of control or control function is extremely useful. Let us recall that a control (on [0, T ]) is a continuous map ω of s, t ∈ [0, T ] , s ≤ t, into the non-negative reals, 0 on the diagonal, and super-additive, i.e for all s ≤ t ≤ u ∈ [0, T ], ω(s, t) + ω(t, u) ≤ ω(s, u). p

The perhaps most important example of a control is given by |x|p-var;[s,t] for x ∈ C p-var ([0, T ], E). This is the content of the following proposition: Proposition 5.8 Let (E, d) be a metric space, p ≥ 1 and x : [0, T ] → E be a continuous path of ﬁnite p-variation. Then p

ω x,p (s, t) := |x|p-var;[s,t] deﬁnes a control. Proof. We dealt with the case p = 1 in Proposition 1.12 and thus can focus on p > 1. Step 1: The same argument which gave super-additivity in the case p = 1 gives super-additivity of ω x,p in the present setting. The proof of continuity of ω x,p splits up into showing (i) “continuity from inside” ω x,p (s+, t−) ≡

lim

h 1 ,h 2 0

ω x,p (s + h1 , t − h2 ) = ω x,p (s, t) ,

which follows from the same argument as in the case p = 1, and (ii) “continuity from outside” ω x,p (s−, t+) ≡

lim

h 1 ,h 2 0

ω x,p (s − h1 , t + h2 ) = ω x,p (s, t) ,

5.1 H¨ older and p-variation paths on metric spaces

81

for all s < t. Remark that ω x,p (s, t+), ω x,p (s−, t), etc. are deﬁned in the obvious way and that all limits here exist by monotonicty of ω x,p . In fact, this reduces the proof of (ii) to showing ω x,p (s−, t+) ≤ ω x,p (s, t) and this requires a careful analysis which is not covered by our previous “p = 1” discussion.1 As a further reduction, it is enough to establish “onesided continuity from outside”, i.e. ω x,p (s, t) ≥ ω x,p (s, t+) and ω x,p (s, t) ≥ ω x,p (s−, t) .

(5.5)

We only discuss ω x,p (s, t) ≥ ω x,p (s, t+), the other inequality following from the same argument, and show how to deduce it from continuity of ω x,p at the diagonal, i.e. (5.6) ω x,p (t, t+) = 0. (The proof of (5.6) is left to step 2 below.) Fixing s < t and h, ε > 0 we consider D = (s = t0 < t1 < · · · < tn −1 < tn = t + h) such that n −1

p d xt i , xt i + 1 > ω x,p (s, t + h) − ε;

i=0

splitting D = D1 ∪D2 so that all points in [s, t] are contained in D1 (clearly, D1 is a dissection of [s, t]) yields p d xt i , xt i + 1 + ω x,p (t, t + h) > ω x,p (s, t + h) − ε t i ∈D 1

and after sending h to 0, using ω x,p (t, t+) = 0, p ω x,p (s, t) ≥ d xt i , xt i + 1 > ω x,p (s, t+) − ε i:t i ∈D 1

and upon sending ε to 0 we see that it is indeed enough to prove rightcontinuity of ω x,p at the diagonal. Step 2: To see (5.6) we seek a contradiction to lim ω x,p (t, t + h) =: δ > 0.

h0

Observe that the limit exists by monotonicity. Keeping t ﬁxed throughout, thanks to continuity of x, we can ﬁnd h1 such that for all h ∈ [0, h1 ] , p

d (xt , xt+ h ) < δ/8.

(5.7)

1 In the case p = 1 we used additivity of ω x , 1 to obtain continuity from outside. In general, when p > 1 a control ω x , p is not additive.

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82

Fix h0 ∈ [0, h1 ] and a dissection (t = τ 0 < τ 1 < · · · < τ k −1 < τ k = t + h0 ) of [t, t + h0 ] such that k −1

p d xτ i , xτ i + 1 > 7δ/8,

i=0

which is possible since ω x,p (t, t + h0 ) ≥ ω x,p (t, t+) = δ. Using (5.7), we have k −1 p d xτ i , xτ i + 1 > 7δ/8 − δ/8 = 3δ/4. i=1

Doing the same with τ 1 in place of t+h0 yields (t = σ 0 < σ 1 < · · · < σ l−1 < σ l = τ 1 ), a dissection of [t, τ 1 ], such that l−1 p d xσ j , xσ j + 1 > 3δ/4. j =1

Combining the previous two sums, over non-overlapping intervals of the form [σ j , σ j +1 ] , [τ i , τ i+1 ] ⊂ [t, t + h0 ], yields ω x,p (t, t + h0 ) ≥ 3δ/4 + 3δ/4 = 3δ/2, which implies ω (t, t+) ≥ 3δ/2, which contradicts limh0 ω x,p (t, t+h). That concludes the proof. Proposition 5.9 Let (E, d) be a metric space, p ≥ 1 and x : [0, T ] → E be a continuous path of ﬁnite p-variation and δ > 0. Then (i) p p sup d xt i , xt i + 1 ≤ |x|p-var;[s,t] ω x,δ ,p (s, t) := (t i )∈Dδ ([s,t])

deﬁnes a control. (ii) We have

i

p d xt i , xt i + 1 =

sup (t i )∈Dδ ([0,T ])

sup |t−s|< δ

(t i )∈Dδ ([0,T ])

i

as well as

sup

d(xs , xt ) 1/p

|t − s|

p

|x|p-var;[t i ,t i + 1 ]

i

= sup |x|1/p-H¨o l;[s,t] . |t−s|< δ

Proof. (i) The proof follows along the same lines as the proof of Proposition 5.8. (ii) In both cases, the ≤ part is obvious. Using the fact that ω x,δ ,p is a control, we obtain p p d xt i , xt i + 1 ≤ sup |x|p-var;[t i ,t i + 1 ] sup (t i )∈Dδ ([0,T ])

(t i )∈Dδ ([0,T ])

i

≤

sup (t i )∈Dδ ([0,T ])

≤

i

ω x,δ ,p ([0, T ])

i

ω x,δ ,p (ti , ti+1 )

5.1 H¨ older and p-variation paths on metric spaces

83

and, by the very deﬁnition of ω x,δ ,p ([0, T ]), equality must hold throughout. The 1/p-H¨older statement is also simple to prove and left to the reader. The following proposition is extremely important. We shall use part (i) below (without further notice) throughout the book; part (ii) says that a modulus of continuity on small intervals gives quantitative control over large intervals. Proposition 5.10 Let (E, d) be a metric space, ω a control on [0, T ] , p ≥ 1, C > 0, and x : [0, T ] → E a continuous path. (i) The pointwise estimate 1/p

d (xs , xt ) ≤ C ω (s, t)

for all s < t in [0, T ]

implies the p-variation estimate |x|p-var;[s,t] ≤ C ω (s, t)

1/p

for all s < t in [0, T ] .

(We say that x is of ﬁnite p-variation controlled by ω.) (ii) Under the weaker assumption 1/p

d (xs , xt ) ≤ C ω (s, t)

for all s < t in [0, T ] such that ω (s, t) ≤ 1

we have |x|p-var;[s,t] ≤ 2C

1/p

ω (s, t)

∨ ω (s, t) for all s < t in [0, T ] .

Proof. (Remark that only the super-additivity of ω is used in the proof.) p Ad (i). By assumption, d (xs , xt ) ≤ C p ω (s, t). Then for any dissection D = {ti } of [s, t], super-additivity implies p d xt i , xt i + 1 ≤ C p ω (ti , ti+1 ) ≤ C p ω (s, t) . i

i

Taking the supremum over all such dissections ﬁnishes the proof of the ﬁrst part. (ii) Deﬁning φp (x) = x∨xp we see (cf. Exercise 1.8) that (s, t) → φp (ω (s, t)) is a control. In view of part (i) we only need to prove 1/p

d (xs , xt ) ≤ 2Cφp (ω (s, t))

.

If s, t are such that ω (s, t) ≤ 1, there is nothing to prove, so we ﬁx s, t such that ω(s, t) > 1. Deﬁne t0 = s, and ti+1 = inf {u > ti , ω(ti , u) = 1} ∧ t. From super-additivity of ω it follows that tN = t for N ≥ ω(s, t). We

Variation and H¨ older spaces

84

conclude with d (xs , xt )

≤

d xt i , xt i + 1

0≤i< ω (s,t)

≤

Cω (ti , ti+1 )

1/p

0≤i< ω (s,t)

≤ C (1 + ω (s, t)) ≤ 2Cω (s, t) .

Exercise 5.11 Let x ∈ C p-var ([0, T ], E) , p ≥ 1, with associated control function p ω x,p (s, t) = |x|p-var;[s,t] . Show that, for any s < t < u in [0, T ], ω x,p (s, t) + ω x,p (t, u) ≤ ω x,p (s, u) ≤ 2p−1 [ω x,p (s, t) + ω x,p (t, u)] . Solution. The ﬁrst inequality is immediate. For the second, if s < s < t < t < u we have d (xs , xt ) ≤ d (xs , xt ) + d (xt , xt ) p

and since (a + b) ≤ 2p−1 (ap + bp ) for a, b ≥ 0 the conclusion follows.

The very same argument used in Lemma 1.15 shows lower semi-continuity of x → |x|p-var in the following sense. Lemma 5.12 Let (xn ) be a sequence of paths from [0, T ] → E of ﬁnite p-variation. Assume xn → x pointwise on [0, T ]. Then, for all s < t in [0, T ], |x|p-var;[s,t] ≤ lim inf |xn |p-var;[s,t] . n →∞

In particular, |x|1/p-H¨o l; [s,t] ≤ lim inf |xn |1/p-H¨o l;[s,t] . n →∞

In a similar spirit, the following lemma says that |x|p-var is a rightcontinuous function of p. Lemma 5.13 Let x : [0, T ] → E be a continuous path of ﬁnite p-variation. Then, for all s < t in [0, T ] , the map p ∈ [p, ∞) → |x|p -var;[s,t] is nonincreasing and |x|p -var;[s,t] = |x|p-var;[s,t] . (5.8) lim p p

5.1 H¨ older and p-variation paths on metric spaces

85

Proof. The non-increasing statement was proved in Proposition 5.3. In particular, it implies that ω (s, t)

1/p

:= lim |x|p -var;[s,t] p p

1/p

exists and satisﬁes ω (s, t) ≤ |x|p-var;[s,t] ; we only need to show the converse inequality. Clearly, d (xs , xt ) ≤ |x|p -var;[s,t] and sending p p we have 1/p (5.9) d (xs , xt ) ≤ ω (s, t) for all s < t in [0, T ]. Let us show that ω is super-additive. First observe that p p |x|p -var;[s,t] = lim |x|p -var;[s,t] . ω (s, t) = lim p →p

p p

p

Then, for s ≤ t ≤ u and using super-additivity of (s, t) → |x|p -var;[s,t] , ω (s, t) + ω (t, u)

= ≤

lim

p →p

p

p

|x|p -var;[s,t] + |x|p -var;[t,u ]

p

|x|p -var;[s,u ] = ω (s, u) . lim

p →p

But (5.9) and super-additivity of ω imply |x|p-var;[s,t] ≤ ω (s, t) 1/p

conclude that ω (s, t)

1/p

. We

= |x|p-var;[s,t] as required.

5.1.2 On some path-spaces contained in C p-var ([0, T ] , E) Observe that x is a path of ﬁnite p-variation controlled by (s, t) → |t − s| if and only if x is 1/p-H¨older. Hence, 1/p-H¨older paths are of ﬁnite pvariation. Conversely, we now show that every ﬁnite p-variation path is the time-change of a 1/p-H¨older path. Proposition 5.14 Let (E, d) be a metric space, and let x : [0, T ] → E be a continuous path. Then x is of ﬁnite p-variation if and only if there exists a continuous increasing function h from [0, T ] onto [0, 1] and a 1/p-H¨ older path g such that x = g ◦ h. Proof. Let x be of ﬁnite p-variation, non-zero. Then, h(t) =

ω x,p (0, t) ω x,p (0, T )

deﬁnes a continuous (from Proposition 5.8) increasing function from [0, T ] onto [0, 1] . Then, there exists a function g such that g ◦ h (t) = x (t) , as

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86

h (t1 ) = h (t2 ) =⇒ x (t1 ) = x (t2 ) . Now, sup u ,v ∈[0,1]

|g (u) − g (v)|

=

1/p

|u − v|

|g (h (u)) − g (h (v))|

sup

1/p

u ,v ∈[0,T ]

|h (u) − h (v)|

ω x,p (0, T )

1/p

1/p

≤

ω x,p (u, v)

1/p

|ω x,p (0, u) − ω x,p (0, v)|

.

From the sub-additivity of ω x,p , |ω x,p (0, u) − ω x,p (0, v)| ≥ |ω x,p (u, v)| , so that |g (u) − g (v)| 1/p ≤ ω x,p (0, T ) , sup 1/p u ,v |u − v| i.e. g is 1/p-H¨ older. Exercise 5.15 (absolute continuity of order p) We say that x : [0, T ] → E is “absolutely continuous of order p” if, for all ε > 0, there exists δ> 0, such that for all s1 < t1 ≤ s2 < t2 ≤ · · · < sn < tn in [0, T ] with p i |ti − si | < δ, we have p d (xs i , xt i ) < ε. (5.10) i

(i) Assume p ≥ 1. Show that in the deﬁnition of absolute continuity of order p one can replace (5.10) by p |x|p-var;[s i ,t i ] < ε. i

(ii) Assume p > 1, and show that x is absolutely continuous of order p if and only if p limδ →0 sup d xt i , xt i + 1 = 0. (5.11) D ∈Dδ ([0,T ])

i

Solution. (i) Consider is1 < t1 ≤ s2 < t2 ≤ · · · < sn < tn in [0, T ] with p |t − s | < δ. Let uj ∈ D ([si , ti ]) be a dissection of [si , ti ] and observe i i i that 1/p  uij +1 − uij p  uij +1 − uij = ti − si .  ≤ j

j

p p It follows that i j uij +1 − uij ≤ i |ti − si | < δ and so p d xu ij + 1 , xu ij <ε i

j

and we conclude by taking the supremum over all possible dissections of [si , ti ] , i = 1, . . . , n.

5.1 H¨ older and p-variation paths on metric spaces

87

(ii) “ ⇐ ” : Condition (5.11) implies that ∀ε > 0 : ∃˜δ :

sup D ∈Dδ˜ ([0,T ])

p d xu i , xu i + 1 < ε, i

for any dissection D with |D| ≤ ˜δ. Fix ε > 0 and take s1 < t1 ≤ s2 < t2 ≤ · · · < sn < tn in [0, T ] such that

p

p

|ti − si | < δ := ˜δ

i

which plainly implies maxi |ti − si | ≤ ˜δ. Take D = (ui ) to be a reﬁnement of {0 ≤ s1 < t1 ≤ · · · < sn < tn ≤ T } with mesh |D| ≤ ˜δ, without adding any (unnecessary) points in the intervals [si , ti ]. It then follows that

p

d (xs i , xt i ) ≤

p d xu i , xu i + 1 < ε,

i:u i ∈D

i

which shows that x is absolutely continuous of order p. “ =⇒ ” : Fix x an absolutely continuous path of order p, and ε > 0. We may write an arbitrary dissection D = (si ) of [0, T ] in the form D = {0 ≤ s1 < t1 = s2 < · · · < tn −1 = sn < tn ≤ T } p p−1 and furthermore assume |D| is small enough so that i |ti − si | ≤ T |D| < δ, where δ is chosen so that this implies, using the assumption of absolute continuity of order p of the path x, p d xs i , xs i + 1 < ε. i 1/(p−1)

This estimate is uniform over all dissections p < (δ/T ) D with |D| It follows that limδ →0 supD ∈Dδ ([0,T ]) i d xt i , xt i + 1 = 0.

= ˜δ.

Example 5.16 (Besov spaces) In Section 1.4 we introduced the (Sobolev) path spaces W 1,q ([0, T ] , E) which provided examples of ﬁnite 1−1/q . We now intro1-variation paths with precise H¨ older modulus |t − s| duce the fractional Sobolev – or Besov – spaces W δ ,q ([0, T ] , E) with δ < 1 whose elements are paths having ﬁnite p-variation with p = 1/δ > 1 and δ −1/q . More precisely, we make the following precise H¨older modulus |t − s| deﬁnition. Given q ∈ [1, ∞) and δ ∈ (1/q, 1), the space W δ ,q ([0, T ] , E) is the set of all x ∈ C ([0, T ] , E) for which |x|W δ , q ;[0,T ] :=

0

T

0

T

d (xu , xv ) δ +1/q

|v − u|

q

1/q dudv

< ∞.

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Following Section A.2 in Appendix A, the Garsia–Rodemich–Rumsey estimate leads quickly to a Besov–H¨older resp. variation “embedding”, by which we mean |x|δ −1/q -H¨o l;[0,T .] ≤ |x|(1/δ )-var;[0,T ]

≤

(const) |x|W δ , q ;[0,T ] , (const) |x|W δ , q ;[0,T ]

5.2 Approximations in geodesic spaces For a continuous path x : [0, T ] → Rd , and a dissection D = (ti )i of [0, T ] , we constructed the piecewise linear approximation xD by deﬁning d xD t i = xt i and connecting by straight lines in between. Straight lines in R are geodesics in the sense of the following deﬁnition. Deﬁnition 5.17 In a metric space (E, d) a geodesic (or geodesic path) joining two points a, b ∈ E is a continuous path Υa,b : [0, 1] → E such that Υa,b (0) = a, Υa,b (1) = b and a,b d Υa,b = |t − s| d (a, b) (5.12) s , Υt for all s < t in [0, 1]. If any two points in E are joined by a (not necessarily unique) geodesic, we call E a geodesic space. Equation (5.12) expresses that there are no shortcuts between any two points on the geodesic path. Even if E is complete and connected, it need not be a geodesic space; for example, the unit circle S 1 ⊂ R2 with metric induced from R2 is not geodesic. However, S 1 is a geodesic space under arclength distance. Readers with some background in Riemannian geometry will recall the Hopf–Rinow theorem;2 it says precisely that a complete connected Riemannian manifold is a geodesic space. The main example of a geodesic space to have in mind for our purposes is the free step-N nilpotent group equipped with Carnot–Caratheodory metric, to be discussed in detail later on. Geodesic spaces have exactly the structure that allows us to generalize the idea of piecewise linear approximations. To simplify, when considering a geodesic space E and two points a, b ∈ E, we will deﬁne Υa,b to be an arbitrary geodesic between a and b. Deﬁnition 5.18 (piecewise geodesic approximation) Let x be a continuous path from [0, T ] into some geodesic space (E, d) . Given a dissection D = {t0 = 0 < t1 < · · · < tn = T } of [0, T ] we deﬁne xD as the concatenation of geodesics connecting xt i and xt i + 1 for i = 1, . . . , n−1. More precisely, set xD t = xt for all t ∈ D 2 For

example, Bishop and Crittenden [14, p. 154].

5.2 Approximations in geodesic spaces

and for t ∈ (ti , ti+1 ),

xD t

x t i ,x t i + 1

=Υ

t − ti ti+1 − ti

89

.

Lemma 5.19 Let E be a geodesic space and x ∈ C([0, T ], E). Then, x converges to x uniformly on [0, T ]. That is, sup d xD t , xt → 0 as |D| → 0.

D

t∈[0,T ]

Proof. Fix two consecutive points ti < ti+1 in D and note that it is enough → 0 uniformly for t ∈ [ti , ti+1 ]. To see this, ﬁx ε > 0 , x to show that d xD t t and pick δ = δ (ε) so that osc (x; δ) ≡

sup

d (xs , xt ) < ε/2

s< t in [0,T ]: t−s< δ

(which is possible since x is continuous on the compact [0, T ] and hence uniformly continuous). Then, for t ∈ [ti , ti+1 ] and provided that |D| < δ, we have ≤ d xD d xD t , xt t , xt i + d (xt i , xt ) t − ti d xt , xt + d (xt i , xt ) = i i+ 1 ti+1 − ti ≤ 2osc (x; δ) < ε which already ﬁnishes the proof. Proposition 5.20 Let E be a geodesic space and x ∈ C p-var ([0, T ], E), p ≥ 1 and D = {0 = t0 < t1 < · · · < tn = T } a dissection of [0, T ] . Then, D x ≤ 31−1/p |x| . (5.13) p-var;[0,T ]

p-var;[0,T ]

If x is 1/p-H¨ older, D x ≤ 31−1/p |x|1/p-H¨o l;[0,T ] . 1/p-H¨o l;[0,T ]

(5.14)

Remark 5.21 D induces a dissection of any interval [ti , tj ] with endpoints ti , tj ∈ D. It follows that [0, T ] in (5.13) may be replaced by any interval [ti , tj ] with ti , tj ∈ D. Proof. p-variation estimate: To prove (5.13) we use the control ω (s, t) = p |x|p-var;[s,t] and then deﬁne ω D ﬁrst on the intervals of D by p t−s ω (ti , ti+1 ) for ti ≤ s ≤ t ≤ ti+1 , 1 ≤ i < #D ω D (s, t) = ti+1 − ti

Variation and H¨ older spaces

90

and then for arbitary s < t, say 1 ≤ i < j < #D and ti ≤ s ≤ ti+1 ≤ tj ≤ t ≤ tj +1 , by ω D (s, t) = ω D (s, ti+1 ) + ω(ti+1 , tj ) + ω D (tj , t).

(5.15)

Clearly, if ti ≤ s ≤ t ≤ ti+1 , D D s − ti t − ti x t i ,x t i + 1 x t i ,x t i + 1 = d Υ ,Υ d xs , xt ti+1 − ti ti+1 − ti t−s d xt , xt using (5.12) = i i+ 1 ti+1 − ti ≤

ω D (s, t)1/p .

On the other hand, if ti ≤ s ≤ ti+1 ≤ tj ≤ t ≤ tj +1 , D D D D ≤ d x d xD , x , x s t s t i + 1 + d xt i + 1 , xt j + d(xt j , xt ) ≤ ω D (s, ti+1 )1/p + ω(ti+1 , tj )1/p + ω D (tj , t)1/p ≤

31−1/p (ω D (s, ti+1 ) + ω(ti+1 , tj ) + ω D (tj , t))

=

31−1/p ω D (s, t)1/p

using (5.15) in the last line. Hence, for all s < t in [0, T ] , D p d xD ≤ 3p−1 ω D (s, t). s , xt

1/p

(5.16)

It now suﬃces to show that ω D is a control (only super-additivity is nontrivial) to obtain the desired conclusion, namely D x ≤ 31−1/p ω D (0, T )1/p p-var;[0,T ] =

31−1/p |x|p-var;[0,T ] .

To see super-additivity, ω D (s, t) + ω D (t, u) ≤ ω D (s, u) for s ≤ t ≤ u in [0, T ] we ﬁrst consider the case when s, t, u are contained in one interval, say ti ≤ s < t < u ≤ ti+1 . Then p p t−s u−t ω (ti , ti+1 ) + ω (ti , ti+1 ) ω D (s, t) + ω D (t, u) = ti+1 − ti ti+1 − ti p u−s ω (ti , ti+1 ) = ω D (s, u) . ≤ ti+1 − ti Consider the case that s, t are contained in one interval, say ti ≤ s < t ≤ ti+1 ≤ tk ≤ u ≤ tk +1 . Then ω D (s, t) + ω D (t, u) = ω D (s, t) + ω D (t, ti+1 ) + ω (ti+1 , tk ) + ω D (tk , u)

≤ω D (s,t i + 1 )

5.2 Approximations in geodesic spaces

91

(using the ﬁrst case!) and conclude the deﬁning equality ω D (s, u) = ω D (s, ti+1 )+ω (ti+1 , tk )+ω D (tk , u). The case that t, u are contained in one interval is similar. At last, if s, t, u are in three diﬀerent intervals, say ti ≤ s ≤ ti+1 ≤ tj ≤ t ≤ tj +1 ≤ tk ≤ u ≤ tk +1 , then ω D (s, t) + ω D (t, u) equals ω D (s, ti+1 ) + ω(ti+1 , tj ) + ω D (tj , t) + ω D (t, tj +1 ) + ω(tj +1 , tk ) + ω D (tk , u)

≤ω D (t j ,t j + 1 )=ω (t j ,t j + 1 )

≤ω (t i + 1 ,t k )

and we conclude again with the deﬁning equality for ω D (s, u). This covers all cases and we have established that ω D is a control. older then 1/p-H¨ older estimate: If x is actually 1/p-H¨ 1/p

ω (s, t)

= |x|p-var;[s,t] ≤ |x|1/p-H¨o l;[0,T ] |t − s|

1/p

and so for ti ≤ s ≤ t ≤ ti+1 p t−s p ω D (s, t) = ω (ti , ti+1 ) ≤ |x|1/p-H¨o l;[0,T ] |t − s| . ti+1 − ti For general s < t, say ti ≤ s ≤ ti+1 ≤ tj ≤ t ≤ tj +1 , we have ω D (s, t)

= ω D (s, ti+1 ) + ω(ti+1 , tj ) + ω D (tj , t) p ≤ |x|1/p-H¨o l;[0,T ] (|ti+1 − s| − |tj − ti+1 | − |t − tj |) p

= |x|1/p-H¨o l;[0,T ] |t − s| The claimed estimate (5.14) now follows immediately from 1/p 1/p D ≤ 31−1/p ω D (s, t) ≤ 31−1/p |x|1/p-H¨o l;[0,T ] |t − s| . d xD s , xt

Remark 5.22 The above proof actually shows that D p x

p-var;[0,T ]

≤ 3p−1

p

sup (t i )∈D|D | ([0,T ])

p

|x|p-var;[t i ,t i + 1 ] ,

(5.17)

i

and a slight extension shows that for |D| < δ one has the estimate p p xD sup ≤ 3p−1 sup |x|p-var;[t i ,t i + 1 ] . ] p-var;[t ,t (t j )∈Dδ ([0,T ])

j

i

j+1

(t i )∈Dδ ([0,T ])

i

(5.18)

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92

Combining Lemma 5.19 and Proposition 5.20 gives immediately the following important approximation result. Theorem 5.23 Let E be a geodesic space and x ∈ C p-var ([0, T ], E), p ≥ 1. Let (Dn ) be a sequence of dissection of [0, T ] such that its mesh |Dn | converges to 0. Then, xD n converges to x “uniformly with uniform p-variation bounds”. That is, n →0 , x sup d xD t t t∈[0,T ]

and

sup xD n p-var;[0,T ] ≤ 31−1/p |x|p-var;[0,T ] . n

If x is 1/p-H¨ older then sup xD n 1/p-H¨o l;[0,T ] ≤ 31−1/p |x|1/p-H¨o l;[0,T ] . n

Exercise 5.24 Let E be a geodesic space, p ∈ (1, ∞) and x ∈ W 1,p ([0, T ], E) as deﬁned in Section 1.4.2. Show that d xt i , xt i + 1 p D p x 1 , p = p−1 . ;[0,T ] W |ti+1 − ti | i:t i ∈D

5.3 H¨older and p-variation paths on Rd 5.3.1 H¨ older and p-variation Banach spaces We now turn to Rd (equipped with Euclidean distance) as our most familiar example of a metric (and geodesic) space. Theorem 5.25 (i) C p-var [0, T ] , Rd is Banach with normd x → |x (0)| + p-var [0, T ] , R started at 0 |x|p-var;[0,T ] . The closed subspace of paths in C is also Banach under x → |x|p-var;[0,T ] . (ii) C 1/p-H¨o l [0, T ] , Rd is Banach with norm x → |x (0)| + |x|1/p-H¨o l;[0,T ] . The closed subspace of paths in C 1/p-H¨o l [0, T ] , Rd started at 0 is also Banach under x → |x|1/p-H¨o l;[0,T ] . These Banach spaces are not separable. Proof. The case p = 1 was dealt with in Section 1.3. Leaving straightforward details to the reader, let us say that completeness in the case p > 1 is proved as in the case p = 1; non-separability follows from the following example.

5.3 H¨ older and p-variation paths on Rd

93

Example 5.26 We construct an uncountable family of functions so that the distance of any two f = f remains bounded below by a ﬁxed positive real. An uncountable subset of C ([0, 1] , R) is given by εk 2−k /p sin 2k πt , t ∈ [0, 1] , fε (t) = k ≥1

where ε is a ±1 sequence, that is, εk ∈ {−1, 1} for all k. We show (i) that fε ∈ C 1/p-H¨o l [0, 1] , Rd ⊂ C p-var [0, 1] , Rd and (ii) if ε = ε then 2 < |fε − fε |p-var;[0,1] ≤ |fε − fε |1/p-H¨o l;[0,1] .

Proof. Ad (i). For 0 ≤ s < t ≤ 1 we have |fε (t) − fε (s)| ≤ εk 2−k /p sin 2k πt − sin 2k πs 1≤k ≤|log ( 2 ) (t−s) | εk 2−k /p sin 2k πt − sin 2k πs + k > |log ( 2 ) (t−s) | is the logarithm with base 2. Using |ε|l ∞ ≤ 1, we obtain where log k (2) sin 2 πt − sin 2k πs ≤ 2k π |t − s| for the ﬁrst sum and |sin (· · · )| ≤ 1 for the second, and hence 2−k /p 2k + 2.2−k /p |fε (t) − fε (s)| ≤ π |t − s| 1≤k ≤|log ( 2 ) (t−s) | k > |log ( 2 ) (t−s) | 1/p

≤ c1 |t − s|

for some constant c1 = c1 (p), independent of s, t and ε. This proves (i). Ad (ii). Assume ε = ε and let j ≥ 1 be the ﬁrst index for which εj = ε j , i.e. ε1 = ε 1 , . . . , εj −1 = ε j −1 but εj = ε j . Consider then a dissection D of [0, 1] given by ti = i2−j −1 : i = 0, . . . , 2j +1 . From j sin 2 πti+1 − sin 2j πti p = 1 it follows readily that sin 2j π· p-var;[0,1] ≥ 2j /p . Moreover, |(fε −fε ) (ti+1 ) − (fε −fε ) (ti )| = εj −ε j 2−j /p sin 2j πti+1 − sin 2j πti = 2.2−j /p . This shows that |fε − fε |p-var;[0,1] ≥ 2.

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94

5.3.2 Compactness

Lemma 5.27 Consider (xn ) ⊂ C [0, T ] , Rd and assume xn → x ∈ C [0, T ] , Rd uniformly. (i) Assume supn |xn |p-var;[0,T ] < ∞. Then xn → x in p -variation for any p > p. older norm for (ii) Assume supn |xn |α -H¨o l;[0,T ] < ∞. Then xn → x in α -H¨ any α < α. Proof. By Lemma 5.12 we see that x is of ﬁnite p-variation. It then sufﬁces to apply the interpolation result (Proposition 5.5) to the diﬀerence x − xn . Proposition 5.28 (compactness) Consider (xn ) ⊂ C [0, T ] , Rd . (i) Assume (xn ) is equicontinuous, bounded and supn |xn |p-var;[0,T ] < ∞. Then xn converges (in p > p variation, along a subsequence) to some p-var d [0, T ] , R . x∈C (ii) Assume (xn ) is bounded and supn |xn |α -H¨o l;[0,T ] < ∞. Then xn converges (in α < α H¨ older topology, along a subsequence) to some x ∈ C α -H¨o l [0, T ] , Rd . Proof. An obvious consequence of Arzela–Ascoli and the previous lemma. The following corollary will be useful, for example, in the proof of the forthcoming Theorem 6.8. Corollary 5.29 (i) Assume (xn ) , x are in C p-var [0, T ] , Rd such that supn |xn |p-var;[0,T ] < ∞ and xn → x uniformly, then for p > p, sup (s,t)∈∆ T

n |x |p -var;[s,t] − |x|p -var;[s,t] → 0 as n → ∞

where ∆T = {(s, t) : 0 ≤ s ≤ t ≤ T }. Furthermore, |xn |p -var; [·,·] : n ∈ N is equicontinuous in the sense that for every ε > 0 there exists δ such that |t − s| < δ implies (5.19) sup |xn |p -var; [s,t] < ε. n

(ii) If (x ) , x are in C ([0, T ] , E) such that supn |xn |α -H¨o l;[0,T ] < ∞ n and x → x uniformly, then for all s < t in [0, T ], then for α < α, sup |xn |α -H¨o l;[s,t] − |x|α -H¨o l;[s,t] → 0 as n → ∞ n

α -H¨o l

(s,t)∈∆ T

and

|xn |α -H¨o l; [·,·] : n ∈ N

is equicontinuous, similar to part (i).

5.3 H¨ older and p-variation paths on Rd

95

Proof. (i) Proposition 5.5, applied to xn − x, actually shows that lim sup |xn − x|p -var;[s,t] = 0.

n →∞ s,t

Hence, |xn |p -var;[·,·] converges uniformly on ∆T and, by Arzela–Ascoli’s theorem, is equicontinuous. That is, for any ε > 0 there exists δ such that |(s, t) − (s , t )| < δ implies sup |xn |p -var;[s,t] − |xn |p -var;[s ,t ] < ε. n

In particular, this applies to (s, t) ∈ ∆T with |t − s| < δ and s := t := s, and using |xn |p -var;[s ,s ] = 0 we see that sup |xn |p -var;[s,t] < ε n

which concludes the proof of (i). The proof of (ii) follows similar lines.

5.3.3 Closure of smooth paths in variation norm

For p ≥ 1 we deﬁne C 0,p-var [0, T ] , Rd resp. C 0,1/p-H¨o l [0, T ] , Rd as the closure of smooth paths from [0, T ] → Rd in p-variation resp. 1/p-H¨older norm. In symbols, C 0,p-var [0, T ] , Rd C 0,1/p-H¨o l [0, T ] , Rd

p-var

: = C ∞ ([0, T ] , Rd )

,

1/p-H¨o l

: = C ∞ ([0, T ] , Rd )

.

p-var

[0, T ] , Rd resp. Obviously, these are closed, linear subspaces of C C 1/p-H¨o l [0, T ] , Rd and thus Banach spaces and so is the restriction to 0,1/p-H¨o l paths with x (0) = 0, denoted by Co0,p-var [0, T ] , Rd resp. Co ([0, T ] , Rd . The case p = 1 was already discussed earlier in Section 1.3 where, 0,1-var among otherthings, we identiﬁed C d as absolutely continuous paths 0,1-H¨o l d 1 [0, T ] , R as C [0, T ] , R . For p > 1 we have and C Lemma 5.30 Let p > 1. (i) Let Ω be a set in C 1-var [0, T ] , Rd such that C 0,1-var [0, T ] , Rd ⊂ 1-var

Ω

. Then,

p-var

Ω

= C 0,p-var [0, T ] , Rd .

1-H¨o l (ii) Let Ω be a set in C 1-H¨o l [0, T ] , Rd such that C 1 [0, T ] , Rd ⊂ Ω . Then, 1/p-H¨o l Ω = C 0,1/p-H¨o l [0, T ] , Rd .

96

Variation and H¨ older spaces p-var

Proof. (i) First, C 0,p-var ⊂ Ω

follows immediately from

C ∞ ⊂ C 0,1-var ⊂ Ω

1-var

p-var

⊂Ω

.

The converse inclusion follows readily from C 1-var ⊂ C ∞ Ω ⊂ C 1-var ⊂ C ∞

p-var

p-var

=⇒ Ω

⊂ C∞

p-var

p-var

; indeed,

.

p-var

, recall from Exercise 2.6 that any x ∈ C 1-var can To see C 1-var ⊂ C ∞ be approximated by xn ∈ C ∞ in uniform norm with uniform 1-variation bounds, i.e. |x − xn |∞;[0,T ] → 0, sup |xn |1-var;[0,T ] < ∞; n

then interpolation (Proposition 5.5 applied to x − xn ) gives xn → x in p-variation, which is what we had to prove. (ii) Similar and left to the reader. Theorem 5.31 (Wiener’s characterization) Let x ∈ C p-var ([0, T ], Rd ), with p > 1. The following statements are equivalent. (i.1) x ∈ C 0,p-var ([0, T ], Rd ). p (i.2a) limδ →0 supD =(t i ),|D |< δ i |x|p-var;[t i ,t i + 1 ] = 0. p (i.2b) limδ →0 supD =(t i ),|D |< δ i d xt i , xt i + 1 = 0. D (i.3) lim|D |→0 dp-var x , x = 0. Secondly, let x ∈ C 1/p-H¨o l [0, T ], Rd , with p > 1. The following statements are equivalent. (ii.1) x ∈ C 0,1/p-H¨o l [0, T ], Rd . (ii.2a) limδ →0 sup|t−s|< δ |x|1/p-H¨o l;[s,t] = 0. (ii.2b) limδ →0 sup|t−s|< δ d(xs , xt )/|t − s|1/p = 0. (ii.3) lim|D |→0 d1/p-H¨o l xD , x = 0. Remark 5.32 From purely metric considerations, we have seen in Exercise 5.15 that (i.2b) is equivalent to “absolute continuity of order p”. Remark also that the case p = 1 requires special care: by Corollary 1.34 (i.1) ⇔ (i.3) holds true. On the other hand, Proposition 1.14 tells us that in the case p = 1 condition (i.2) is tantamount to saying x is constant; in particular, conditions (i.1), (i.3) do not imply (i.2). Similar comments apply in the H¨older case. Proof. We only prove the p-variation statements, as the 1/p-H¨ older ones follow the same logic. From Lemma 5.30, the dp-var -closure of C 1-var [0, T ] , Rd is C 0,p-var ([0, T ], Rd ), which implies (i.3) ⇒ (i.1). The reverse proof of (i.1) ⇒ (i.3) follows the same lines as the proof in the case p = 1, i.e. the proof of Corollary 1.34.

5.3 H¨ older and p-variation paths on Rd

97

We already proved in Proposition 5.9 that (i.2a) ⇔ (i.2b) and now turn to (i.1) ⇒ (i.2b). p Let us ﬁx ε > 0, and a smooth path y such that dp-var (x, y) ≤ ε2−p . For a dissection D, we obtain from the triangle inequality p p d xt i , xt i + 1 ≤ 2p−1 d yt i , yt i + 1 + 2p−1 dp-var;[0,T ] (y, x)p . t i ∈D

t i ∈D

Since y is smooth, there exists δ > 0 (that depends on n) such that for all dissections |D| < δ implies that p d yt i , yt i + 1 < ε2−p . t i ∈D

Hence, we obtain that for all dissections D with |D| < δ, p d xt i , xt i + 1 ≤ ε. t i ∈D

We ﬁnish by proving (i.2a) ⇒ (i.3). First, if x and y are two paths, and δ is some ﬁxed positive real, observe that for all subdivisions D = (ti ) , p d xt i ,t i + 1 , yt i ,t i + 1 i

≤

i,|t i + 1 −t i |≤δ



≤

2p−1 

p d xt i ,t i + 1 , yt i ,t i + 1 +

i,|t i + 1 −t i |> δ



p

|x|p-var;[t i ,t i + 1 ] +

i,|t i + 1 −t i |≤δ

+

p d xt i ,t i + 1 , yt i ,t i + 1 |y|p-var;[t i ,t i + 1 ]  p

i,|t i + 1 −t i |≤δ

T d0 (x, y) . δ

Taking the supremum over all dissections, we obtain p p dp-var (x, y) ≤ 2p−1 sup |x|p-var;[t i ,t i + 1 ] (t i )∈Dδ ([0,T ])

p−1

+2

i

sup

(t i )∈Dδ ([0,T ])

p

|y|p-var;[t i ,t i + 1 ]

i

T + d0 (x, y) . δ Taking a bounded variation path x and its piecewise linear approximation xD for some dissection D with |D| < δ, we obtain, using inequality (5.18), p p dp-var x, xD ≤ cp sup |x|p-var;[t i ,t i + 1 ] (t i )∈Dδ ([0,T ])

T + d0 x, xD . δ

i

Variation and H¨ older spaces

98

p First ﬁx δ > 0 such that cp sup(t i )∈Dδ ([0,T ]) i |x|p-var;[t i ,t i + 1 ] < ε/2. Then as xD converges to x in uniform topology when |D| → 0, there exists δ 2 < δ such that for all dissections D with |D| < δ 2 , T d0 x, xD < ε/2. δ p Hence, for all dissections D with |D| < δ 2 , we have dp-var x, xD < ε and the proof is ﬁnished. Corollary 5.33 For p > 1, we have the following set inclusions: ⊂ C 0,p-var [0, T ] , Rd C q -var [0, T ] , Rd 1≤q < p

⊂

. q -var [0, T ] , Rd . C p-var [0, T ] , Rd ⊂ C q>p

Proof. Recalling basic inclusions between p- and q-variation spaces (Proposition 5.3), only the inclusion C q -var ([0, T ] , R) ⊂ C 0,p-var ([0, T ] , R) q
requires an argument. Thanks to Proposition 5.6, p x∈ C q -var ([0, T ] , R) =⇒ lim sup d xt i , xt i + 1 = 0 1≤q < p

δ →0 D =(t i ),|D |< δ

i

and we conclude using Theorem 5.31. Example 5.34 An example of a function in C 1/2-H¨o l ([0, 1], R) but not in C 0,1/2-H¨o l ([0, 1], R) is given by t → t1/2 , as follows immediately from Wiener’s characterization, Theorem 5.31. ∞ Exercise 5.35 (i) Deﬁne g (x) = i=1 c−i/p sin ci x . If c is a suﬃciently / C 0,p-var ([0, 1] , large positive integer, show that g ∈ C p-var ([0, 1] , R) but g ∈ R) . (ii) Deﬁne h (x) = x1/p cos2 (π/x) / log x for x > 0, h (0) = 0. Show that h ∈ C 0,p-var ([0, 1] , R) and h∈ / ∪q < p C q -var ([0, 1] , R) . 0,1/p-H¨o l 0,p-var d [0, T ], R ,C Proposition 5.36 Let p ≥ 1. The spaces C ([0, d T ] , R are separable Banach spaces (and hence Polish). Proof. From 1.35, there is a countable space Ω that is dense Proposition in C 0,1-var [0, T ] , Rd . We conclude using Lemma 5.30.

5.4 Generalized variation

99

5.4 Generalized variation 5.4.1 Deﬁnition and basic properties The concept of variation (and then p-variation) allows for an obvious generalization: Deﬁnition 5.37 Let (E, d) be a metric space, ϕ ∈ C ([0, ∞), [0, ∞)), 0 at 0, strictly increasing and onto. A path x : [0, T ] → E is said to be of ﬁnite ϕ-variation on the interval [s, t] if $ # / 0 d xt i , xt i + 1 ≤ 1 < ∞. ϕ |x|ϕ-var;[0,T ] := inf M > 0, sup M D ∈D([0,T ]) t i ∈D

We will use the notation C ϕ-var ([0, T ], E) for the set of continuous paths x : [0, T ] → E of ﬁnite ϕ-variation. The set of paths pinned at time zero to some ﬁxed element o ∈ E is denoted by Coϕ-var ([0, T ], E). For ϕ (x) = xp , the deﬁnition of |x|ϕ-var;[0,T ] coincides with |x|p-var;[0,T ] . If (E, d) is a normed space and ϕ is (globally) convex, |.|ϕ-var;[0,T ] is a seminorm. Several variation functions of interest are not convex (including the class ψ p,q to be introduced in the forthcoming Deﬁnition 5.45, which will be convenient for our later applications). A ﬁrst interest in ϕ-variation comes from the fact that (sharp) sample path properties for stochastic processes are often available in this form. For example, a classical result of Taylor (cf. the forthcoming Theorem 13.15) states that Brownian motion has a.s. ﬁnite ψ 2,1 -variation on any compact interval [0, T ] and this is optimal (cf. Theorem 13.69). A wide class of (enhanced) Gaussian processes have a.s. ﬁnite ψ p,p/2 -variation, while (enhanced) Markov processes (with uniformly elliptic generator in divergence form) have the “Brownian” ψ 2,1 -variation regularity. The other reason for our interest in ϕ-variation is that it is intimately related to the uniqueness of solutions to rough diﬀerential equations under minimal regularity assumptions; as will be discussed in Section 10.5. Lemma 5.38 Let x ∈ C ϕ-var ([0, T ], E). Then, for all M ≥ |x|ϕ-var;[0,T ] we have / 0 d xt i , xt i + 1 ≤ 1. (5.20) ϕ sup M D ∈D([0,T ]) t i ∈D

Proof. Only the case M = |x|ϕ-var;[0,T ] requires a proof. By deﬁnition, there exists a sequence Mn ↓ M such that (5.20) holds with M replaced by Mn . In particular, for a ﬁxed dissection D, / / 0 0 d xt i , xt i + 1 d xt i , xt i + 1 ≤1 ≤ 1, and hence ϕ ϕ Mn M t i ∈D

t i ∈D

Variation and H¨ older spaces

100

by continuity of ϕ. Taking the supremum over all D ∈ D ([0, T ]) ﬁnishes the proof. Just as in the common case of p-variation, controls are a very useful concept. Proposition 5.39 Let (E, d) be a metric space, ϕ ∈ C ([0, ∞), [0, ∞)), 0 at 0, strictly increasing and onto. Then the following are equivalent. (i) x ∈ C ϕ-var ([0, T ], E) with |x|ϕ-var;[0,T ] ≤ M for some M ≥ 0. (ii) There exists a control ω with ω (0, T ) ≤ 1 such that for all s < t in [0, T ] , d (xs , xt ) ≤ M ϕ−1 (ω (s, t)) . Proof. Deﬁne ω x,ϕ (s, t) :=

sup

D ∈D([s,t]) t ∈D i

/ 0 d xt i , xt i + 1 . ϕ M

Working as in the proof of Proposition 5.8, we see that ω x,ϕ is a control with ω x,ϕ (0, T ) ≤ 1. We then, by deﬁnition of ω x,ϕ , have that d (xs , xt ) ≤ M ϕ−1 (ω x,ϕ (s, t)) . Conversely, assume that for all s < t in [0, T ] , d (xs , xt ) ≤ M ϕ−1 (ω (s, t)) for some M > 0 and control ω with ω (0, T ) ≤ 1. Then, for a dissection D, we have / 0 d xt i , xt i + 1 ≤ ϕ ϕ ◦ ϕ−1 [ω (ti , ti+1 )] M t i ∈D

t i ∈D

≤ ω (0, T ) ≤ 1, and hence |x|ϕ-var;[0,T ] ≤ M. For simplicity, we will only look at ϕ-variation of paths for functions ϕ satisfying the following condition. Condition 5.40 [∆c ] Assume ϕ ∈ C ([0, ∞), [0, ∞)), 0 at 0, strictly increasing and onto. We say ϕ satisﬁes condition ∆c if for all c > 0, there exists ∆c ≥ 0 such that ∀x ∈ [0, ∞) : ϕ (cx) ≤ ∆c ϕ (x) and limc→0 ∆c = 0. The condition ∆c leads to the following convenient equivalences. Proposition 5.41 Let (E, d) be a metric space, and let x : [0, T ] → E be a continuous path. Assume the variation function ϕ satisﬁes condition (∆c ). Then the following conditions are equivalent. (i) The path x is of ﬁnite ϕ-variation. (ii) There exists M > 0 such that / 0 d xt i , xt i + 1 < ∞. ϕ sup M D ∈D([0,T ]) t i ∈D

5.4 Generalized variation

101

(iii) for all K > 0, sup

D ∈D([0,T ]) t ∈D i

/ 0 d xt i , xt i + 1 < ∞. ϕ K

Proof. Trivially, (i) =⇒ (ii) and (iii) =⇒ (ii). We show that (ii) implies (i) and (iii). For any K > 0, using condition ∆c , we have / / 0 0 d xt i , xt i + 1 d xt i , xt i + 1 ≤ ∆M/K sup <∞ ϕ ϕ sup K M D ∈D([0,T ]) D ∈D([0,T ]) t i ∈D

t i ∈D

which proves (ii) =⇒ (iii). Similarly, / 0 d xt i , xt i + 1 ≤ ∆M /K ϕ sup K D ∈D([0,T ]) t i ∈D

sup

D ∈D([0,T ]) t ∈D i

/ 0 d xt i , xt i + 1 ϕ M

≤ 1 for K large enough (as ∆M/K →K→∞ 0) and so (ii) =⇒ (i). The proof is ﬁnished. ˜ Recall C p-var ([0, T ] , E) ⊂ C p-var ([0, T ] , E), p˜ ≥ p. This generalizes to ˜ = O (ϕ) at 0+. Lemma 5.42 Assume ϕ, ϕ ˜ satisﬁes condition (∆c ) and ϕ Then C ϕ-var ([0, T ] , E) ⊂ C ϕ˜ -var ([0, T ] , E) . Proof. Let x ∈ C ϕ-var ([0, T ] , E) , then for all K > 0, and in particular K := |x|0;[0,T ] , we have sup

D ∈D([0,T ]) t ∈D i

/ 0 d xt i , xt i + 1 < ∞. ϕ K

For all i, d xt i , xt i + 1 /K ≤ 1, and by assumption, on [0, 1] , there exists a ﬁnite constant c such that ϕ ˜ (u) ≤ cϕ (u) for all u ∈ [0, 1] . Hence / / 0 0 d xt i , xt i + 1 d xt i , xt i + 1 ≤ c sup <∞ ϕ ˜ ϕ sup K K D ∈D([0,T ]) D ∈D([0,T ]) t i ∈D

t i ∈D

and so x ∈ C ϕ˜ -var ([0, T ] , E), using Proposition 5.41. We now make some quantitative relations between ϕ-variation and pvariation. As will be seen in Corollary 5.44 below, p-variation estimates often imply ϕ-variation estimates with no extra work. Theorem 5.43 Fix p ≥ 1, and assume that ϕ satisﬁes condition ∆c . p Assume also that ϕ−1 (·) is convex on [0, δ] for some δ ∈ (0, 1]. Let

Variation and H¨ older spaces

102

x ∈ C ϕ-var ([0, T ], E). Then, the control3 ω (s, t) :=

sup

D ∈D([s,t]) t ∈D i

/ 0 d xt i , xt i + 1 ϕ |x|ϕ-var;[0,T ]

satisiﬁes ω (0, T ) ≤ 1. Moreover, for C = C (ϕ, p) and all s < t in [0, T ] , |x|p-var;[s,t] ≤ C |x|ϕ-var;[0,T ] ϕ−1 (ω (s, t)) . Proof. It suﬃces to consider the case of non-constant x (·) so that |x|ϕ-var;[0,T ] > 0. By Lemma 5.38, ω (0, T ) ≤ 1. Then, as ϕ satisﬁes condition ∆c , there exists κ > 0 such that ∆κ ≤ δ, and we have 0 0 / / d (xs , xt ) d (xs , xt ) ≤ ∆κ ϕ ≤ δω (s, t) ϕ κ |x|ϕ-var;[0,T ] |x|ϕ-var;[0,T ] from which d (xs , xt ) ≤ κ−p |x|ϕ-var;[0,T ] ϕ−1 (δω (s, t)) . Note that p δω (s, t) ∈ [0, δ] for any s < t in [0, T ] and so, from convexity of ϕ−1 (·) on [0, δ], p (s, t) → ϕ−1 (δω (s, t)) p

p

p

is a control. It then follows from basic super-additivity properties of controls that p

p

|x|p-var;[s,t] ≤

|x|ϕ-var;[0,T ] κp

p

ϕ−1 (δω (s, t)) ≤ p

|x|ϕ-var;[0,T ] κp

ϕ−1 (ω (s, t))

p

where we used δ ≤ 1 in the ﬁnal step, and the proof is ﬁnished. ˜ ˜ d), We now consider a second path y with values in some metric space (E, whose p-variation is dominated by the p-variation of x. (This situation will be typical for solutions of (rough) diﬀerential equations.) Corollary 5.44 Fix p ≥ 1, and assume that ϕ satisﬁes condition ∆c . p Assume also that ϕ−1 (·) is convex on [0, δ] for some δ ∈ (0, 1]. Let ˜ be such that for all s < t in [0, T ] , x ∈ C([0, T ], E), y ∈ C([0, T ], E) |y|p-var;[s,t] ≤ K |x|p-var;[s,t] . Then, for some constant C = C (p, ϕ) and all s < t in [0, T ] , |y|ϕ-var;[s,t] ≤ CK |x|ϕ-var;[s,t] . Proof. From Theorem 5.43, we have d (ys , yt )

3 With

convention 0/0 = 0.

≤

|y|p-var;[s,t] ≤ K |x|p-var;[s,t]

≤

CK |x|ϕ-var;[0,T ] ϕ−1 (ω (s, t)) .

(5.21)

5.4 Generalized variation

103

Hence, if D = (ti ) is a dissection of [0, T ], we have / 0 d yt i , yt i + 1 ≤ ϕ ω (ti , ti+1 ) ≤ ω (0, T ) ≤ 1 CK |x|ϕ-var;[0,T ] t i ∈D

t i ∈D

which implies that |y|ϕ-var;[0,T ] ≤ CK |x|ϕ-var;[0,T ] . There is nothing special about the interval [0, T ] and by a simple reparametrization argument we see that |y|ϕ-var;[s,t] ≤ CK |x|ϕ-var;[s,t] .

5.4.2 Some explicit estimates for ψ p,q We now apply all these abstract considerations to the following class of variation function. Deﬁnition 5.45 For any (p, q) ∈ R+ × R set ψ p,q (0) = 0 and ψ p,q (t) :=

tp (ln ln 1/t) q p

for t ∈ (0, e−e ) t for t ≥ e−e

or, equivalently, ψ p,q (t) = tp / (ln∗ ln∗ 1/t) where ln∗ = max (1, ln) . q

Exercise 5.46 Show that, for any (p, q) ∈ R+ × R, the function ψ p,q (·) satisﬁes condition ∆c . Solution. A possible choice is ∆c = 4/ψ p,q (1/c). The details are left to the reader. Exercise 5.47 For (p1 , q1 ) , (p2 , q2 ) in R+ × R, we say that (p1 , q1 ) ≤ (p2 , q2 ) if p1 ≤ p2 or if p1 = p2 and q1 ≤ q2 . Show that for (p1 , q1 ) ≤ (p2 , q2 ) , C ψ p 1 , q 1 -var ([0, T ] , E) ⊂ C ψ p 2 , q 2 -var ([0, T ] , E) . Solution. In all cases, we have ψ p 2 ,q 2 (t) = lim sup tp 2 −p 1 lim sup t→0+ ψ p 1 ,q 1 (t) t→0+

1 ln ln 1/t

q 2 −q 1

and this limit is bounded as t → 0+ if (p1 , q1 ) ≤ (p2 , q2 ) .The result follows from Lemma 5.42. The following estimates on the inverse of ψ −1 will be useful to us later p,q on. Lemma 5.48 There exists C = C (p, q) such that for all t ∈ [0, ∞), 1 ψ (t) ≤ ψ −1 p,q (t) ≤ Cψ 1/p,−q /p (t) . C 1/p,−q /p

104

Variation and H¨ older spaces

Proof. For t large enough ψ p,q (t) = tp , ψ 1/p,−q /p = t1/p and there is nothing to show. For t small it suﬃces to observe that ψ 1/p,−q /p is the asymptotic inverse of ψ p,q at 0 + . Finally, the following proposition will allow us to use Theorem 5.43 with the functions ψ p,q . p Proposition 5.49 For any p > p > 0 and q ∈ R, the function ψ −1 p,q (·) is locally convex in a positive neighbourhood of 0. Proof. Obvious from an explicit computation of the second derivative of p ψ −1 near 0 + . p,q (·)

5.5 Higher-dimensional variation 5.5.1 Deﬁnition and basic properties We now discuss p-variation regularity of a function 2 f : [0, T ] → Rd , |·| s s → f . u u n

The generalization to [0, T ] with n > 2 follows the same arguments but 2 will not be relevant to us. Given a rectangle R = [s, t] × [u, v] ⊂ [0, T ] we write s, t s t s t f (R) := f := f +f −f −f . (5.22) u, v u v v u

2 If d = 1 and f is smooth, this is precisely R ∂∂a∂fb (a, b) dadb. Also, if f (s, t) = gs ⊗ ht , then s, t f = gs,t ⊗ hu ,v . u, v We will also use the following notations, consistent with our 1-dimensional increment notation. t t t f := f −f , u, v v u s, t t s f := f −f . v v v We will also frequently use the notation |R| for the area of R.

5.5 Higher-dimensional variation

105

2 Deﬁnition 5.50 Let f : [0, T ] → Rd , |·| and p ∈ [1, ∞). We say that f has ﬁnite p-variation if |f |p-var;[0,T ] 2 < ∞, where  |f |p-var;[s,t]×[u ,v ] =

sup (t i )∈D([s,t]) (t i )∈D([u ,v ])

1/p ti , ti+1 p  f  t j , t j +1 i,j

and write f ∈ C p-var [0, T ] , Rd .

In the 1-dimensional (1D) case, i.e. functions deﬁned on [0, T ], the notion of control is fundamental. In the 2-dimensional (2D) case, controls are deﬁned on ∆T × ∆T where we recall that ∆ := ∆T = {(s, t) : 0 ≤ s ≤ t ≤ T } . We think of elements in ∆T × ∆T as rectangles contained in the square 2 [0, T ] and write [s, t]×[u, v] rather than ((s, t) , (u, v)) for a generic element. Deﬁnition 5.51 Let ∆T = {(s, t) : 0 ≤ s ≤ t ≤ T }. A 2D control (more 2 precisely, 2D control function on [0, T ] ) is a continuous map ω : ∆T × ∆T → [0, ∞) which is super-additive in the sense that for all rectangles R1 , R2 , and R with R1 ∪ R2 ⊂ R and R1 ∩ R2 = ∅, ω (R1 ) + ω (R2 ) ≤ ω (R) and such that for all rectangles of zero area, ω (R) = 0. A 2D control ω is said to be H¨older-dominated if there exists a constant C such that for all s < t in [0, T ] , 2 ω [s, t] ≤ C |t − s| .

The proof of the following lemma is a straightforward adaption of the 1D case treated in Section 5.1 and left to the reader. 2 Lemma 5.52 Let f ∈ C [0, T ] , Rd . Then (i) If f is of ﬁnite p-variation for some p ≥ 1, 2

p

R rectangles in [0, T ] → |f |p-var;R is a 2D control. 2 (i) f is of ﬁnite p-variation on [0, T ] if and only if there exists a 2D control 2 ω such that for all rectangles R ⊂ [0, T ] , p

|f (R)| ≤ ω (R) and we say that “ω controls the p-variation of f ”.

Variation and H¨ older spaces

106

2

Remark 5.53 If f : [0, T ] → Rd is symmetric (i.e. f (s, u) = f (u, s) p for all s, u) and of ﬁnite p-variation then [s, t] × [u, v] → |f |p-var;[s,t]×[u ,v ] is symmetric. In fact, one can always work with symmetric controls, it suﬃces to replace a given ω with [s, t]×[u, v] → ω ([s, t] × [u, v])+ω ([u, v] × [s, t]). 2 Lemma 5.54 A function f ∈ C [0, T ] , Rd is of ﬁnite p-variation if and only if ti , ti+1 p < ∞. f sup tj , tj +1 (t j )∈D([0,T ]) i,j

Moreover, the p-variation of f is controlled by 3p−1 times p ti , ti+1 f . ω ([s, t] × [u, v]) := sup tj , tj +1 (t j )∈D([0,T ]) i,j [t i ,t i + 1 ]⊂[s,t] [t j ,t j + 1 ]⊂[u ,v ]

2 Proof. Assuming that ω [0, T ] is ﬁnite, it is easy to check that ω is a 2D control. Then, for any given [s, t] and [u, v] which do not intersect or such that [s, t] = [u, v] , p s, t f ≤ ω ([s, t] × [u, v]) . u, v Take now s ≤ u ≤ t ≤ v, then, s, t s, u u, t f = f +f u, v u, v u, v s, u u, t s, u = f +f +f . u, v u, t t, v Hence, p s, t f u, v

≤

2 3p−1 ω ([s, u] × [u, v]) + ω [u, t] + ω ([s, u] × [t, v])

≤

3p−1 ω ([s, t] × [u, v]) .

The other cases are dealt with similarly, and we ﬁnd at the end that for all s ≤ t, u ≤ v, p s, t f ≤ 3p−1 (ω [s, t] × [u, v]) . u, v This concludes the proof.

Example 5.55 Given two functions g, h ∈ C p-var [0, T ] , Rd we can deﬁne s (g ⊗ h) := g (s) ⊗ h (t) ∈ Rd ⊗ Rd t

5.5 Higher-dimensional variation

107

and g ⊗ h has ﬁnite 2D p-variation.4 More precisely, p s, t p p (g ⊗ h) ≤ |g|p-var;[s,t] |h|p-var;[u ,v ] =: ω ([s, t] × [u, v]) u, v and since ω is indeed a 2D control function (as a product of two 1D control functions!) we see that |g ⊗ h|p-var;[s,t]×[u ,v ] ≤ |g|p-var;[s,t] |h|p-var;[u ,v ] .

2 Exercise 5.56 Given f ∈ C p-var [0, T ] , Rd , for any ﬁxed [s, t]×[u, v] ∈ r 2 [0, T ] , prove that the (1-dimensional) p-variation of r ∈ [s, t] → f u, v is bounded by |f |p-var;[s,t]×[u ,v ] . Similarly, prove that the (1-dimensional) p s, t variation of r ∈ [u, v] → f is bounded by |f |p-var;[s,t]×[u ,v ] . r Remark 5.57 If ω is a 2D control function, then 2 (s, t) → ω [s, t] 2 2 2 is a 1D control function, i.e. ω [s, t] + ω [t, u] ≤ ω [s, u] , and 2 (s, t) → ω [s, t] is continuous and zero on the diagonal. 2 Remark 5.58 A function f ∈ C [0, T ] , Rd of ﬁnite p-variation can also be considered as a path t → f (t, ·) with values in the space C p-var [0, T ] , Rd with p-variation (semi-)norm. It is instructive to observe that t → f (t, ·) has ﬁnite p-variation if and only if f has ﬁnite 2D p-variation.

5.5.2 Approximations to 2D functions Piecewise linear-type approximations Recall from Section 5.3 that a continuous path of ﬁnite p-variation can be approximated by smooth and/or piecewise linear paths in the sense of “uniform convergence with uniform p-variation bounds”, but not, in general, in p-variation norm. The same is true in the 2D case and the approximations deﬁned below are the natural 2D analogue of piecewise linear approximations. 4X

⊗ X is equipped with a compatible tensor norm.

Variation and H¨ older spaces

108

˜ Deﬁnition 5.59 (D, linear-type approximation of a 2D func D piecewise 2 tion) Assume f ∈ C [0, T ] , E where E is a normed space. Let

˜

A function f D , D f

˜ = (˜ τ j ) ∈ D [0, T ] . D = (τ i ) , D 2 ∈ C [0, T ] , E with the property that

˜ D ,D

s t

=f

s t

˜ for all (s, t) ∈ D × D

is uniquely deﬁned by requiring that ˜

f (D , D ) (·, 0) ˜ f (D , D ) (0, ·)

= f D (·, 0) , ˜

= f D (0, ·) ,

τ j , τ˜j +1 ), and, for (s, t) × (u, v) ⊂ (τ i , τ i+1 ) × (˜ v−u t−s τ i , τ i+1 ˜) s, t D ,D ( f . × f = τ˜j , τ˜j +1 u, v τ i+1 − τ i τ˜j +1 − τ˜j

2 ˜ ∈ D [0, T ]. Then, Proposition 5.60 Let f ∈ C ρ-var [0, T ] , E and D, D ˜ we have for all s < t in D and u < v in D ρ D , D˜ ρ ≤ 9ρ−1 |f |ρ-var;[s,t]×[u ,v ] . (5.23) f ρ-var;[s,t]×[u ,v ]

˜ ˜ Moreover, f D , D → f uniformly as |D| , D → 0. ˜ ˜ Remark 5.61 It need not be true that f D , D → f as |D| , D → 0 in ρvariation. However, by interpolation this holds true when ρ is replaced by ρ > ρ. 2

Proof. Without loss of generality, [s, t] × [u, v] = [0, 1] . Given D = 2 ˜ = (˜ τ j ) ∈ D [0, 1] we now deﬁne ω D , D˜ on [0, 1] as follows: for (τ i ) , D τ j , τ˜j +1 ] we set small rectangles [s, t] × [u, v] ⊂ I × J ≡ [τ i , τ i+1 ] × [˜ ω D ([s, t] × [u, v]) :=

(t − s) (v − u) ρ |f |ρ-var;I ×J |I × J|

(with s, t ∈ I; u, v ∈ J);

then, for vertical “strips” of the form [s, t] × (J1 ∪ · · · ∪ Jn ) with s, t ∈ I ≡ [τ i , τ i+1 ] and Jl = [τ j + l−1 , τ j + l ], ω D ([s, t] × (J1 ∪ · · · ∪ Jn )) := for s, t ∈ I; u, v ∈ J.

(t − s) ρ |f |ρ-var;I ×(J 1 ∪···∪J n ) |I|

5.5 Higher-dimensional variation

109

We use a similar deﬁnition for horizontal strips; at last, for a (possibly) large rectangle with endpoints in D we set ρ

ω D ((I1 ∪ · · · ∪ Im ) × (J1 ∪ · · · ∪ Jn )) := |f |ρ-var;(I 1 ∪···∪I m )×(J 1 ∪···∪J n ) . 2

Now, an arbitrary rectangle A = [a, b] × [c, d] ⊂ [0, 1] decomposes uniquely into (at most) 9 rectangles A1 , . . . , A9 of the above type (4 small rectangles in the corners, 2 vertical and 2 horizontal 9 strips and 1 rectangle with endpoints in D) and we deﬁne ω D , D˜ (A) = i=1 ω D , D˜ (Ai ). We leave it to the reader to check that ω D , D˜ is indeed a 2D control function on ˜

2

D ,D that [0, 1] . On ρthe other hand, it is clear from the deﬁnition of f D , D˜ (Ai ) ≤ ω D , D˜ (Ai ) for i = 1, . . . , 9 and so f

9 ρ 9 ρ ρ ˜ D , D˜ D , D˜ D ,D ρ−1 (A) = f (Ai ) ≤ 9 (Ai ) = 9ρ−1 ω D , D˜ (A) . f f i=1

i=1

2 The proof of (5.23) is then ﬁnished with the remark that ω D , D˜ [0, 1] = ˜ ˜ ρ |R|ρ-var;[0,1] 2 . At last, uniform convergence of f D , D → f as |D| , D → 0 is 2

a simple consequence of (uniform) continuity of f on [0, 1] . Molliﬁer approximations We now turn to another class of well-known smooth approximations: molliﬁer approximations. Notation 5.62 (continuous extension of 2D functions) Whenever neces2 sary, we shall extend a continuous function f deﬁned on [0, T ] to a con2 tinuous function f = f (s, t) deﬁned on R by setting f (0, 0) for s, t f (T, T ) for s, t

< 0, f (0, T ) for s < 0, t > T, > T, f (T, 0) for s > T, t < 0

and, for s ∈ [0, T ] resp. t ∈ [0, T ], f (s, t) =

f (s, 0) if t < 0 f (s, T ) if t > T

resp. f (s, t) =

f (0, t) if s < 0 . f (T, t) if s > T

Note that, as a consequence of this deﬁnition, we have 2 f (R) = f R ∩ [0, T ] for all rectangles in R2 .

Variation and H¨ older spaces

110

Deﬁnition 5.63 (µ, µ ˜ molliﬁer approximation of a 2D function) Assume 2 ˜ be two compactly f ∈ C [0, T ] , E where E is a normed space. Let µ, µ 2 supported probability measures on R. We deﬁne f µ, µ˜ ∈ C [0, T ] , E by f

µ, µ ˜

s u

=

f

s−a u−b

dµ (a) d˜ µ (b) ,

noting that the same relation remains valid for rectangular increments, f

µ, µ ˜

s, t u, v

=

f

s − a, t − a u − b, v − b

dµ (a) d˜ µ (b) .

Proposition 5.64 Let µ, µ ˜ be two compactly supported probability mea 2 sures on R and f ∈ C ρ-var [0, T ] , E , extended to a continuous function on R2 (cf. notation above) with ρ-variation controlled by ω

s, t u, v

ρ

= |f |ρ-var;[s,t]×[u ,v ] .

Then f µ, µ˜ is also of ﬁnite ρ-variation, controlled by the 2D control ω µ, µ˜ and

s, t u, v

=

µ, µ˜ ρ f

ω

ρ-var;[0,T ]

2

s − a, t − a u − b, v − b

dµ (a) d˜ µ (b) ,

2 ρ ≤ ω µ, µ˜ [0, T ] ≤ |f |ρ-var;[0,T ] 2 .

(5.24)

2

˜ n converge weakly Moreover, f µ n , µ˜ n → f uniformly on [0, T ] whenever µn , µ to the Dirac measure at zero.5 Remark 5.65 There is nothing special about the interval [0, T ]. However, we cannot deduce from (5.24) that µ, µ˜ ρ f

ρ-var;[s,t] 2

ρ

= |f |ρ-var;[s,t] 2 for all s < t in [0, T ] . 2

The reason is that f µ, µ˜ depends on our extension of f from [0, T ] to R2 . Thus, if we construct fˆµ, µ˜ from fˆ = f |[s,t] 2 , extended to R2 , it will not, in general, coincide with f µ, µ˜ . 5 That

is,

ϕdµ n → ϕ (0) for all continuous, bounded ϕ.

5.6 Comments

111

Proof. Given (si ) ∈ D [0, T ],(tj ) ∈ D [0, T ] we have, using Jensen’s inequality, ρ ρ µ, µ˜ si , si+1 si − a, si+1 − a f dµ (a) d˜ µ (b) = f tj , tj +1 tj − b, tj +1 − b ρ si − a, si+1 − a f ≤ dµ (a) d˜ µ (b) tj − b, tj +1 − b ρ si − a, si+1 − a dµ (a) d˜ f µ (b) = tj − b, tj +1 − b ≤

ω µ, µ˜ ([si , si+1 ] × [tj , tj +1 ]) ,

µ, µ ˜ which shows that f has ﬁnite ρ-variation controlled by ω . Moreover, 2 since f (R) = f R ∩ [0, T ] for all rectangles, ρ

ρ

|f |ρ-var;[0−a,T −a]×[0−b,T −b] ≤ |f |ρ-var;[0,T ] 2 2 ρ and so ω µ, µ˜ [0, T ] ≤ |f |ρ-var;[0,T ] 2 , which concludes the proof.

5.6 Comments Proposition 5.8 appears in Lyons and Qian [120]; our (complete) proof partially follows Dudley and Norvaiˇsa [47], p. 93. Continuity properties of the type discussed in Lemma 5.13 appear in Musielak and Semadeni [132]. The notion of paths which are “absolutely continuous of order p”, cf. Exercise 5.15, is due to Love [114]. Fractional Sobolev spaces (discussed in Example 5.16) are also known as Besov or Slobodetzki spaces and arise in many areas of analysis. The notion of “geodesic space” and its variations (length space, etc.) is now well understood, for example Gromov [72] or Burago et al. [21] and the references cited therein. Exercise 5.35 is taken from Dudley and Norvaiˇsa [48], p. 28. An almost complete list of references for generalized variation, or ϕ-variation, is found in Dudley and Norvaiˇsa [47]. Comments on higher-dimensional p-variation will be given in Chapter 6.

6 Young integration We construct

·

ydx, the Young integral of y against x where x∈C [0, T ] , Rd , y ∈ C q -var [0, T ] , L Rd , Re 0

p-var

with 1/p+1/q > 1. Although the results here are well known, our approach is novel and extends – without much conceptual eﬀort! – to rough path estimates for ordinary – and then rough – diﬀerential equations.

6.1 Young–L´oeve estimates We start with two elementary analysis lemmas, tailor-made for obtaining the Young–L´ oeve estimate in Proposition 6.4 below. Lemma 6.1 Let ξ > 0 and θ > 1. Consider : [0, T ] → R with (r) ≤ 2 (r/2) + ξrθ for all r ∈ [0, T ]

(6.1)

and such that (r) = o (r) as r → 0+, i.e. lim

r →0+

(r) = 0. r

(6.2)

Then, for all r ∈ [0, T ] , ξ (r) ≤ rθ . 1 − 21−θ Proof. We deﬁne φ (r) = (r) / ξrθ and note that (6.1) implies φ (r) ≤ 1 + 21−θ φ (r/2) . Iterated use of this inequality shows that for all n ∈ N, φ (r) ≤ 1 +

n −1 k =1

21−θ

k

n r + 21−θ φ n . 2

(6.3)

We now send n → ∞. The last term on the right-hand side above tends to zero since 1−θ n (r2−n ) 1−θ n r = 2 φ n 2 2 ξrθ 2−θ n 1−θ r (r2−n ) ≤ ξ r2−n → 0 by assumption (6.2)

6.1 Young–L´ oeve estimates

113

and the proof is ﬁnished with φ (r) ≤

∞

2−θ k 2k =

k =0

1 . 1 − 21−θ

Lemma 6.2 Let Γ : ∆ ≡ {0 ≤ s < t ≤ T } → Re and assume (i) there exists a control ω ˆ such that sup

lim

r →0 (s,t)∈∆ : ω ˆ (s,t)≤r

|Γs,t | = 0; r

(6.4)

(ii) there exists a control ω and θ > 1, ξ > 0 such that θ

|Γs,u | ≤ |Γs,t | + |Γt,u | + ξω (s, u)

(6.5)

holds for 0 ≤ s ≤ t ≤ u ≤ T . Then, for all 0 ≤ s < t ≤ T , |Γs,t | ≤

ξ θ ω (s, t) . 1 − 21−θ

Remark 6.3 It is important to notice that the control ω ˆ does not appear in the conclusion. Proof. At the cost of replacing Γ by Γ/ξ, we can and will take ξ = 1. We assume that ω ˆ ≤ 1ε ω for some ε > 0. If this is not the case, we replace ω by ω + εˆ ω and let ε tend to 0 at the end. Deﬁne for all r ∈ [0, ω (0, T )] , (r) =

sup

|Γs,t | .

s,t such that ω (s,t)≤r

Consider any ﬁxed pair (s, u) with 0 ≤ s < u ≤ T such that ω (s, u) ≤ r, and pick t such that ω (s, t) , ω (t, u) ≤ ω (s, u) /2. (This is possible thanks to basic properties of a control function, see Exercise 1.10). By deﬁnition of , |Γs,t | ≤ (r/2) ,

|Γs,u | ≤ (r/2)

and it follows from the assumption (6.5) that r + rθ . |Γs,u | ≤ 2 2 Taking the supremum over all s < u for which ω (s, u) ≤ r yields that for r ∈ [0, ω (0, T )], r + rθ . (r) ≤ 2 2

Young integration

114

On the other hand, assumption (6.4) implies that lim

r →0

(r) = 0. r

It then suﬃces to apply Lemma 6.1 to see that for all r ∈ [0, ω (0, T )] , (r) ≤

1 rθ 1 − 21−θ

and this readily translates to the statement that, for all 0 ≤ s < t ≤ T, |Γs,t | ≤

1 θ ω (s, t) . 1 − 21−θ

Proposition 6.4 (Young–L´ oeve estimate) Assume x ∈ C 1-var [0, T ] , Rd , y ∈ C 1-var [0, T ] , L Rd , Re for p, q ≥ 1 with θ := 1/p + 1/q > 1.With the deﬁnition t t Γs,t := yu dxu − ys xs,t = ys,u dxu s

we have |Γs,t | ≤

s

1 |x|p-var;[s,t] |y|q -var;[s,t] . 1 − 21−θ

(6.6)

Remark 6.5 It is instructive to think of ys xs,t as a ﬁrst-order Euler ap t proximation to the Riemann–Stieltjes integral s ydx so that (6.6) is nothing but a “ﬁrst-order Euler error” estimate. The point of this estimate is its uniformity: although 1-variation was assumed to have a well-deﬁned Riemann–Stieltjes integral, the ﬁnal estimate only depends on the respec tive p, q-variation. Proof. From Exercise 1.9, 1/θ

1/θ

ω (s, t) := |x|p-var;[s,t] |y|q -var;[s,t] is a control. For all s < t in [0, T ] we deﬁne t t yu dxu − ys xs,t = ys,u dxu . Γs,t = s

s

Then, for ﬁxed s < t < u in [0, T ], we have u t ys,r dxr − ys,r dxr − Γs,u − Γs,t − Γt,u = =

s ys,t xt,u

s

t

u

yt,r dxr

6.2 Young integrals

115

and hence |Γs,u | ≤ |Γs,t | + |Γt,u | + |x|p-var;[s,t] |y|q -var;[t,u ] =

θ

|Γs,t | + |Γt,u | + ω (s, u) .

Deﬁning ω ˜ (s, t) = |x|1-var;[s,t] + |y|1-var;[s,t] , elementary Riemann–Stieltjes integral estimates show that 2

˜ (s, t) . |Γs,t | ≤ |ys,· |∞;[s,t] |x|1-var;[s,t] ≤ ω It only remains to apply Lemma 6.2 and the proof is ﬁnished. In the following section we shall use the Young–L´ oeve estimate to deﬁne the Young integral for x ∈ C p-var and y ∈ C q -var . Remark 6.6 We could have assumed y ∈ C q -var right away in Proposition 6.4. Indeed, as long as x ∈ C 1-var , Γs,t remains a well-deﬁned Riemann– Stieltjes integral and the only change in the argument is to use q

1+ 1/q

ω ˜ (s, t) := |x|1-var;[s,t] + |y|q -var;[s,t] =⇒ |Γs,t | ≤ ω ˜ (s, t)

in the ﬁnal lines of the proof.

6.2 Young integrals The Young–L´ oeve estimate clearly implies that · (x, y) → ydx 0

is bilinear (as a function of smooth Rd resp. L Rd , Re -valued paths x, y) and continuous in the respective p- and q-variation norm. The (unique, continuous) extension of this map to x ∈ C 0,p-var [0, T ] , Rd , y ∈ C 0,q -var [0, T ] , L Rd , Re is immediate from general principles and by squeezing p, q to p + ε, q + ε so that 1/ (p + ε) + 1/ (q + ε) > 1 one covers genuine p-variation and qvariation regularity, i.e. x ∈ C p-var and y ∈ C q -var . That said, we shall proceed in a slightly diﬀerent way which will motivate our later deﬁnition of p-var ([0, T ] , rough diﬀerential equations. To this end, recall that any x ∈ C d R can be approximated “uniformly with uniform variation bounds” by bounded variation paths xn , i.e. d∞;[0,T ] (xn , x) → 0 and sup |xn |p-var;[0,T ] < ∞. n

(For instance, piecewise geodesic=linear approximations will do.)

Young integration

116

Deﬁnition 6.7 (Young integral) Given x ∈ C p-var [0, T ] , Rd and y ∈ C q -var [0, T ] , L Rd , Re we say that z ∈ C ([0, T ] , Re ) is an (indeﬁnite) Young integral of y against a sequence (xn , y n ) ⊂ x if there exists C 1-var [0, T ] , Rd × C 1-var [0, T ] , L Rd , Re which converges uniformly with uniform variation bounds in the sense

and

|xn − x|∞;[0,T ]

→

0 and sup |xn |p-var;[0,T ] < ∞,

|y n − y|∞;[0,T ]

→

0 and sup |y n |q -var;[0,T ] < ∞,

·

n n

y n dxn → z uniformly on [0, T ] as n → ∞.

0

If z is unique we write

s − 0 ydx.

· 0

ydx instead of z and set

t s

ydx :=

p-var

t 0

ydx

d [0, T ] , R and Theorem 6.8 (Young–L´ o eve) Given x ∈ C with θ = 1/p + 1/q > 1, there exists a y ∈ C q -var [0, T ] , L Rd , Re

· unique (indeﬁnite) Young integral of y against x, denoted by 0 ydx and the Young–L´ oeve estimate t 1 ydx − ys xs,t ≤ |x|p-var;[s,t] |y|q -var;[s,t] ∀0 ≤ s ≤ t ≤ T : 1 − 21−θ s remains valid. Moreover, the indeﬁnite Young integral has ﬁnite p-variation and · ydx ≤ C |x|p-var;[s,t] |y|q -var;[s,t] + |y|∞;[s,t] 0 p-var;[s,t] (6.7) ≤ 2C |x|p-var;[s,t] |y|q -var;[0,T ] + |y0 | where C = C (p, q).

· Proof. Let us ﬁrst argue that any limit point z of 0 y n dxn (in uniform topology on [0, T ]) satisﬁes the Young–L´ oeve estimate. For every ε > 0 small enough so that θε := 1/ (p + ε) + 1/ (q + ε) > 1 the Young–L´ oeve estimate of Proposition 6.4 gives t 1 n n n n yr dxr − ys xs,t ≤ |xn |(p+ε)-var;[s,t] |y n |(q + ε)-var;[s,t] . (6.8) 1 − 21−θ ε s By Corollary 5.29, the right-hand side above can be made arbitrarily small, uniformly in n, provided t−s is small enough; this readily leads to equicontinuity of the indeﬁnite Riemann–Stieltjes integrals · n n yr dxr : n ∈ N . 0

6.2 Young integrals

117

Boundedness is clear and so, by Arzela–Ascoli, we have uniform convergence along a subsequence to some z ∈ C ([0, T ] , Re ), which proves existence of the Young integral. Using the ﬁrst part of Corollary 5.29, we let n tend to ∞ in (6.8) and obtain |zs,t − ys xs,t | ≤

1 |x|(p+ε)-var;[s,t] |y|(q + ε)-var;[s,t] . 1 − 21−θ ε

Then, an application of Lemma 5.13 justiﬁes the passage ε 0 which shows validity of the Young–L´ oeve estimate, |zs,t − ys xs,t | ≤

1 |x|p-var;[s,t] |y|q -var;[s,t] . 1 − 21−θ 1/θ

(6.9) 1/θ

To prove uniqueness we use the control ω (s, t) := |x|p-var;[s,t] |y|q -var;[s,t]

· (cf. Exercise 1.9). Assume z, z˜ are two limit points of 0 y n dxn so that z0 = z˜0 = 0. Fix a dissection (ti ) of [0, T ] and observe zt ,t − xt yt ,t + xt yt ,t − z˜t ,t |zT − z˜T | ≤ i i+ 1 i i i+ 1 i i i+ 1 i i+ 1 i

≤

2 θ ω (ti , ti+1 ) 1 − 21−θ i

≤

2 θ −1 ω (0, T ) max ω (ti , ti+1 ) . i 1 − 21−θ

Applying this to a sequence of dissections with mesh (=maxi |ti+1 − ti |) tending to zero we see that |zT − z˜T | can be made arbitrarily small and hence must be zero. This shows zT = z˜T and, as T was arbitrary, z ≡ z˜. At last, set c = 1/ 1 − 21−θ and observe that (6.9) implies p θp p p |zs,t | ≤ 21−1/p cp ω (s, t) + |y|∞;[s,t] |x|p-var;[s,t] . We observe that the right-hand side above is super-additive in (s, t) and so p · θp p p 1−1/p p ydx c ≤ 2 ω (s, t) + |y| |x| ∞;[s,t] p-var;[s,t] 0

p-var;[s,t]

and the proof is easily ﬁnished. Exercise 6.9 The purpose of this exercise is to show that our Deﬁnition 6.7 is consistent with the “usual” deﬁnition of Young integrals as limits of Riemann–Stieltjes sums. To this end, let x ∈ C p-var ([0, T ] , R) , y ∈ C q -var ([0, T ] , R) with θ := 1/p + 1/q > 1 and let Dn = (tni )i be a sequence of dissections of [0, T ] with |Dn | → 0, and ξ ni some points in tni , tni+1 . |D n |−1 Show that i=0 y (ξ ni ) xt i ,t i + 1 converges when n tends to ∞ to a limit I independent of the choice of ξ ni and the sequence (Dn ). Identify I as a

T Young integral 0 ydx in the sense of Deﬁnition 6.7.

Young integration

118

Exercise 6.10 Let ϕ = (ϕi )i=1,...,d be a collection of maps from Rd to Re and assume x ∈ C p-var [0, T ] , Rd . Show that · ϕ (x) dx 0

is a well-deﬁned Young integral provided ϕ, viewed as a map Rd → d e older provided L R , R is (γ − 1)-H¨ γ>p and γ − 1 ∈ (0, 1], to avoid trivialities.(We shall encounter this type of regularity assumption in our forthcoming discussion of rough integrals.) Solution. The path ϕ (x· ) has ﬁnite q = p/ (γ − 1)-variation and we see that the Young integral is well-deﬁned since γ 1 1 + = > 1. q p p

6.3 Continuity properties of Young integrals p-var [0, T ] , Rd and y ∈ C q -var ([0, T ] , Proposition d e 6.11 Given x ∈ C with 1/p + 1/q > 1 the map L R ,R · (x, y) → ydx

d

q -var

0

[0, T ] , R × C [0, T ] , L Rd , Re → C p-var ([0, T ] , Re ) from C equipped with the respective p , q-variation norms is a bilinear and continuous map. As a consequence, it is Lipschitz continuous on bounded sets and Fr´echet smooth.

Proof. Bilinearity of (x, y) → ydx follows from bilinearity of the approximations in the deﬁnition and uniqueness; continuity in the sense of bilinear maps is immediate from Young–L´ oeve and Fr´echet smoothness for bilinear, continuous maps is trivial. The following property deals with continuity with respect to “uniform convergence with uniform bounds”. n p-var d [0, T ] , R and y n , y ∈ Proposition 6.12 Assume given x , x ∈ C d e q -var [0, T ] , L R , R such that C p-var

lim |xn − x|∞;[0,T ]

=

0 and sup |xn |p-var;[0,T ] < ∞,

lim |y n − y|∞;[0,T ]

=

0 and sup |y n |q -var;[0,T ] < ∞

n →∞

n →∞

n n

6.4 Young–L´ oeve–Towghi estimates and 2D Young integrals

and 1/p + 1/q > 1. Then · · n n lim y dx − ydx n →∞

0

∞;[0,T ]

0

· n n = 0 and sup y dx n 0

119

< ∞. p-var;[0,T ]

Proof. Increase p, q by ε small enough so that 1/ (p + ε)+1/ (q + ε) > 1. By interpolation, xn → x in (p + ε)-variation and similarly y n → y in (q + ε)· · variation. By the preceding proposition, 0 y n dxn converges to 0 ydx in (p + ε)-variation and hence in ∞-norm. The uniform p-variation bounds · on 0 y n dxn follow immediately from the estimate in the Young–L´oeve theorem. Exercise 6.13 Using continuity properties, establish an integration-byparts formula for Young integrals. Exercise 6.14 Fix p, q ≥ 1 with > 1, R > 0, and ﬁx x ∈ 1/p + 1/q p-var d q -var C [0, T ], R . Deﬁne BR = y ∈ C [0, T ] , L Rd , Re , |y|q -var;[0,T ] ≤ R} . Prove that the map BR y

→

C p-var ([0, T ] , Re ) · ydx → 0

is Lipschitz using the dq -var -metric. Prove it is uniformly continuous with respect to dq -var -metric with q > q. Prove also it is uniformly continuous with respect to d∞ -metric. oeve esti[Hint: Lipschitz with respect to dq -var -metric is just the Young–L´ mate. For q > q, use the ﬁrst case plus interpolation.]

6.4 Young–L´oeve–Towghi estimates and 2D Young integrals Young integrals extend naturally to higher dimensions but only the 2D case will be relevant to us. 2D statements which extend line by line from the 1D statements will not be discussed in detail. In particular, we shall

use the Riemann–Stieltjes integral [0,T ] 2 ydx of a continuous function y ∈ 2 with respect to a bounded variation function x ∈ C [0, T ] , L Rd , Re 2 C 1-var [0, T ] , Rd . We will also show that the estimate ydx ≤ |y| ∞;R |x|1-var;R R

2

is valid for any rectangle R ⊂ [0, T ] . We ﬁrst need to extend Lemma 6.2 to its 2-dimensional version.

Young integration

120

2

Lemma 6.15 Let Γ : {0 ≤ s < t ≤ T } → Re be such that1 (i) for some control ω ˆ, sup

lim

r →0 rectangle R s.t. ω ˆ (R )≤r

|Γ (R)| = 0; r

(6.10)

(ii) for some control ω and some real θ > 1, for all rectangles R being the union of two (essentially) disjoint rectangles R1 , R2 , θ

|Γ (R)| ≤ |Γ (R1 )| + |Γ (R2 )| + ξω (R) .

(6.11)

2

Then, for all rectangles R ∈ [0, T ] , |Γ (R)| ≤

ξ θ ω (R) . 1 − 21−θ

Proof. The proof follows exactly the same lines as the 1-dimensional proof. We now proceed as in the 1D case and prove uniform Young–L´ oeve-type estimates, for integrand and driving signal which are assumed to be of bounded variation.

Proposition 6.16 Let 2 2 x ∈ C 1-var [0, T ] , Rd , y ∈ C 1-var [0, T ] , L Rd , Re . 2

Given R = [s, t] × [u, v] ⊂ [0, T ] , deﬁne

y

Γ (R) = R

s, . u, .

dx.

2

Then, for all such rectangles R ⊂ [0, T ] and p, q ≥ 1 such that θ := p−1 + q −1 > 1, |Γ (R)| ≤

1 1 − 21−θ

2 |y|q -var;R |x|p-var;R . 1

1

θ |y|qθ -var;R . We consider a generic Proof. Deﬁne the control ω (R) = |x|p-var;R rectangle R = [r, t] × [u, v] cut into two non-intersecting rectangles, say R1 = [r, s] × [u, v] and R2 = [s, t] × [u, v] . Observe that from the deﬁnition

1 We

identify once again {0 ≤ s < t ≤ T }2 with rectangles contained in [0, T ]2 .

6.4 Young–L´ oeve–Towghi estimates and 2D Young integrals

of Γ (R), we see

y

r, . u, .

121

dx r, . r, . y dx + y dx = u, . u, . R1 R2 r, . s, . y −y dx = Γ (R1 ) + Γ (R2 ) + u, . u, . R 2 r, s = Γ (R1 ) + Γ (R2 ) + y dx u, . R2 r, s s, t = Γ (R1 ) + Γ (R2 ) + y d x . u, . . [u ,v ] But from Exercise 5.56, the 1-dimensional q-variation of y r ,. s over [u, v] is bounded by |y|q -var;[r,s],[u ,v ] and the 1-dimensional p-variation of x s ,. t oeve 1 diover [u, v] is bounded by |x|p-var;[s,t],[u ,v ] . Hence, using Young–L´ mensional estimates, we obtain Γ (R) =

R

|Γ (R)| ≤ |Γ (R1 )| + |Γ (R2 )| +

1 θ ω (R) . 1 − 21−θ

Deﬁning ω ˜ to be a 2D control dominating the 1-variation of x and y, we see that 2 |Γ (R)| ≤ ω ˜ (R) . It only remains to apply Lemma 6.15. With the notation of the above theorem, we see s, . ydx = y dx + y u, . R R R s, . y dx + y + u R

that

s dx u, . s x (R) . u

We see that the second and third integrals will be well deﬁned when y and x are of ﬁnite q- and p-variation if their 1D projections y s. and y u. are of ﬁnite (1D) q-variation. This is actually satisﬁed if y 0. and y 0. are of ﬁnite (1D) q-variation and y is of ﬁnite (2D) q-variation. To . simplify, 0 = y = 0. In we therefore restrict ourselves to paths y such that y . 0 d e 2 q -var the set of functions y such [0, T ] , L R , R particular, we deﬁne C0,0 that y0, .= y.,0 = 0. 2 Deﬁnition 6.17 (Young integral) Given x ∈ C p-var [0, T ] , Rd , y ∈ 2 2 q -var we say that z ∈ C [0, T ] , Re is an (indef[0, T ] , L Rd , Re C0,0 inite) Young integral of y against x if there exists a sequence (xn , y n ) ⊂

Young integration

122

2 2 1-var [0, T ] , L Rd , Re which converges uniformly C 1-var [0, T ] , Rd ×C0,0 with uniform variation bounds in the sense (xn , x)

=

0 and sup |xn |p-var;[0,T ] 2 < ∞,

lim d∞;[0,T ] 2 (y n , y)

=

0 and sup |y n |q -var;[0,T ] 2 < ∞,

2 lim d n →∞ ∞;[0,T ]

n

n →∞

and

n

2

y n dxn = z uniformly on (s, u) ∈ [0, T ] as n → ∞.

lim

n →∞

[0,s]×[0,u ]

If z is unique we write

· 0

ydx instead of z.

Following the same lines as the 1-dimensional case (which involves generalizing a few analysis and p-variation lemmas from 1D to 2D), we obtain 2 Theorem 6.18 (Young–L´ oeve–Towghi) Given x ∈ C p-var [0, T ] , Rd , 2 q -var with θ = 1/p + 1/q > 1, there exists a [0, T ] , L Rd , Re y ∈ C0,0

· unique (indeﬁnite) Young integral of y against x, denoted by 0 ydx and we have 2 1 y s , u dx ≤ |x|p-var;R |y|q -var;R (6.12) · · 1 − 21−θ R 2

for all rectangles R = [s, t] × [u, v] ⊂ [0, T ] .

· One can also check, just as in the 1D case, that (x, y) → 0 yu dxu is a bilinear continuous map from 2 2 2 q -var [0, T ] , L Rd , Re → C p-var [0, T ] , Re C p-var [0, T ] , Rd × C0,0

(and hence Lipschitz on bounded sets and Fr´echet smooth).

6.5 Comments Young integration goes back to Young [177]. The higher-dimensional case was partially discussed in Young [177] and then in Towghi [170] in the form which is relevant to us.

Part II

Abstract theory of rough paths

7 Free nilpotent groups Motivated by simple higher-order Euler schemes for ODEs we give a systematic and self-contained account of the “algebra of iterated integrals”. Tensor algebras play a natural role. However, thanks to algebraic relations between iterated integrals the “correct” state-space will be seen to be a (so-called) free nilpotent Lie group, faithfully represented as a subset of the tensor algebra. It becomes a metric (and even geodesic) space under the so-called Carnot–Caratheodory metric and will later serve as a natural state-space for geometric rough paths.

7.1 Motivation: iterated integrals and higher-order Euler schemes Let x be an Rd -valued continuous path of bounded variation and deﬁne the kth iterated integrals of the path segment x|[s,t] as t g

k ;i 1 ,...,i k

:=

uk

... s

s

s

u2

dxiu11 . . . dxiukk .

The collection of all such iterated integrals, g = gk ;i 1 ,...,i k : 1 ≤ k ≤ N ; i1 , . . . , ik ∈ {1, . . . , d}

(7.1)

(7.2)

is called the step-N signature of the path segment x|[s,t] and is denoted by SN (x)s,t . Postponing (semi-obvious) algebraic formalities to the next section, let us consider (higher-order) Euler schemes for the ODE dy = V (y) dx =

d

Vi (y) dxi

i=1

with V ∈ C Re , L Rd , Re and recall that π (V ) (0, y0 ; x) stands for any (not necessarily unique) solution started at y0 , possibly only deﬁned up to some explosion time. Let I denote the identity function on Re and recall T the identiﬁcation of a vector ﬁeld W = W 1 , . . . , W e : Re → Re with the ﬁrst-order diﬀerential operator

e k =1

W k (y)

∂ . ∂y k

Free nilpotent groups

126

Granted suﬃcient regularity of V , a simple Taylor expansion suggests, at least for 0 < t − s << 1, a step-N approximation of the form yt

≈

ys +

d

Vi (ys ) xis,t

i=1

+... +

t Vi 1 · · · Vi N I (ys )

u2

... s

i 1 ,...,i N ∈{1,...,d}

uN

s

s

dxiu11 . . . dxiuNN .

Having made plain the importance of iterated integrals in higher-order Euler schemes, let us observe the presence of non-linear constraints between the iterated integrals (7.1). This happens already at the “second” level of iterated integrals. Example 7.1 Let x ∈ C01-var [0, T ] , Rd and write x = S2 (x) for its step2 lift. (i) Using integration by parts it readily follows that, for all i, j ∈ {1, . . . , d} t t 2;j,i i j + x = x dx + xjs,r dxir = xis,t xjs,t . (7.3) x2;i,j s,t s,t s,r r s

(ii) As a trivial consequence of

s

x1s,t

:= xt − xs we have

1;i 1;i x1;i s,t + xt,u = xs,u , i = 1, . . . , d.

(7.4)

More interestingly, an elementary computation1 gives 2;i,j 2;i,j 1;i 1;j x2;i,j s,u = xs,t + xt,u + xs,t xt,u , i, j = 1, . . . , d.

(7.5)

This (matrix) equation can be expressed in terms of equations of the respective symmetric and anti-symmetric parts. Adding the equation obtained by interchanging i, j we see, using (7.3) xis,u xjs,u = xis,t xjs,t + xit,u xjt,u + xis,t xjt,u + xjs,t xit,u which just (re-)expresses the additivity of vector increments (7.4). On the other hand, subtracting the equation obtained by interchanging i, j, followed by multiplication with 1/2, yields i,j i,j Ai,j s,u = As,t + At,u +

1 . . . to

1 i j xs,t xt,u − xjs,t xit,u 2

be compared with the forthcoming Theorem 7.11.

(7.6)

7.1 Motivation

127

Figure 7.1. We plot (xi· , xj· ). The triangle connects the points: (xis , xjs ) on the lower left side, (xit , xjt ) in the middle and (xiu , xju ) on the right side. Note that the respective area increments (signed area between the path and a linear chord) are not additive. In fact, Ais,,ju = Ais,,jt + Ait ,, ju + ∆is,,jt , u , where ∆is,,jt , u is the area of the triangle as indicated in the ﬁgure.

where Ai,j s,t

1 1 2;i,j xs,t − x2;j,i = := s,t 2 2

t

xis,r dxjr s

−

t

xjs,r dxir s

has an appealing geometric interpretation as seen in Figure 7.1. Let us draw some ﬁrst conclusions. 2

N

(i) The naive state space Rd+d +···+d for iterated integrals of the form (7.2) is too big. For instance, there is no need to store the symmetric part of x2s,t .

(ii) Any analysis which is based on higher-order Euler approximations over ﬁner and ﬁner intervals, e.g. starting with some interval [s, u] then [s, t] , [t, u], etc., must acknowledge the non-linear nature of “higher-order increments” as seen in (7.5) and (7.6).

Free nilpotent groups

128

7.2 Step-N signatures and truncated tensor algebras 7.2.1 Deﬁnition of SN We have seen that expressions of iterated integrals of type dxiu11 . . . dxiukk , x ∈ C 1-var [0, T ] , Rd s< u 1 < ...< u k < t

appear naturally when considering Euler schemes. A ﬁrst-order Euler scheme over the interval [s, t] involves

↔

dxiu s< u < t

i=1,...,d

d i=1

dxiu

ei ∈ Rd

s< u < t

where (ei )i=1,...,d denotes the canonical basis of Rd . For a second-order scheme one needs additionally dxiu dxjv , s< u < v < t

↔

d

i,j =1,...,d

dxiu dxjv ,

(ei ⊗ ej ) ∈ Rd ⊗ Rd

s< u < v < t

i,j =1

where Rd ⊗ Rd with basis (ei ⊗ ej )i,j =1,...,d can be viewed as the set of realvalued d × d matrices with canonical basis.2 The (obvious) generalization to k ≥ 2 reads dxiu11 . . . dxiukk s< u 1 < ...< u k < t

↔

i 1 ,...,i d

s< u 1 < ...< u k < t

i 1 ,...,i k

dxiu11

. . . dxiukk

(ei 1 ⊗ · · · ⊗ ei k )

(7.7)

where (ei 1 ⊗ · · · ⊗ ei k ), i1 , . . . , ik ∈ {1, . . . , d}, is the canonical basis of d ⊗k R , the space of k-tensors over Rd . Life is easier without many indices and we shall write the right-hand side of (7.7) simply as ⊗k dxu 1 ⊗ · · · ⊗ dxu k ∈ Rd . s< u 1 < ...< u k < t 2 e ⊗ e corresponds to the matrix with entry 1 in the ith line, jth column and 0 i j everywhere else.

7.2 Step-N signatures and truncated tensor algebras

129

⊗k k ∼ We note that, as vector spaces Rd = Rd , and it is a convenient convention to set d ⊗0 R := R. To any Rd -valued path γ of ﬁnite length deﬁned on some interval [s, t], we may associate the collection of its iterated integrals. We have Deﬁnition 7.2 The step-N signature of γ ∈ C 1-var [s, t] , Rd is given by SN (γ)s,t ≡ 1, dxu , . . . , dxu 1 ⊗ · · · ⊗ dxu k ⊕N k =0

∈

s< u < t d ⊗k

R

s< u 1 < ...< u k < t

.

The path u → SN (γ)s,u is called the (step-N ) lift of x.

Remark 7.3 This notation is further justiﬁed by Chen’s theorem below: the step-N lift of some γ ∈ C 1-var [0, T ] , Rd , t → SN (γ)t takes values in a group so that SN (γ)s,t is the natural increment of this path, i.e. the −1 product of (SN (γ)s ) with SN (γ)t . Given two vectors a, b ∈ Rd with coordinates ai i=1,...,d and bi i=1,...,d one can construct the matrix ai bj i,j =1,...,d and hence the 2-tensor a ⊗ b :=

d

ai bj ei ⊗ ej ∈ Rd ⊗ Rd .

i,j =1

(In fact, this is the linear extension of the map ⊗ : Rd × Rd → Rd ⊗ Rd which maps the pair (ei , ej ) to the (i, j) th basis element of Rd ⊗ Rd , for which we already used the suggestive notation ei ⊗ ej .) More generally, given ⊗k ai 1 ,...,i k ei 1 ⊗ · · · ⊗ ei k ∈ Rd (7.8) a= i 1 ,...,i k

⊗l then, with similar notation b ∈ Rd , we agree that a ⊗ b is deﬁned by a⊗b = ∈ We now deﬁne

ai 1 ,...,i k bj 1 ,...,j l ei 1 ⊗ · · · ⊗ ei k ⊗ ej 1 ⊗ . . . ej l d ⊗k d ⊗l d ⊗(k + l) ∼ ⊗ R . R = R

(7.9)

d ⊗k , (7.10) T N Rd := ⊕N k =0 R ⊗k for the projection to the kth tensor and write π k : T N Rd → Rd level. We shall also use the projection (7.11) π 0,k : T N Rd → T k Rd , for k ≤ N ,

Free nilpotent groups

130

which maps g = g 0 , . . . , g N ∈ T N Rd into g 0 , . . . , g k ∈ T N Rd . ⊗k Here, g k = π k (g) ∈ Rd is sometimes refered to as the kth level of g. Given g, h ∈ T N Rd , one extends (7.9) to T N Rd by setting g⊗h=

g ⊗ h ⇔ ∀k ∈ {0, . . . , N } : π k (g ⊗ h) = i

j

i+ j ≤N i,j ≥0

k

g k −i ⊗ hi .

i=0

The vector space T N Rd becomes an (associative) algebra under ⊗. More precisely, we have Proposition 7.4 The space T N Rd , +, .; ⊗ is an associative algebra with neutral element 1 := (1, 0, . . . , 0) = 1 + 0 + · · · + 0 ∈ T N Rd . (The unit element for + is 0 = (0, 0, . . . , 0), of course.) We will call T N Rd the truncated tensor algebra of level N . Proof. Straightforward and left to the reader. Remark 7.5 We shall see below that the set of all g ∈ T N Rd with π 0 (g) = 1 forms a (Lie) group ⊗ with unit element 1. When N = 1 under this group is isomorphic to Rd , + with the usual unit element 0 and this persuades us to set o := (1, 0, . . . , 0) . Similar to the algebra of square matrices, the algebra product is not commutative (unless N = 1 or d = 1), indeed 1 1 1 1 1 1 1 1 ⊗ = = ⊗ = . 0 1 0 0 1 0 1 0 d ⊗k Let us now deﬁne a norm on T N Rd := ⊕N . To this end, we k =0 R d ⊗k equip each tensor level R with Euclidean structure, which amounts to declaring the canonical basis {ei 1 ⊗ · · · ⊗ ei k : i1 , . . . , ik ∈ {1, . . . , d}} to ⊗k be orthonormal so that for any a ∈ Rd of form (7.8) 1 2 |ai 1 ,...,i k | |a|(Rd ) ⊗k = i 1 ,...,i k

and when no confusion is possible we shall simply write |a|. Let us also observe that for 0 ≤ i ≤ k ≤ N , ⊗i d ⊗k −i × R , |a ⊗ b| (a, b) ∈ Rd

( Rd ) ⊗k

= |a|

( Rd ) ⊗i

|b|

( Rd ) ⊗( k −i )

7.2 Step-N signatures and truncated tensor algebras

131

which is a compatibility relation between the tensornorms on the respective N tensor levels. Then for any g = k =0 π k (g) ∈ T N Rd we set |g|T N (Rd ) :=

max |π k (g)|

k =0,...,N

which makes T N Rd a Banach space (of ﬁnite dimension 1 + d + d2 + · · · + dN ); again we shall write |g| if no confusion is possible. We remark that there are other choices of norms on T N Rd , of course all equivalent, but this one will turn out to be convenient later on (cf. deﬁnition of ρp−ω later). Exercise 7.6 Consider the (inﬁnite) tensor algebra T ∞ Rd := ⊕∞ k =0 d ⊗k R . Show thatT N Rd is the algebra obtained by factorization by the ideal g ∈ T ∞ Rd : π i (g) = 0 for 0 ≤ i ≤ N . Exercise 7.7 Identify T ∞ Rd ≡ R !e1 , . . . , ed " with the algebra of polynomials in d non-commutative indeterminants e1 , . . . , ed . Conclude that T N Rd can be viewed as an algebra of polynomials in d non-commutative indeterminants for which ei 1 . . . ei n ≡ 0 whenever n ≥ N .

7.2.2 Basic properties of SN Given x : [0, T ] → Rd , continuous of bounded variation, and a ﬁxed s ∈ N [0, T ) the path SN (x)s,· takes values in T N Rd ∼ = R1+d+···+d , as vector spaces. Almost by deﬁnition, the path SN (x)s,t then satisﬁes an ODE on T N Rd driven by x. Proposition 7.8 Let x : [0, T ] → Rd be a continuous path of bounded variation. Then, for ﬁxed s ∈ [0, T ),

dSN (x)s,t = SN (x)s,t ⊗ dxt , SN (x)s,s = 1.

d Remark 7.9 If we write x (·)= i=1 xi (·) ei and deﬁne the (linear!) vec tor ﬁelds Ui : T N Rd → T N Rd by g → g ⊗ ei , i ∈ {1, . . . , d} this ODE can be rewritten in the more familiar form

dSN (x)s,t =

d i=1

Ui SN (x)s,· dxit .

Free nilpotent groups

132

Proof. Let us look at level k, k ≥ 1, of SN (x)s,t , dxr 1 ⊗ · · · ⊗ dxr k s< r 1 < ...< r k < t t

= rk =s

t

= r=s

s< r 1 < ...< r k −1 < r k

dxr 1 ⊗ · · · ⊗ dxr k −1

⊗ dxr k

π k −1 SN (x)s,r ⊗ dxr .

Hence, we see that SN (x)s,t = 1 +

s

t

SN (x)s,r ⊗ dxr .

Proposition 7.8 tells us that SN (x) satisﬁes an ODE of the type discussed in Part I of this book. A number of interesting properties of signatures are then direct consequences of the corresponding ODE statements. We ﬁrst describe what happens under reparametrization. Proposition 7.10 Let x : [0, T ] → Rd be a continuous path of bounded variation, φ : [0, T ] → [T1 , T2 ] a non-decreasing surjection, and write xφt := xφ(t) for the reparametrization of x under φ. Then, for all s, t ∈ [0, T ] , SN (x)φ(s),φ(t) = SN xφ s,t . Proof. A consequence of Propositions 3.10 and 7.8. Simple as it is, Proposition 7.10 has an appealing If [x] interpretation. denotes the equivalence class of some x ∈ C 1-var [0, T ] , Rd obtained by all possible reparametrizations, then the signature-map x ∈ C 1-var [0, T ] , Rd → SN (x)0,T is really a function of [x]. We now discuss the signature of the con 1-var [0, T ] , Rd , η ∈ catenation of two paths. Recall that, given γ ∈ C 1-var d [T, 2T ] , R , we set C γ (·) on [0, T ] γη ≡ η (·) − η (0) + γ (T ) on [T, 2T ] so that γ η ∈ C 1-var [0, 2T ] , Rd . If η was deﬁned on [0, T ] , rather than [T, 2T ], one may prefer to reparametrize such that γ η too is deﬁned on [0, T ]. Whatever parametrization one chooses, thanks to the previous proposition the signature of the entire path γ η is intrinsically deﬁned. The following theorem says the signature of γ η is precisely the tensor product of the respective signatures of γ and η.

7.2 Step-N signatures and truncated tensor algebras

133

Theorem 7.11 (Chen) Given γ ∈ C 1-var [0, T ] , Rd , η ∈ C 1-var ([T, 2T ] , Rd , SN (γ η)0,2T = SN (γ)0,T ⊗ SN (η)T ,2T . Equivalently, given x ∈ C 1-var [0, T ] , Rd and 0 ≤ s < t < u ≤ T we have SN (x)s,u = SN (x)s,t ⊗ SN (x)t,u . Proof. By induction on N . For N = 0, it just reads 1 = 1.1. Assume it is true for N and all s < t < u ∈ [0, T ], and let us prove it is true for N + 1. First observe, in T (N +1) Rd , SN +1 (x)s,u = 1 +

u

s

SN +1 (x)s,r ⊗ dxr = 1 +

u

s

SN (x)s,r ⊗ dxr ,

where the second equality follows from truncation beyond level (N + 1). For similar reasons, u u SN (x)t,r ⊗ dxr = SN +1 (x)s,t ⊗ SN (x)t,r ⊗ dxr . SN (x)s,t ⊗ t

t

Hence, using the induction hypothesis to split up SN (x)s,r when s < t < r < u, t u SN (x)s,r ⊗ dxr + SN (x)s,t ⊗ SN (x)t,r ⊗ dxr SN +1 (x)s,u = 1 + s t u = SN +1 (x)s,t + SN +1 (x)s,t ⊗ SN (x)t,r ⊗ dxr t = SN +1 (x)s,t ⊗ 1 + SN +1 (x)t,u − 1 =

SN +1 (x)s,t ⊗ SN +1 (x)t,u .

We now show that the inverse (with respect to ⊗) of the signature of a path is precisely the signature of that path with time reversed. − denotes the path x Proposition 7.12 Let x ∈ C 1-var [0, T ] , Rd . Then, if ← d x : t ∈ [0, T ] → xT −t ∈ R , −) −) ⊗ S (x) S (x) ⊗ S (← x = S (← x = 1. N

0,T

N

0,T

N

0,T

N

0,T

Proof. Using the fact that t → SN (x)0,t is the solution to an ODE driven by x (Proposition 7.8) this follows immediately from the corresponding results on ODEs with time-reversed driving signal (Proposition 3.13). Deﬁnition 7.13 For λ ∈ R, we deﬁne the dilation map δ λ : T N Rd → T N Rd such that π k (δ λ (g)) = λk π k (g) .

Free nilpotent groups

134

Exercise 7.14 Check that if λ is a real, x : [0, 1] → Rd is a continuous path of bounded variation, λx the path x scaled by λ, then SN (λx)s,t = δ λ SN (x)s,t . Proposition 7.15 Let (xn ) ⊂ C 1-var [0, 1] , Rd with supn |xn |1-var;[0,1] < ∞, uniformly convergent to some x ∈ C 1-var [0, 1] , Rd . Then, SN (xn )0,· converges uniformly to SN (x)0,· . In particular, lim SN (xn )0,1 = SN (x)0,1 .

n →∞

Proof. Using the fact that t → SN (x)0,t is the solution to an ODE driven by x (Proposition 7.8) this follows immediately from the ODE results on continuity of the solution map (cf. Corollary 3.16).

7.3 Lie algebra tN Rd and Lie group 1 + tN Rd The space T N Rd , +, ., ⊗ is an associative algebra. We introduce two simple subspaces (linear and aﬃne-linear, respectively) which will guide us towards the (crucial) free nilpotent Lie algebra and group and their Lie algebra. Let us set tN Rd ≡ g ∈ T N Rd : π 0 (g) = 0 so that

7.3.1

1 + tN Rd = g ∈ T N Rd : π 0 (g) = 1 .

The group 1 + tN Rd

We ﬁrst show that elements in 1 + tN Rd are invertible with respect to the tensor product ⊗. Lemma 7.16 Any g = 1 + a ∈ 1 + tN Rd has an inverse with respect to ⊗ given by N −1 k (−1) a⊗k , g −1 = (1 + a) = k =0

that is, g ⊗ g

−1

=g

−1

⊗ g = 1.

Proof. We have (1 + a)

N k =0

k

(−1) ak

=

N +1

k =1 N +1

= a

k +1

(−1)

ak +

N k =0

+ 1.

k

(−1) ak

7.3 Lie algebra and Lie group

135

⊗n Now because we set to zero any elements in Rd for n > N , we see that aN +1 = 0. It is also obvious that if g and h are in 1 + tN Rd , then g⊗ h ∈ 1 + tN Rd . Finally, 1 + tN Rd is an aﬃne-linear subspace of T N Rd hence N ˙ a smooth manifold, trivially diﬀeomorphic to tN Rd ∼ = Rd +···+ d . Noting that the group operations ⊗, −1 are smooth maps (in fact, polynomial when written out in coordinates) we have Proposition 7.17 The space 1 + tN Rd is a Lie group3 with respect to tensor multiplication ⊗. Let us remark that the manifold topology of 1 + tN Rd is, of course, induced by the metric ρ (g, h) := |g − h|T N (Rd ) = |g − h| = max |π i (g − h)| , i=1,...,N

(7.12)

which arises from the norm on T N Rd ⊃ 1 + tN Rd . We end this subsection with a simple observation. d N Proposition 7.18 Assume −1gn , g ⊂ 1 + t R . Then, limn →∞ |gn − g| = 0 if and only if limn →∞ gn ⊗ g − 1 = 0. Proof. All group operations in the Lie group 1 + tN Rd are continuous in the manifold topology of 1 + tN Rd . In particular, gn → g ⇔ gn−1 → g −1 ⇔ gn−1 ⊗ g → g −1 ⊗ g = 1 and we conclude with the remark that, as n → ∞, gn → g ⇔ |gn − g| → 0 and gn−1 ⊗ g → 1 ⇔ gn−1 ⊗ g − 1 → 0.

7.3.2 The Lie algebra tN Rd and the exponential map

The vector space tN Rd , +, . becomes itself an algebra under ⊗. As in every algebra, the commutator, in our case (g, h) → [g, h] := g ⊗ h − h ⊗ g ∈ tN Rd for g, h ∈ tN Rd , deﬁnes a bilinear map which is easily seen to be anticommutative, i.e. [g, h] = − [h, g] , for all g, h ∈ tN Rd 3 Recall that a Lie group is by deﬁnition a group which is also a smooth manifold and in which the group operations are smooth maps.

Free nilpotent groups

136

and to satisfy the Jacobi identity for all g, h, k ∈ tN Rd ; that is, [g, [h, k]] + [h, [k, g]] + [k, [g, h]] = 0 for all g, h, k ∈ tN Rd . Recalling that a vector space V = (V, +, .) equipped with a bilinear, anticommutative map [·, ·] : V × V → V which satisﬁes the Jacobi identity is called a Lie algebra (the map [·, ·] is called the Lie bracket), this can be summarized as Proposition 7.19 tN Rd , +, ., [·, ·] is a Lie algebra. We now deﬁne the exponential and logarithm maps via their power series: Deﬁnition 7.20 The exponential map is deﬁned by exp : tN Rd → 1 + tN Rd a →

1+

N a⊗k

k!

k =1

while the logarithm map is deﬁned by log (1 + a)

: →

1 + tN Rd → tN Rd N

(−1)

k =1

k +1

a⊗k . k

The deﬁnitions of exp and log are precisely via their classical power series with (i) usual powers replaced by “tensor powers” and (ii) the inﬁnite N sums replaced by k =1 , thanks to working within the tensor algebra with truncation beyond level N . A direct calculation shows that exp (log (1 + a)) = a, log (exp (a)) = a, for all a ∈ tN Rd . We emphasize that log (·) is globally deﬁned and there are no convergence issues whatsoever. Example 7.21 Fix a ∈ Rd ∼ = π 1 tN Rd . The step-N signature of x (·) given by t ∈ [0, 1] → ta computes to N dxr 1 ⊗ · · · ⊗ dxr k SN (x)0,1 = 1 + 0< r 1 < ...< r k < 1

k =1

=

1+

N

⊗k

a

1+

N a⊗k k =1

dr1 . . . drk 0< r 1 < ...< r k < 1

k =1

=

k!

= exp (a) .

(7.13)

7.3 Lie algebra and Lie group

7.3.3

137

The Campbell–Baker–Hausdorﬀ formula

Recall that exp maps tN Rd one-to-one and onto 1 + tN Rd . In general, ea eb = ea+ b , but one has 1 1 1 (7.14) ∀a, b ∈ tN Rd : ea eb = ea+b+ 2 [a,b]+ 1 2 [a,[a,b]]+ 1 2 [b,[b,a]]+... where . . . stands for (a linear combination, with universal coeﬃcients, of) higher iterated brackets of a and b. Thanks to truncation beyond tensorlevel N , all terms involving N or more iterated brackets must be zero. For N = 2, this formula is obtained by simple computation; indeed, given a,b ∈ t2 Rd we have b⊗2 a⊗2 ⊗ 1+b+ exp (a) ⊗ exp (b) = 1+a+ 2 2 = =

1 (a + b) 1 + a + b + (a ⊗ b − b ⊗ a) + 2 2 1 exp a + b + [a, b] . 2

⊗2

The same computation is possible, if tedious, for N = 3 and allows us to recover the next set of bracket terms as seen in (7.14). The general case, however, requires a diﬀerent argument and we shall give a proof based on ordinary diﬀerential equations. Given a linear operator A : tN Rd → tN Rd deﬁne A0 as identity map and set An = A ◦ · · · ◦ A (n times). An arbitrary real-analytic function n ≥0 γ n z n gives rise to the operator n ≥0 γ n An provided this series converges in operator norm. Fix a ∈ tN Rd and deﬁne the linear map tN Rd $ d → (ad a) d ≡ [a, d] ∈ tN Rd . n

Observe that (ad a) ≡ 0 for n > N. Lemma 7.22 For all a, d ∈ tN Rd we have exp (a) ⊗ d ⊗ exp (−a) = ead a d where ead a ≡

N 1 1 n n (ad a) = (ad a) . n! n! n =0

n ≥0

As a consequence we have the following operator identity on tN Rd : ead c = ead a ◦ ead b where c = log (exp (a) ⊗ exp (b)) .

Free nilpotent groups

138

Proof. Deﬁne the linear map ρt from tN Rd into itself by ρt : d → exp (t a) ⊗ d ⊗ exp (−t a) . Obviously, t → ρt (d) ∈ tN Rd is diﬀerentiable and d ρ (d) = [a, ρt (d)] = (ad a) ρt (d) . dt t Noting ρ0 (d) = d the solution is given by ρt (d) = et ad a d and we conclude by setting t = 1. The last statement is immediately veriﬁed by considering the image of some d ∈ tN Rd under both ead c and ead a ◦ ead b . Lemma 7.23 Assume t → ct ∈ tN Rd is continuously diﬀerentiable. Then d exp (−ct ) = −G (ad ct ) c˙t exp (ct ) ⊗ dt where c˙t = dct /dt and G (z) =

ez − 1 1 = zn . z (n + 1)! n ≥0

Proof. Set

d exp (−s ct ) . dt Then b0,t = 1 ⊗ 0 = 0. Taking derivatives with respect to s, a short computation shows that bs,t = exp (s ct ) ⊗

d bs,t ds

=

[ct , bs,t ] − c˙t

=

(ad ct ) bs,t − c˙t

where c˙t = dct /dt. Keeping t ﬁxed, the solution is given by bs,t

= =

es ad c t b0,t − f (s, ad ct ) c˙t −f (s, ad ct ) c˙t

where f (s, z) = (es z − 1) /z, entire in z. Setting s = 1 ﬁnishes the proof. Theorem 7.24 (Campbell–Baker–Hausdorﬀ ) Let a, b ∈ tN Rd . Then

1

log [exp (a) ⊗ exp (b)] = b +

H et

ad a

◦ ead b a dt

0

where H (z) =

(−1)n ln z n = (z − 1) . z−1 n+1 n ≥0

(7.15)

7.3 Lie algebra and Lie group

139

In particular, log (exp (a) ⊗ exp (b)) equals a sum of iterated brackets of a and b, with universal4 coeﬃcients. Proof. First observe that the linear operator H et ad a ◦ ead b is in fact given by a ﬁnite series. Indeed, et ad a ◦ ead b minus the identity map is a ﬁnite sum of N or less iterated applications of ad a, ad b. Consequently, only the ﬁrst N terms of the expansion of H are needed. We also observe that both sides of (7.15) of degree less than or equal to are polynomials N in the coordinates an ;i 1 ,...,i n , bn ,;i 1 ,...,i n , 1 ≤ n ≤ N . Therefore, it is suﬃcient to check (7.15) for a,b in a neighbourhood of 0. For t ∈ [0, 1] deﬁne ct = log et a ⊗ e b . Then

d −c t d −b e e ⊗ e−t a = −a = et a ⊗ eb ⊗ dt dt which, combined with Lemma 7.23, shows that ec t ⊗

a = G (ad ct ) c˙t .

(7.16)

On the other hand, Lemma 7.22 implies the operator identity ad ct = ln et ad a ◦ ead b which certainly holds when a,b are small enough. Since G(ln z)H(z) = 1, at least near z = 1, the operator identity −1 G (ad ct ) = G ln et ad a ◦ ead b = H et ad a ◦ ead b holds for a,b small enough. This allows us to rewrite (7.16) as c˙t = H et ad a ◦ ead b a and by integration c1 = b+

1

H et ad a ◦ ead b a dt

0

which is exactly what we wanted to show. d ⊂ tN Rd as the smallest sub-Lie algebra Deﬁnition 7.25 Deﬁne gN R of tN Rd which contains π 1 tN Rd ∼ = Rd . That is, gN Rd = Rd ⊕ Rd , Rd ⊕ · · · ⊕ Rd , . . . , Rd , Rd .

(N −1) brackets

We call it the free step-N nilpotent Lie algebra.

4 The coeﬃcients are given by numerical constants, computable from H . In particular, they do not depend on N or d = dim Rd .

140

Free nilpotent groups

Remark 7.26 (universal property of free Lie algebras) Let i denote the (linear) inclusion map i : π 1 tN Rd ∼ = Rd → gN Rd . Then g = gN Rd has the property that for any (step-N nilpotent) Lie algeba a and any linear map f : Rd → a there is a Lie algebra homomorphism φ : g → a so that f = φ ◦ i. Indeed, we can take a basis of g where each element is of the form eα = eα 1 , eα 2 , . . . eα k −1 , eα k d d ⊗k ∈ R , . . . , Rd , Rd ⊂ R ,k ≤ N and one checks that cα f (eα 1 ) , f (eα 2 ) , . . . f eα k −1 , f (eα k ) φ cα eα := has the desired properties.

Corollary 7.27 Let a, b ∈ gN Rd . Then log ea ⊗ eb ∈ gN Rd . In other words, exp gN Rd is a subgroup of 1 + tN Rd , ⊗, 1 Proof. An obvious corollary of the Campbell–Baker–Hausdorﬀ formula.

7.4 Chow’s theorem As was seen in Example 7.21, the step-N signature of the path t ∈ [0, 1] → vt, v ∈ Rd , is precisely exp (v) ∈ 1 + tN Rd . A piecewise linear path is just the concatenation of such paths (up to irrelevant reparametrization) and by Chen’s theorem its step-N signature is of the form ev 1 ⊗ · · · ⊗ ev m ∈ 1 + tN Rd with v1 , . . . , vm ∈ Rd . Conversely, any element of this form arises as the step-N signature of a piecewise linear path, e.g. of x : [0, m] → Rd with xi−1,i = vi , i = 1, . . . , m and linear between integer times. (If one prefers, the trivial reparametrization x ˜ (t) = x (tm) deﬁnes a piecewise linear path on [0, 1] with identical signature.) Theorem 7.28 (Chow) Let g ∈ exp gN Rd . Then, there exist v1 , . . . , vm ∈ Rd such that g = ev 1 ⊗ · · · ⊗ ev m . Equivalently, there exists a piecewise linear path x : [0, 1] → Rd with signature g, by which we mean g = SN (x)0,1 .

7.4 Chow’s theorem

141

Proof. It isenough to show this for log (g) in an open neighbourhood of A of the 0 ∈ gN Rd or, equivalently, for g in an open neighbourhood unit element 1 = exp (0). Indeed, given g ∈ exp gN Rd , it is clear that δ ε g ∈ A for ε small enough and if x denotes the (piecewise linear) path whose signature equals δ ε g, the scaled path x (·) /ε, which is still piecewise linear, has signature g. d With i: i = 1, . . . , d) denoting the standard basis in R , let us deﬁne the (e N d exp g R -valued paths φit = exp (tei ) . An easy application of the Campbell–Baker–Hausdorﬀ formula shows that5 φij t

: = exp (−tej ) ⊗ exp (−tei ) ⊗ exp (tej ) ⊗ exp (tei ) = exp t2 [ei , ej ] + o t2 .

For a multi-index I = iJ, we deﬁne by induction φIt

= φJ−t ⊗ exp (−tei ) ⊗ φJt ⊗ exp (tei ) exp t|I | eI + o t|I |

: =

k where eI = ei 1 , . . . ei k −1 , ei k for I = (i1 , . . . , ik ) ∈ {1, . . . , d} . We I I then deﬁne ψ t := φt 1 / |I | for t ≥ 0 and for t < 0, # ψ It

:=

φI−|t|1 / |I |

φJ−|t|1 / |I | ⊗

1/|I |

exp − |t|

ei ⊗

φJ|t|1 / |I |

1/|I |

⊗ exp |t|

ei

for |I| odd for |I| even

so that ψ It = exp (teI + o (t)) . We now choose a vector space basis (eI k : k = 1, . . . , n) of gN deﬁne ϕ (t1 , . . . , tn ) = log ψ It nn ⊗ · · · ⊗ ψ It 11 ∈ gN Rd

(7.17) Rd and

and note that ϕ(0) = log (1) = 0. We also observe that, thanks to (7.17), ϕ : Rn → gN Rd ∼ = Rn (as vector spaces) is continuously diﬀerentiable near 0, with non-degenerate derivative at the origin.6 This implies that the (one-to-one) image under ϕ of a small enough neighbourhood of 0 ∈ Rn , say Ω, contains an open neighbourhood of 0 ∈ gN Rd . On 5 Observe

⊗3 ⊗N . that the o t2 term is an element in g N Rd ∩ Rd ⊕ · · · ⊕ Rd

6 The diﬀerential of log at 1 is the identity map and hence plays no role in the nondegeneracy at 0.

142

Free nilpotent groups

the other hand, unwrapping the very deﬁnition of ϕ shows any element of the form exp ϕ (t1 , . . . , tn ) ∈ exp gN Rd , (t1 , . . . , tn ) ∈ Ω, is the concatenation of piecewise linear path segments in basis directions (ei : i = 1, . . . , d) and so the proof is ﬁnished. N d converges to 1 ∈ exp Corollary N d 7.29 Assume (gk ) ⊂ exp g R as k → ∞. Let xk denote the piecewise linear path with sigg R nature gk as constructed in the previous theorem. Then the length of xk converges to 0 as k → ∞. Proof. proof of the previous theorem shows that any element g ∈ The exp gN Rd , close enough to 1, can be written as g = exp ϕ (t1 , . . . , tn ) and the length of the associated (piecewise linear) path x = xg is bounded by 1/|I | 1/|I | K |t1 | 1 + · · · + |tn | n where K counts the maximal number of concatenations involved in the I ψ t j ’s. Observe that K depends on the space gN Rd but can be chosen independent of g. Since g → 1 is equivalent to (t1 , . . . , tn ) → 0, we obtain the desired continuity statement length (xg ) → 0 as g → 1.

7.5 Free nilpotent groups 7.5.1 Deﬁnition and characterization Let us consider (i) the set of all step-N signatures of continuous paths of ﬁnite length, GN Rd := SN (x)0,1 : x ∈ C 1-var [0, 1] , Rd ; (ii) the image of the sub-Lie algebra gN Rd ⊂ tN Rd , cf. Deﬁnition 7.25, under the exponential map exp gN Rd ⊂ 1 + tN Rd ;

7.5 Free nilpotent groups

143

2 3 of 1 + tN Rd generated by elements in (iii) the subgroup exp Rd exp Rd , i.e. 2

exp R

d

3

:=

# m 4

$ exp (vi ) : m ≥ 1, v1 , . . . , vm ∈ R

d

.

i=1

Theorem 7.30 (and deﬁnition) We have 2 3 GN Rd = exp gN Rd = exp Rd and GN Rd is a (closed) sub-Lie group of 1 + tN Rd , ⊗ , called the free nilpotent group of step N over Rd . 2 3 Proof. Step 1: We show exp gN Rd = exp Rd . Obviously, exp Rd N d and it follows from the CBH formula that ⊂ exp g R 2

3 exp Rd ⊂ exp gN Rd .

Conversely, Chow’s theorem tells us that any element of exp gN Rd can be expressed in the form ev 1 ⊗ · · · ⊗ ev m with vi ∈ Rd , which plainly implies 3 2 exp gN Rd ⊂ exp Rd . 2 3 Step 2: We show that exp Rd is a closed subset of 1 + tN Rd . By the N d previous step, it suﬃces to check that exp this g R is closed. But follows from (obvious!) closedness of gN Rd ⊂ tN Rd , exp gN Rd = log−1 gN Rd and continuity 2 of log.3 Step 3: We show GN Rd = exp Rd . The inclusion 2

3 exp Rd ⊂ GN Rd

is clear from Example 7.21 and Chen’s theorem. Indeed, any ev 1 ⊗ · · · ⊗ ev m is the step-N signature of x ˜ : [0, 1] → Rd where x ˜ (t) = x (tm) and x : d [0, m] → R with xi−1,i = vi , i = 1, . . . , m and linear between integer times. (This was already pointed out in our discussion of Theorem 7.28.) 2 3 By step 2, !exp (Rd )" = exp Rd and so the other inclusion will follow from GN Rd ⊂ !exp (Rd )". To prove this inclusion, take g = SN (x)0,1 ∈ GN Rd , for some x ∈ C 1-var [0, 1] , Rd . We know that piecewise linear approximations (xn ) satisfy supn |xn |1-var;[0,1] < ∞ and converge to x, uniformly on [0, 1]. By Chen’s 2 3 theorem, gn = SN (xn )0,1 ∈ exp Rd and from Proposition 7.15, gn converges to g. This shows that g ∈ !exp (Rd )", which is what we wanted.

Free nilpotent groups

144

2 3 Step 4: We show that the set G := GN Rd = exp gN Rd = exp Rd is a closed sub-Lie group of 1 + tN Rd , ⊗ . Topological 2 d 3closedness was is an abstract already seenin step 2. It is also clear that G = exp R sub-Lie group subgroup of 1 + tN Rd , ⊗ . To see that we have an actual we have to check that G is a submanifold of 1 + tN Rd . It is not hard to deduce this from d (log ◦ exp) = Id, so that by the chain rule, d exp is one-to-one at every point. Alternatively, one can appeal to a standard theorem in Lie group theory7 which asserts that a closed abstract subgroup is automatically a (closed) Lie subgroup. Remark 7.31 (manifold topology of GN Rd ) In equation (7.12) we deﬁned the metric ρ on 1 + tN Rd , induced from the norm |·|T N (Rd ) . By trivial restriction, ρ is also a metric on GN Rd , given by ρ (g, h) = max |π i (g − h)| , i=1,...,N

which induces the (sub)manifold topology on GN Rd .

7.5.2 Geodesic existence

Chow’s theorem tells us that for all elements g ∈ GN Rd , there exists a continuous path x of ﬁnite length such that SN (x)0,1 = g. One may ask for the shortest path (and its length) which has the correct signature. For instance, given a > 0, we can ask for the shortest path with signature 0 0 a exp + ∈ G2 R2 , 0 −a 0 or, equivalently, the shortest path in R2 which ends where it starts and wipes out area a. As is well known from basic isoperimetry, the shortest such path √ is given by a circle (with area a) with easily computed length given by 2 πa. With this motivating example in mind we now state Theorem 7.32 (geodesic existence) For every g ∈ GN Rd , the socalled “Carnot–Caratheodory norm”8 1 g := inf |dγ| : γ ∈ C 1-var [0, 1] , Rd and SN (γ)0,1 = g 0

is ﬁnite and achieved at some minimizing path γ ∗ , i.e. 1 g = |dγ ∗ | and SN (γ ∗ )0,1 = g. 0

7 See,

for example, Warner [175].

d usual, with Euclidean structure so that 01 |dγ| is the length of R isd equipped 1 -va r d [0, 1] , R , based on the Euclidean distance on R . γ∈C 8 As

7.5 Free nilpotent groups

145

Moreover, this minimizer can (and will) be parametrized to be Lipschitz (i.e. 1-H¨ older) continuous and of constant speed, i.e. |γ˙ ∗ (r)| ≡ (const) for a.e. r ∈ [0, 1]. Remark 7.33 By invariance of length and signatures under reparametrization, γ ∗ need not by deﬁned on [0, 1] but may be deﬁned for any interval [s, t] with non-empty interior. Proof. From Chow’s theorem, the inf is taken over a non-empty set so that g < ∞. By deﬁnition of inf, there is a sequence (γ n ) with signature g and we can assume (by reparametrization, cf. Proposition 1.38) that each γ n = γ n (t) has a.s. constant speed |γ˙ n | ≡ |γ n |1-H¨o l;[0,1] = cn where cn is the length of the path γ n· and cn ↓ g . Clearly, sup |γ n |1-H¨o l;[0,1] = sup cn < ∞ n

n

and from Arzela–Ascoli, after relabelling the sequence, γ n converges uniformly to some (continuous) limit path γ ∗ . By Lemma 1.23, |γ ∗ |1-H¨o l;[0,1] ≤ lim inf |γ n |1-H¨o l;[0,1] n

(7.18)

which shows in particular that γ ∗ itself is 1-H¨older, hence absolutely continuous, so that 1 1 |dγ ∗ | = |γ˙ ∗t | dt. 0

0

From basic continuity properties of the signature (Proposition 7.15) g ≡ SN (γ n )0,1 → SN (γ ∗ )0,1 which shows that SN (γ ∗ )0,1 = g. It remains to see that g =

1

|γ˙ ∗t | dt.

0

1 First, g ≤ 0 |γ˙ ∗t | dt is obvious from the deﬁnition of g . On the other hand, using (7.18) we have 0

1

|γ˙ ∗t | dt = |γ ∗ |1-H¨o l;[0,1] ≤ lim inf cn = g

and the proof is ﬁnished.

n

Free nilpotent groups

146

7.5.3 Homogenous norms

Let us now deﬁne the important concept of a homogenous norm on GN Rd . Deﬁnition 7.34 A homogenous norm is a continuous map |||.||| : GN Rd → R+ which satisﬁes (i) |||g||| = 0 if and only if g equals the unit element 1 ∈ GN Rd , (ii) homogeneity with respect to the dilation operator δ λ , |||δ λ g||| = |λ| . |||g||| for all λ ∈ R.

A homogenous norm is said to be symmetric if |||g||| = g −1 , and sub additive if |||g ⊗ h||| ≤ |||g||| + |||h|||. Remark 7.35 If |||.||| is a non-symmetric homogenous norm, then g → |||g||| + g −1 is a symmetric homogenous norm. Sub-additivity is pre served under such symmetrization. Proposition 7.36 Every symmetric sub-additive homogenous norm |||.||| leads to a genuine metric GN Rd via (g, h) → g −1 ⊗ h . Moreover, this metric is left-invariant.9 Proof. We write d (g, h) = g −1 ⊗ h . Property (i) in Deﬁnition 7.34 implies d (g, h) = 0 iﬀ g = h. Sub-additivity of |||.||| implies the triangle inequality for d and symmetry |||.||| implies d (g, h) = d (h, g). At last, left-invariance of d, i.e. d (g ⊗ h, g ⊗ k) = d (h, k) , follows from (g ⊗ h)

−1

⊗ (g ⊗ k) = h−1 ⊗ k.

Example 7.37 The simplest example of a homogenous norm is the map 1/i g ∈ GN Rd → |||g||| = max |π i (g)| . i=1,...,N

In general, it is neither symmetric nor sub-additive.

Exercise 7.38 Prove that −1/N 1/i maxi=1,...,N (i! |π i (g)|) is a sub(i) |||.|||1 : g ∈ GN Rd → (N !) additive homogenous norm. 1/i is a symmetric (ii) |||.|||2 : g ∈ GN Rd → maxi=1,...,N |π i (log g)| homogenous norm. 9 in general not right-invariant. A right-invariant metric could be deﬁned by . . . but g ⊗ h −1 .

7.5 Free nilpotent groups

147

Exercise 7.39 Compute the minimal length of all paths with signature x 0 a exp + . y −a 0 (This is precisely the Carnot–Caratheodory norm of (x, y, a) ∈ H, the 3dimensional Heisenberg group.) Check that this gives 5 x2 + y 2 when a = 0, 5 2 π |a| when x = y = 0. (See [157] for instance.)

7.5.4 Carnot–Caratheodory metric We now check that the Carnot–Caratheodory norm ·, which we introduced in Theorem 7.32, deﬁnes a homogenous norm in the sense of Deﬁnition 7.34. The geodesic existence result came with a map, the Carnot–Caratheodory d N R → [0, ∞). In conjunction with the group structure norm, from G of GN Rd , it is only a small step to deﬁne a genuine metric on GN Rd . Proposition 7.40 Let g, h ∈ GN Rd . We have (i) g = 0 if and only if g = 1, the unit element in GN Rd ; (ii) homogeneity δ λ g g for all λ ∈ R; = |λ| (iii) symmetry g = g −1 ; (iv) sub-additivity g ⊗ h ≤ g + h; (v) continuity: g → g is continuous. Proof. Notation: for g ∈ G let γ ∗g = γ ∗ denote an arbitrary minimizer from the geodesic existence theorem. (i) If g = 0, γ ∗g has almost everywhere zero derivative, hence g = SN γ ∗g 0,1 = 1. If g = 1, it is obvious that g = 0. (ii) The case λ = 0 is easy, so we assume λ = 0. The path λγ ∗g satis ﬁes SN λγ ∗g 0,1 = δ λ g. Hence δ λ g ≤ length λγ ∗g = |λ| × length γ ∗g = |λ| g. The opposite inequality follows from replacing λ by 1/λ and g by δ λ g. ← − (iii) Using the fact that SN γ ∗g = g −1 we obtain 0,1

− −1 g ≤ length ← γ ∗g = length γ ∗g = g . The opposite inequality follows from replacing g by g −1 . (iv) If γ ∗g , γ ∗h denote the resp. geodesics then, from Chen’s theorem, g ⊗ h = SN γ ∗g ,h 0,1

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where γ ∗g ,h is the (Lipschitz continuous) concatenation of γ ∗g and γ ∗h with obvious length g + h. Hence, g ⊗ h must be less than or equal to the length of γ ∗g ,h . (v) Consider a sequence gn such that |gn − g| →n →∞ 0. (Here |·| denotes a norm topology on on the tensor algebra which induces the “original” −1 GN Rd .) By continuity of the group operations ⊗ and (·) , all of which are polynomial in the coordinates, gn−1 ⊗ g → 1 is an obvious consequence. From sub-additivity, |gn − g| ≤ gn−1 ⊗ g and since gn−1 ⊗ g is dominated by the length of any path with correct signature (namely gn−1 ⊗ g), it follows from Corollary 7.29 that −1 gn ⊗ g → 0. As a consequence, |gn − g| → 0 which implies continuity. Deﬁnition 7.41 The Carnot–Caratheodory norm on GN Rd induces (viaProposition 7.36) a genuine (left-invariant, continuous)10 metric d on N d G R , called the Carnot–Caratheodory metric. N d The space G R , d is not only a metric but a geodesic space (in the sense of Deﬁnition 5.17). To this end recall that, given g ∈ GN Rd , Theorem 7.32 provides us with an associated Lipschitz path γ ∗ : [0, 1] → Rd of minimal length11 equal to g such that t ∈ [0, 1] → SN (γ ∗ )0,t ∈ GN Rd connects the unit element in GN Rd with g. Proposition 7.42 GN Rd equipped with Carnot metric d is a geodesic space. Given g,h ∈ GN Rd , a connecting geodesic is given by t ∈ [0, 1] → Υt := g ⊗ SN (γ ∗ )0,t where γ ∗ is the geodesic associated with g −1 ⊗ h. Proof. Obviously, Υ is continuous and Υ0 = g, Υ1 = h. For any s < t in [0, 1] , d (Υs , Υt ) := SN (γ ∗ )s,t t |dγ ∗ | (7.19) ≤ s

1

|dγ ∗ |

=

(t − s)

=

(t − s) g −1 ⊗ h = |t − s| d (g, h) . 0

1 0 By 1 1 By

Proposition 7.40, part (v). reparametrization, the speed |γ˙ ∗t | may be taken constant for a.e. t ∈ [0, 1].

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In fact, the inequality cannot be strict; there would be a strict inequality in d (g, h)

≤ d (Υ0 , Υs ) + d (Υs , Υt ) + d (Υt , Υ1 ) ≤ (|s| + |t − s| + |1 − t|) d (g, h) = d (g, h)

which is not possible. We conclude that equality holds in (7.19), which shows that Υ is the desired connecting geodesic. Remark 7.43 [sub-Riemannian structure of GN Rd ] The geodesic constructed above satisﬁes the diﬀerential equation dΥt =

d

Ui (Υt ) dγ i

i=1

where the Ui (g) = g ⊗ei , i = 1, . . . , d are easily seen to be left-invariant vector ﬁelds on GN Rd . In fact, Lie [U1 , . . . , Ud ] |g = Tg GN Rd Lie algefor all g ∈ GN Rd , where Lie[. . . ] stands for the generated bra and Tg GN Rd denotes the tangent space to GN Rd at the point g. Chow’s theorem can now be understood as the statement that any two tangent to the points in GN Rd can be joined by a path that remains {U1 , . . . , Ud }. A sub-Riemannian metric on Tg GN Rd is given by declaring the {U1 , . . . , Ud } to be orthonormal. This induces a natural length for any path which remains tangent to span{U1 , . . . , Ud }. This applies in particular to the geodesic Υ of the previous proposition (Υ0 = g, Υ1 = h) and this natural length is precisely the Carnot–Caratheodory distance d (g, h).

7.5.5 Equivalence of homogenous norms

Similar to the case of norms on Rd , all homogenous norms on GN Rd are equivalent. The proof relies crucially on the continuity of homogenous norms (which was part of their deﬁnition). Theorem 7.44 All homogenous norms on GN Rd are equivalent. More precisely, if .1 and .2 are two homogenous norms, there exists C ≥ 1 such that for all g ∈ GN Rd , we have 1 g1 ≤ g2 ≤ C g1 . C

(7.20)

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Proof. It is enough to consider the case when g1 is given by 1/i

|||g||| := max |π i (g)| i=1,...,N

.

Let B = g ∈ GN Rd , |||g||| = 1 . Clearly, B is a compact set by continuity and .2 attains a (positive) minimum m and maximum M, i.e. for all g ∈ B, m ≤ g2 ≤ M. Since (7.20) holds trivially true when g = 1, the unit element of GN Rd , we only need to consider g = 1. We deﬁne ε = 1/ |||g||| so that |||δ ε g||| = 1. In particular, m ≤ ||δ ε g||2 ≤ M and by using homogeneity of .2 we obtain m ≤ ||g||2 / |||g||| ≤ M and (7.20) follows. Let us recall that the metric ρ on GN Rd , induced from the norm |·|T N (Rd ) , is given by ρ (g, h) = |g − h| = max |π i (g) − π i (h))| . i=1,...,N

(7.21)

When h = 1 this reduces to |g − 1| = max |π i (g)| . i=1,...,N

d N R . Then, Proposition 7.45 Let |||.||| be a homogenous norm on G there exists a constant C > 0 such that for all g ∈ GN Rd 1 N N min |||g||| , |||g||| ≤ |g − 1| ≤ C max |||g||| , |||g||| C and

1 1/N 1/N min |g − 1| , |g − 1| ≤ |||g||| ≤ C max |g − 1| , |g − 1| . C Proof. By equivalence of homogenous norms, it suﬃces to consider the case when 1/i |||g||| := max |π i (g)| . i=1,...,N 1/N which implies that But then, obviously, |||g||| ≤ max |g − 1| , |g − 1| N min |||g||| , |||g||| ≤ |g − 1| . On the other hand

i N |g − 1| = max |π i (g)| ≤ max |||g||| = max |||g||| , |||g||| i=1,...,N

i=1,...,N

and together these imply all the stated inequalities.

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Corollary 7.46 The topology on GN Rd induced by Carnot–Caratheodory distance (in fact, by any metric associated with a symmetric sub-additive homogenous norm) coincides with the original12 topology of GN Rd . Proof. homogenous norm on −1 sub-additive Let |||.||| be any symmetric g ⊗ h for the associated metric. Given GN Rd and write d (g,h) = d N d N a sequence −1 (gn ) ⊂ G R and g ∈ G R , Proposition 7.45 implies that gn ⊗ g − 1 → 0 if and only if d (gn , g) = gn−1 ⊗ g → 0. On the other hand, we saw in Proposition 7.18 that gn−1 ⊗ g − 1 → 0 if and only if |gn − g| → 0. Remark 7.47 There are more geometric arguments for this. A Riemannian taming argument easily gives that convergence with respect to the CC distance implies convergence in the original topology. For the converse, continuity of the CC norm implies gn−1 ⊗ g → 0 as gn−1 ⊗ g → 1, which (by Proposition 7.18) is equivalent to gn → g in the original topology. We can improve Corollary 7.46 towards a quantitative comparison of the Carnot–Caratheodory distance with the “Euclidean” distance on GN Rd as given in (7.21). To this end, we need Lemma 7.48 Let g, h ∈ GN Rd , of form g = 1 + g 1 + · · · + g N , g i ∈ d ⊗i ⊗k R and similarly for h. The following equations then hold in Rd k=1, . . . , N : k −1 k −i −1 k g g ⊗h = ⊗ (hi − g i ) (7.22) i=1

and hk − g k =

k

i g k −i ⊗ g −1 ⊗ h .

(7.23)

i=1

Proof. Set g 0 = h0 = 1. By deﬁnition of the tensor product in GN Rd ⊂ k k k −i T N Rd , g −1 ⊗ h = i=0 g −1 ⊗hi . The result follows by subtract −1 k −i i k k ing from the previous expression 0 = g ⊗ g = i=0 g −1 ⊗g . The other equality follows from h − g = g ⊗ g −1 ⊗ h − 1 . Proposition 7.49 (ball-box estimate) Consider g, h ∈ GN Rd . There exists a constant C = C (N ) > 0 such that 1/N 1− 1 (7.24) d(g, h) ≤ C max |h − g| , |h − g| max 1, g N 1 2 Cf.

Remark 7.31.

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and

N −1 N , d (g, h) . |h − g| ≤ C max d (g, h) max 1, g

(7.25)

In particular, recalling from (7.21) that ρ (g, h) ≡ |h − g|, Id : GN Rd , d GN Rd , ρ is Lipschitz on bounded sets in → direction and 1/N -H¨ older on bounded sets in ← direction. Proof. Equation (7.22) implies k −1 g ⊗h

≤ c1

k −1 k −i i g . h − g i i=1

=

c1

k

k −i

g

. hi − g i

by symmetry of ·

i=1

k −1 ≤ c2 |h − g| max 1, g . Hence, k 1/k max g −1 ⊗ h

! " 1− 1 1/k max max 1, g k |h − g| k =1,...,N 1/N 1− 1 . max 1, g N ≤ c4 max |h − g| , |h − g|

≤

k

c3

Conversely, from (7.23), k k k −i g −1 ⊗ hi . h − g k ≤ c5 g i=1

Hence, |h − g| ≤ c6

k N

k −i

g

−1 g ⊗ hi

k =1 i=1

≤

c7

N

i N −i d (g, h) max 1, g

i=1

N −1 N , d (g, h) . ≤ c8 max d (g, h) max 1, g

Corollary the Carnot–Caratheodory distance on 7.50 Let d denote GN Rd . Then GN Rd , d is a Polish space in which closed bounded sets are compact.

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Proof. Completeness, and compactness of closed, bounded sets separability are obvious for GN Rd under ρ, the metric induced from |·|T N (Rd ) . It then suﬃces to apply the previous proposition. Exercise 7.51 (i) Let x ∈ Rd . Show that exp (x) = |x|, the Euclidean length of x. (ii) Assume .1 is a sub-additive, homogenous norm on GN Rd such that for all x ∈ Rd , (∗) : exp (x)1 = |x| . Show that g1 ≤ g for all g ∈ GN Rd . This says that the Carnot– Caratheodory norm is the largest sub-additive, homogenous norm which satisﬁes (∗). Solution. Let g ∈ GN Rd , γ a geodesic associated with g, and (γ n ) a sequence of piecewise linear approximations. Then, if (tni ) are the discontinuity points of the derivative of γ n , 4 SN (γ n )t n ,t n SN (γ n )0,1 = i i+ 1 1 i 1 n ≤ SN (γ )t ni ,t ni+ 1 1

i

=

exp γ ntni ,t ni+ 1

=

γ ntni ,t ni+ 1

i

1

i 1

|dγ nu | .

= 0

Letting n tend to ∞, we have by continuity of the map SN (.)0,1 , which follows from Theorem 3.15, 1 |dγ u | , SN (γ)0,1 ≤ 1

0

which reads g1 ≤ g .

7.5.6 From linear maps to group homomorphisms Linear maps from Rn to Rd can always be written as x → Ax where A is a n × d matrix. With a slight abuse of notation, we will call A the linear n map itself. It isobvious that A is a homomorphism from the group (R , +) d into the group R , + (in fact, the linear maps describe the set of all such homomorphisms), i.e. that for all x, y ∈ Rn , A (x + y) = Ax+Ay. It will be useful to extend A to a homomorphism from GN (Rn ) to GN Rd . To this

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end, we recall that tN (Rn ) is generated by Rn in the sense that a vector space basis of tN (Rn ) is given by # m $ 4 N ∪m =1 ej i : ji ∈ {1, . . . , n} i=1 n where e1 , . . . , en is the canonical basisof R . We can then (uniquely) extend N n N d A to a homomorphism t (R ) → t R by requiring that it is compatible with ⊗, i.e. m m 4 4 ⊗m A ej i := (Aej i ) ∈ Rd ⊂ tN Rd i=1

i=1

and then extend A by linearity to all of tN (Rn ). On the other hand, tN (Rn ) is a Lie algebra with bracket [a, b] = a ⊗ b − b ⊗ a and so A is clearly compatible with the bracket, which is to say a Lie algebra homomorphism. From the Campell–Baker–Hausdorﬀ formula, A (·) := exp (A (log (·))) is thena group homomorphism between the Lie groups 1 + tN (Rn ) and N d 1 + t R . Equivalently, one can deﬁne directly A 1 + a1 + · · · + aN := 1 + Aa1 + A⊗2 a2 + · · · + A⊗N aN , ⊗k ⊗k ⊗k where ak = π k (a) ∈ (Rn ) and A⊗k : (Rn ) → Rd is deﬁned by linearity from A⊗k (ej 1 ⊗ · · · ⊗ ej k ) := Aej 1 ⊗ · · · ⊗ Aej k with ji ∈ {1, . . . , n} , and check that this deﬁnes a group homomorphism. By sheer restriction, this yields the group homomorphism A between GN (Rn ) and GN Rd . That said, we will ﬁnd it convenient in the sequel to have a direct construction of A based on step-N signatures. We have n d Proposition 7.52 Let A be a linear map from dR into R . There exists a N n N unique homomorphism from G (R ) to G R , denoted by A, such that for all x ∈ Rd , A exp (x) = exp (Ax) .

For all g ∈ GN (Rn ) we have13 Ag ≤ |A|op g and if g ∈ GN (Rn ) is written as the step-N signature of some x ∈ C 1-var ([0, 1] , Rn ), i.e. g = SN (x)0,1 , then Ag = ASN (x)0,1 = SN (Ax)0,1 . 1 3 |·| op

denotes the operator (matrix) norm.

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155

Proof. Let x, x ˜ be continuous paths of bounded variation such that g = x)0,1 . Then, writing SN (Ax)0,1 and SN (A˜ x)0,1 in coorSN (x)0,1 = SN (˜ dinates shows that they are equal. Hence, it is possible to deﬁne Ag = SN (Ax)0,1 . We establish is a homomorphism, which may be done by checking that A−1 A g −1 ⊗ h = (Ag) ⊗ Ah for arbitrary elements g, h ∈ GN (Rn ) , which we may assume to be of form g = SN (x)0,1 , h = SN (y)0,1 (x, y are continuous paths of bounded variation). We recall that g −1 = −) , the signature of ← − = x (1 − ·), and deﬁne z to be the concatex x SN (← 0,1 − and y. Then, we have nation of ← x A g −1 ⊗ h = SN (Az)0,1 . ←− − and Ay and the On the other hand, Az is the concatenation of Ax = A← x proof is ﬁnished by observing that ←− −1 SN (Az)0,1 = SN Ax ⊗ SN (Ay)0,1 = (Ag) ⊗ Ah. 0,1

Finally, we discuss the estimate on Ag. Let γ : [0, 1] → Rd be a geodesic

1 path associated with g, i.e. a path such that SN (γ)0,1 = g and 0 |dγ| = g . Then, Ag = SN (Aγ)0,1 ≤

0

1

|d (Aγ)| ≤ |A|op

0

1

|dγ| = |A|op g

and the proof is ﬁnished. Example 7.53 One simple linear map from Rn ⊕Rn into Rn is the addition map, i.e. plus (x, y) = x+y where x, y ∈ Rn . It extends to a homomorphism plus from GN (Rn ⊕ Rn ) into GN (Rn ) .

Example 7.54 Another simple linear map from Rd ⊕ Rd onto Rd is the projection ponto the ﬁrst It then extends to a homomorphism d coordinates. d N d d N R ⊕R into G R . For example, if (x, h) is a Rd ⊕ Rd p from G valued path, p ◦ SN (x, h)0,1 = SN (x)0,1 .

Exercise 7.55 Another simple linear map from Rd to Rd is the map x → λx for a given λ ∈ R. Prove that its homomorphism extension is the re striction of the dilation map δ λ to GN Rd .

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Exercise 7.56 Consider for λ = (λi )1≤i≤d ∈ Rd , the map δ λ : 1 + tN Rd → 1 + tN Rd deﬁned by   N δ λ 1 + xi 1 ,...,i k ei 1 ⊗ . . . ⊗ ei k  k =1 1≤i 1 ,...,i k ≤d

=1+

N

xi 1 ,...,i k λi 1 . . . λi k ei 1 ⊗ . . . ⊗ ei k .

k =1 1≤i 1 ,...,i k ≤d

d d (i) Prove that δ λ is the extension of the linear map i=1 xi ei → i=1 xi λi ei . (ii) Provethat if all the λi are equal to some scalar, then the restriction of δ λ to GN Rd is the dilation map from Exercise 7.55. Exercise 7.57 Show that sup g ∈G N (Rn ),g > 0

Ag / g = |A|op .

Solution. ≤ is clear from Proposition 7.52 and equality is achieved at g = exp (x) where x ∈ Rn , non-zero, is such that |Ax| / |x| = |A|op . Exercise 7.58 Prove that for all (λ, g) ∈ Rd+ × GN Rd , δ λ (g) ≤ (maxi=1,...,d λi ) g and g ≤ (maxi=1,...,d 1/λi ) δ λ (g) .

7.6 The lift of continuous bounded variation paths on Rd 7.6.1 Quantitative bound on SN Recall from Section 7.2.1 that SN maps a continuous path x of ﬁnite 1variation with values in Rd to a path {t → SN (x)t ≡ SN (x)0,t } simply by computing all iterated (Riemann–Stieltjes) integrals up to order N . Recall also that SN (x) wasNseen d to take values in the (free, step-N nilpotent) N d R ⊂ T R . We call SN (x) the canonical lift of x to a group G GN Rd -valued path, since14 π 1 SN (x)0,t = x0,t for all t ∈ [0, T ]. As we shall now see, SN (x) is not only of ﬁnite length (i.e. 1-variation) with respect to Carnot–Caratheodory metric on GN Rd but has the same length as the Rd -valued path x. 14 π

1

S N (x)t = xt for all t ∈ [0, T ] only holds if x (0) = 0.

7.6 The lift of continuous bounded variation paths

157

Proposition 7.59 Let x ∈ C 1-var [0, T ] , Rd . Then, SN (x)1-var;[0,T ] = |x|1-var;[0,T ] . Proof. From the very deﬁnition of the Carnot–Caratheodory norm, ≥ |x (x) SN s,t | for all 0 ≤ s < t ≤ T and thus s,t d (SN (x)s , SN (x)t ) = SN (x)s,t ≥ |xs,t | . Clearly then, SN (x)1-var;[s,t] ≥ |x|1-var;[s,t] for all 0 ≤ s < t ≤ T . Conversely, t |dx| = |x|1-var;[s,t] SN (x)s,t ≤ s

and since (s, t) → |x|1-var;[s,t] is a (super-additive) control function, we immediately obtain that for all 0 ≤ s < t ≤ T , SN (x)1-var;[s,t] ≤ |x|1-var;[s,t] and the proof is ﬁnished. Exercise 7.60 The purpose of this exercise is to replace C 1-var -regularity in Proposition 7.59 by W 1,p -regularity with p ∈ (1, ∞). Following Section 1.4.2, the space of all x : [0, T ] → GN Rd with xW 1 , p ;[0,T ] :=

sup

D ⊂[0,T ] i:t ∈D i

xt

i ,t i + 1

p 1/p p−1

|ti+1 − ti |

<∞

is denoted by W 1,p [0, T ] , GN Rd , a subset of C 1-var [0, T ] , GN Rd . Let x ∈ W 1,p [0, T ] , Rd and recall from Section 1.4.1 that its W 1,p -(semi-) norm is given by 1/p T

|x|W 1 , p ;[0,T ] =

p

|x˙ t | dt

.

0

Show that SN (x)W 1 , p ;[0,T ] = |x|W 1 , p ;[0,T ] . [0, T ] , G2 Rd is important as it allows an intrinsic (The case of W deﬁnition of the rate function of enhanced Brownian motion viewed as a symmetric diﬀusion process on G2 Rd .) 1,2

Solution. From the results in Section 1.4.1 we know that x is the indeﬁnite integral of some x˙ ∈ Lp [0, T ] , Rd and that T xt ,t p p p i i+ 1 = |x˙ t | dt. |x|W 1 , p ;[0,T ] = sup p−1 D ⊂[0,T ] i:t ∈D |ti+1 − ti | 0 i

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Next, SN (x)W 1 , p ;[0,T ] ≥ |x|W 1 , p ;[0,T ] follows readily from SN (x)s,t ≥ |xs,t |. For the converse inequality, we ﬁrst observe that t 1−1/p |x˙ u | du ≤ |x|W 1 , p ;[s,t] |t − s| . SN (x)s,t ≤ s

Hence p SN (x)s,t |t − s|

p−1

p

p

p

≤ |x|W 1 , p ;[s,t] =⇒ SN (x)W 1 , p ;[s,t] ≤ |x|W 1 , p ;[s,t] , p

where we used super-additivity of (s, t) → |x|W 1 , p ;[s,t] . We aim now to show that continuous GN Rd -valued paths of bounded 1-var variation d are in one-to-one correspondence with elements of C [0, T ] , R . We ﬁrst need a simple lemma; recall that o ≡ 1 stands for the unit element in GN Rd . We note that the projection map πd 0,k (cf. k(7.11)) N may be restricted to yield a projection map π 0,k : G R → G Rd , whenever k ≤ N . Lemma 7.61 Let x, y be two elements of Co [0, T ] , GN Rd such that π 0,N −1 (x) = π 0,N −1 (y) . Then, the path h deﬁned by d N R ht := log x−1 t ⊗ yt ∈ g

is such that for some constant C depending only on N, for all s, t ∈ [0, T ] , N

|hs,t | ≤ C (xs,t + ys,t ) . d ⊗N is non-zero so that Proof. Note that only the projection of h to R exp (ht ) commutes with all elements in GN Rd . In particular, ys,t

= = = =

ys−1 ⊗ yt exp (−hs ) ⊗ x−1 s ⊗ xt ⊗ exp (ht ) −1 xs ⊗ xt ⊗ exp (−hs ) ⊗ exp (ht ) xs,t ⊗ exp (hs,t ) .

Using the equivalence of homogenous norms, we obtain |hs,t |

N

≤ c exp (hs,t ) N ≤ c x−1 ⊗ y s,t s,t ≤

N

c (xs,t + ys,t ) .

7.6 The lift of continuous bounded variation paths

159

Theorem 7.62 Let N ≥ 1 and x ∈ Co1-var [0, T ] , Rd . Then x = SN (x) is the unique “lift” ofx in the sense that π 1 (x) = x and such that x ∈ Co1-var [0, T ] , GN Rd . Moreover, SN : Co1-var [0, T ] , Rd → Co1-var [0, T ] , GN Rd is a bijection with inverse π 1 and, for all 0 ≤ s < t ≤ T , x1-var;[s,t] = |x|1-var;[s,t] . Proof. It is obvious that x = SN (x) has the lifting property, i.e. that 1-var d [0, T ] , R , and from Proposition 7.59 on C π 1 ◦SN is the identity map o it enough to we see that x ∈ Co1-var [0, T ] , GN Rd . To see uniqueness is show that SN ◦ π 1 is the identity map on Co1-var [0, T ] , GN Rd . This is trivially true when N = 1. By induction, we now assume the statement is true at level N − 1, for N ≥ 2. Deﬁne y = SN ◦ π 1 (x) and the path h by d N R . ht := log x−1 t ⊗ yt ∈ g N

N

From Lemma 7.61, |hs,t | ≤ c (xs,t + ys,t ) ≤ cω (s, t) where ω (s, t) is the super-additive (control) function x1-var;[s,t] + y1-var;[s,t] . In particular, h is of ﬁnite N1 -variation. As 1/N < 1 and h0 = 0, this implies that h ≡ 0, i.e. that y = x.

7.6.2 Modulus of continuity for the map SN

Proposition 7.63 Let x1 , x2 ∈ C 1-var [0, T ] , Rd , and ≥ maxi=1,2 i x . Then, for all N ≥ 1, 1-var;[s,t] ∃CN : ∀0 ≤ s < t ≤ T : π N SN x1 s,t − SN x2 s,t ≤ CN N −1 x1 − x2 1-var;[s,t] .

(7.26)

In particular, if ω is a ﬁxed control and ε is a positive real such that for all s, t ∈ [0, T ] , max xi 1-var;[s,t] ≤ ω (s, t) and x1 − x2 1-var;[s,t] ≤ εω (s, t) , i=1,2

we have15 max

sup

k =1,...,N 0≤s< t≤T

|π k (xs,t − ys,t )| k

ω (s, t)

≤ CN ε.

1 5 In the terminology of the forthcoming Deﬁnition 8.6, this is equivalent to ρ1 , ω S N x1 , S N x2 ≤ C N ε.

Free nilpotent groups

160

Proof. Obviously, (7.26) holds true with C1 = 1 for N = 1. We proceed by induction, assuming (7.26) holds. Then t π N +1 SN +1 x1 s,t −SN +1 x2 s,t = π N SN x1 s,r −SN x2 s,r ⊗dx1r s t + π N SN x2 s,r ⊗d x1r −x2r . s

From the induction hypothesis, sup π N SN x1 s,r − SN x2 s,r r ∈[s,t]

≤ CN

max xi

i=1,2

N −1 1-var;[s,t]

1 x − x2 , 1-var;[s,t]

hence the ﬁrst integral on the right-hand side above is estimated by t 1 2 1 π N SN x s,r − SN x s,r ⊗ dxr s

≤ CN

max xi 1-var;[s,t]

i=1,2

N

1 x − x2 . 1-var;[s,t]

N On the other hand, supr ∈[s,t] π N SN x2 s,r ≤ (1/N !) x2 1-var;[s,t] so that we can estimate the second integral on the right-hand side above: t 2 1 1 2 N 2 x 1-var;[s,t] x1 − x2 1-var;[s,t] . x S ⊗ d x π − x N N r r ≤ s,r N! s Combining the two estimates ﬁnishes the induction step and thus concludes the proof of the ﬁrst part of the proposition. The second part is an obvious corollary of the ﬁrst part. The previous proposition implies in particular that if x1 and x2 are two continuous paths of length bounded by 1, and such that |x1 − x2 |1-var;[0,1] ≤ ε, then for all N ∈ {1, 2, 3, . . . }, max |π k (g1 − g2 )| ≤ CN ε

k =1,...,N

where gi := SN (xi )0,1 ∈ GN Rd , i = 1, 2. We now prove in some sense the converse statement; the basic idea of the proof of the following proposition is illustrated in Figure 7.2. Proposition 7.64 Let g1 , g2 ∈ GN Rd , with g1 , g2 ≤ C1 and max |π k (g1 − g2 )| ≤ ε, k = 1, . . . , N.

k =1,...,N

7.6 The lift of continuous bounded variation paths

161

Then, there exists xi ∈ C 1-var [0, 1] , Rd , such that SN (xi )0,1 = gi , i = 1, 2, and a constant C2 = C2 (C1 , N ), such that max |xi |1-var;[0,1] ≤ C2 ,

i=1,2

|x1 − x2 |1-var;[0,1] ≤ εC2. Proof. First case: Assume that π 1,N −1 (g1 ) = π 1,N −1 (g2 ) = 0. In such a case, we can write gi = exp (i ) = 1 + i ; the hypothesis implies that |i | ≤ c1 , and that 2 = 1 + εm, with |m| ≤ c2 . We write g1 and g2 in the following way: g1 g2

= exp (1 − m) ⊗ exp (m) , = exp (1 − m) ⊗ exp ((1 + ε) m) = exp (1 − m) ⊗ δ (1+ε) 1 / N exp (m) .

Deﬁne z : [0, 1] → Rd to be a geodesic associated with the group element exp (1 − m) (observe that the length of z is bounded by a constant independent of ε). Deﬁne also y : [0, 1] → Rd to be a geodesic associated with the group element exp (m) (the length of y is bounded by a constant independent of ε). Deﬁne x1 : [0, 2] → Rd to be the concatenation of z and y, and x2 : [0, 2] → Rd to be the concatenation of z and 1/N (1 + ε) ! y. Observe that " the 1-variation distance between x1 and x2 is 1/N equal to (1 + ε) − 1 times the length of y, i.e. it is bounded by a con 1/N − 1 ≤ ε. Reparametrizing the paths x1 and x2 to stant times (1 + ε) be from [0, 1] into Rd ﬁnishes the ﬁrst case. General case: We prove the general case by induction. The case N = 1 can be solved using the ﬁrst case (or more simply with d straight lines). N R we now prove Assuming that the proposition holds for elements in G d N +1 R . To this end, take two arbitrary that it also holds for elements in G d N +1 R with elements g1 , g2 ∈ G max

k =1,...,N +1

|π k (g1 ) − π k (g2 )| < ε.

Set c3 := g1 ∨ g2 and deﬁne hi ∈ GN Rd by projection of gi to the ﬁrst N levels, i.e. so that π 0,N (gi ) = π 0,N (hi ). Obviously hi ≤ c3 for i = 1, 2 and max |π k (h1 ) − π k (h2 )| < ε. k =1,...,N

By the induction hypothesis, there exist two paths z1 , z2 (which we may take to be deﬁned on [0, 1]) of length bounded by c4 , with the length of

Free nilpotent groups

162

4 z 2

10

0 0 5

y

5 x 10 0

Figure 7.2. We illustrate the basic idea of the proof of Proposition 7.64. Two points of G2 R2 , identiﬁed with the 3-dimensional Heisenberg group, are given as g1 = (9, 9, 1.5) and g2 = (8, 8.5, 2). The corresponding paths x1 , x2 are the concatenation of straight lines, connecting the origin with (9, 9) and (8, 8.5) respectively, followed by a circle which wipes out the prescribed area, 1.5 and 2 respectively.

z1 − z2 bounded by c4 ε, where c4 = c4 (N, c3 ), and with the property that SN (zi )0,1 = hi , i = 1, 2. We now deﬁne −1 ki = SN +1 (zi )0,1 ⊗ gi ∈ GN +1 Rd ; i = 1, 2. Clearly, ki ≤ c4 + c3 =: c5 and π 1,N (ki ) = 0. Also, from Proposition 7.63, we have, for all j ≤ N + 1, −1 −1 z−1 )0,1 − SN +1 (← z−2 )0,1 π j SN +1 (z1 )0,1 − SN +1 (z2 )0,1 = π j SN +1 (← ≤

c6 ε. It is easy to see that, for any a, b ∈ T N +1 Rd , π 0,N (a) = π 0,N (b) =⇒ a−1 ⊗ b = 1 + π N +1 a−1 + π N +1 (b) . −1 +π N +1 (gi ) , Applied in our context this gives ki = 1+π N +1 SN +1 (zi ) and hence, for all j ≤ N + 1, we have |π j (k1 − k2 )| ≤ (1 + c6 ) ε. Therefore, using the ﬁrst case, there exist two paths y1 , y2 (deﬁned on [0, 1]) of length bounded by c7 , with length of y1 −y2 bounded by c7 ε, and with the property that SN +1 (yi ) = ki ; i = 1, 2. We conclude the proof by observing that the paths xi = zi yi , (re)parametrized to [0, 1], satisfy SN +1 (xi ) = gi , are of length bounded by c8 , and with length of x1 − x2 bounded by c8 ε. The proof is now ﬁnished.

7.7 Comments

163

As a ﬁrst corollary of this powerful lemma, we prove a modulus of continuity for homomorphism on GN Rd that extends the linear map on Rd (see Section 7.5.6). Proposition 7.65 Let A be a linear map from Rn into Rd . Then, for g, h ∈ GN (Rn ) with g and h bounded by 1, we have for some constant C = C (N ) , |Ag − Ah| ≤ C |A|op |g − h| . Proof. Using Lemma 7.64, we take g = SN (x)0,1 and h = SN (y)0,1 , where x and y are some bounded variation paths of length bounded by some constant c1 and such that |x − y|1-var;[0,1] ≤ c2 |g − h| . Then, using Proposition 7.63, |Ag − Ah|

= |SN (Ax)0,1 −SN (Ay)0,1 | ≤ c3 |A (x − y)|1−var;[0,1] ≤

c3 |A|op |(x − y)|1−var;[0,1]

≤

c4 |g − h| .

7.7 Comments Section 7.2: The signature map goes back to the classical work of Chen [28, 29] and it was clear from his work that truncation at step N leads to (a presentation) of the step-N free nilpotent Lie group. This point of view was in particular adopted by Lyons [115, 116]; see Lyons et al. [123] for references. Section 7.3: All this is standard, see Lyons et al. [123] for references. The proof of the Campbell–Baker–Hausdorﬀ formula via diﬀerential equations is also well-known and appears, for instance, in Strichartz [157]. If one only cares about the qualitative statement Corollary 7.27, an algebraic proof is possible via Friedrich’s criterion, see Reutenauer [142] for instance. Section 7.4: In our setting, Chow’s theorem plays the role of a converse to the Campbell–Baker–Hausdorﬀ formula. See Baudoin [7], Folland and Stein [54] for related points of view. It can be formulated in sub-Riemannian geometry (Montgomery [131] and the references cited therein), see also Varopouols et al. [174]. Section 7.5: Again, geodesics are essentially a sub-Riemannian concept but the details are simpler in our setting. Equivalence of homogenous

164

Free nilpotent groups

norms, on the other hand, is a group concept and allows for considerable simplifcation when it comes to topological consistency of the Carnot– Caratheodory metric with the original topology. Exercise 7.39 is taken from Strichartz [157]. Section 7.6 contains some “lifting” estimates which correspond, in essence, to the case p = 1 in the forthcoming estimates for the Lyons lift (see Section 9.1). Proposition 7.64 appears to be new; the construction of “almost” geodesic paths associated with a pair of “nearby” group elements will be a key ingredient (cf. the proof of the forthcoming Theorem 10.26) in establishing local Lipschitzness of the Itˆ o–Lyons map.

8 Variation and H¨ older spaces on free groups In the general setting of a (continuous) path with values in a metric space, say x : [0, T ] → E, we deﬁned its p-variation “norm”over [0, T ] , in symbols |x|p-var;[0,T ] . This applies in particular to a E = GN Rd -valued path x (·), where GN Rd is the free step-N nilpotent group discussed at length in the previous chapter. As a constant reminder that the (Carnot–Caratheodory) metric d on GN Rd was derived from ·, the Carnot–Caratheodory norm, we shall then use the notation 1/p p sup d xt i , xt i + 1 (8.1) xp-var;[0,T ] = (t i )⊂[0,T ]

=

sup

i

1/p xt i ,t i + 1 p

(t i )⊂[0,T ]

i

and, thanks to homogeneity of · with respect to dilation, speak of a homogenous p-variation norm. As a special case of Deﬁnition 5.1, C p-var [0, T ] , GN Rd = x ∈ C [0, T ] , GN Rd : xp-var;[0,T ] < ∞ and we shall assume p ≥ 1 unless otherwise stated. When E = Rd , (8.1) is precisely the usual p-variation (semi-)norm and (x, y) → |x0 − y0 | + |x − y|p-var;[0,T ] deﬁnes a genuine metric, the p-variation metric, on path space. Recalling p that |x − y|p-var;[0,T ] is of the form sup (t i )⊂[0,T ]

xt

i ,t i + 1

p − yt i ,t i + 1

i

(with xs,t ≡ xt − xs ∈ Rd ), a convenient extension to a GN Rd -valued path is to replace |xs,t − ys,t | by d (xs,t , ys,t ), where now d N R . xs,t ≡ x−1 s ⊗ xt ∈ G p p/k Alternatively, we may replace xt i ,t i + 1 − yt i ,t i+ 1 by |π k (xs,t − ys,t )| for all k = 1, . . . , N, using the fact that GN Rd ⊂ T N Rd ; recalling that

166

Variation and H¨ older spaces on free groups

⊗k π k : T N Rd → Rd denotes projection to the kth tensor level. When N = 1, the two notations coincide; for N ≥ 2, they do not. However, both resulting p-variation distances remain “locally uniformly” comparable (and in particular induce the same topology and the same notion of Cauchy sequences) and both are ﬁrst allows us to discuss the properties useful. The of the space C p-var [0, T ] , GN Rd with often identical arguments as in the Rd case. The second arises naturally in Lipschitz estimates of the Lyons lift and the Itˆ o–Lyons map, discussed in later chapters. Of course, everything said here applies in a H¨ older context. In particular, the homogenous 1/p-H¨ older norm is given by x1/p-H¨o l;[0,T ] =

d (xs , xt )

sup 0≤s< t≤T

1/p

|t − s|

=

xs,t

sup

|t − s|

0≤s< t≤T

1/p

(8.2)

and

C 1/p-H¨o l [0, T ] , GN Rd = x ∈ C [0, T ] , GN Rd : x1/p-H¨o l;[0,T ] < ∞ .

8.1 p-Variation and 1/p-H¨older topology 8.1.1 Homogenous p-variation and H¨ older distances As usual in the discussion of p-variation, we assume p ≥ 1. We then have Deﬁnition variation and H¨ older distance) Given 8.1 (homogenous x, y ∈ C [0, T ] , GN Rd we deﬁne dp-var;[0,T ] (x, y) :=

sup D

d xt i ,t i + 1 , yt i ,t i + 1

p

1/p

t i ∈D

and d1/p-H¨o l;[0,T ] (x, y) :=

d (xs,t , ys,t )

sup 0≤s< t≤T

1/p

|t − s|

,

where it is understood that d0;[0,T ] (x, y) := d0-H¨o l;[0,T ] (x, y) :=

sup

d (xs,t , ys,t ) .

0≤s< t≤T

With o = 1, the unit element in GN Rd , we note that dp-var;[0,T ] (x, o) = xp-var;[0,T ] , the homogenous p-variation norm of x and similarly in the H¨ older case. It is obvious that (x, y) → dp-var;[0,T ] (x, y) resp. d1/p-H¨o l;[0,T ] (x, y)

(8.3)

8.1 p-Variation and 1/p-H¨ older topology

167

is non-negative, symmetric and one sees (precisely as the case of Rd -valued paths) that the triangle inequality is satisﬁed. However, when the right hand side above is 0 this only tells us that xt = c ⊗ yt with c = x−1 0 ⊗ y0 . If attention is restricted to paths with ﬁxed starting point, which is the 1/p-H¨o l [0, T ] , GN Rd , then (8.3) case for Cop-var [0, T ] , GN Rd resp. Co deﬁnes a genuine metric. Otherwise, it suﬃces to add the distance of the starting points, in which case (x, y) → d(x0 , y0 ) + dp-var;[0,T ] (x, y) resp. d(x0 , y0 ) + d1/p-H¨o l;[0,T ] (x, y) resp. C 1/p-H¨o l ([0, T ] , gives a genuine metric on C p-var [0, T ] , GN Rd N d G R . Let us observe that for any λ ∈ R, dp-var;[0,T ] (δ λ x, δ λ y) = |λ| dp-var;[0,T ] (x, y), d1/p-H¨o l;[0,T ] (δ λ x, δ λ y) = |λ| d1/p-H¨o l;[0,T ] (x, y), where δ λ denotes dilation by λ on GN Rd , which explains the terminology homogenous p-variation (resp. 1/p-H¨older) distance; we also note that (s, t) → dp-var;[s,t] (x, y)p is a control function. Deﬁnition 8.2 (Homogenous p-ω distance) (i) Given a control function ω on [0, T ] and x ∈ C [0, T ] , GN Rd we deﬁne1 the homogenous p-ω norm xp-ω ;[0,T ] :=

sup 0≤s< t≤T

d (xs , xt ) 1/p

ω (s, t)

=

sup 0≤s< t≤T

xs,t 1/p

(8.4)

ω (s, t)

and

C p-ω [0, T ] , GN Rd = x ∈ C [0, T ] , GN Rd : xp-ω ;[0,T ] < ∞ .

(ii) Given x, y ∈ C p-ω [0, T ] , GN Rd and a control function ω on [0, T ], we deﬁne d (xs,t , ys,t ) dp-ω ;[0,T ] (x, y) = sup . 1/p 0≤s< t≤T ω (s, t) 1 The deﬁnition of · p -ω would make sense for paths with values in p -ω ;[0 , T ] and C an abstract metric space.

168

Variation and H¨ older spaces on free groups

Remark 8.3 Whenever ω (s, t) = 0 for some s < t, the right-hand side of (8.4) is inﬁnity unless xs,t = 0, in which case we use the convention 0/0 = 0. Equivalently, one can deﬁne 1/p for all 0 ≤ s < t ≤ T . xp-ω ;[0,T ] = inf M ≥ 0 : xs,t ≤ M ω (s, t)

Similar remarks apply to our deﬁnition of dp-ω ;[0,T ] (x, y). It is clear from the deﬁnition that

(x, y) → d (x0 , y0 ) + dp-ω ;[0,T ] (x, y) is a metric on C p-ω [0, T ] , GN Rd and, as above, one can omit the term d (x0 , y0 ) if attention is restricted to paths pinned at time 0. Let us also observe, as an elementary consequence of super-additivity of controls, dp-var;[0,T ] (x, y) ≤ ω (0, T )

1/p

dp-ω ;[0,T ] (x, y);

(8.5)

in the special case ω (s, t) = t − s this reads dp-var;[0,T ] (x, y) ≤ T 1/p d1/p-H¨o l;[0,T ] (x, y). Proposition 8.4 For all x1 , x2 ∈ C p-var [0, T ] , GN Rd there exists a control ω with ω (0, T ) = 1 such that i x , ≤ 31/p xi p-var;[0,T ] , i = 1, 2; p-ω ;[0,T ] dp-ω ;[0,T ] (x1 , x2 ) ≤ 31/p dp-var;[0,T ] (x1 , x2 ). Proof. Given x, y ∈ C p-var [0, T ] , GN Rd deﬁne the control ω x,y (s, t) :=

dp-var;[s,t] (x, y) dp-var;[0,T ] (x, y)

p

p

(using the convention 0/0 = 0 if necessary). Note ω x,o (s, t) = xp-var;[s,t] / p xp-var;[0,T ] where o denotes the trivial path constant equal to the unit element in GN Rd . Then ω (s, t) :=

1 1 1 ω x 1 ,x 2 (s, t) + ω x 1 ,o (s, t) + ω x 2 ,o (s, t) 3 3 3

has the desired properties. For instance, 1/p

d(x1s,t , x2s,t ) ≤ dp-var;[s,t] (x1 , x2 ) ≤ dp-var;[0,T ] (x1 , x2 )ω x 1 ,x 2 (s, t)

≤(3ω (s,t)) 1 / p

and similar for x1 , o and x2 , o.

8.1 p-Variation and 1/p-H¨ older topology

169

We now show that a map which is uniformly continuous on bounded sets in the metric dp-ω for all possible control functions ω, is also uniformly continuous on bounded sets in the metric dp-var . To this end, for paths x ∈ C [0, T ] , GM Rd we deﬁne the (homogenous) balls x : dp-ω ;[0,T ] (x, o) < R , Bp-var (R) = x : dp-var;[0,T ] (x, o) < R . Bp-ω (R) =

Corollary 8.5 Consider a map φ : C p-var [0, T ] , GM Rd → C p-var [0, T ] , GN (Re ) . Assume that for any control function ω, any ε > 0 and R > 0 there exists δ = δ (ε, R; ω) such that x, y ∈ Bp-ω (R) , dp,ω (x, y) < δ =⇒ dp-ω (φ (x) , φ (y)) < ε. Then, for any ε > 0 and R > 0 there exists η = η (ε, R) such that x, y ∈ Bp-var (R) , dp-var (x, y) < η =⇒ dp-var (φ (x) , φ (y)) < ε. In fact, we can choose η = δ (ε, CR; ω) /C with ω deﬁned as in Proposition older on bounded 8.4 and C = 31/p . (This shows that if φ is (locally) γ-H¨ sets in the metric dp-ω , for all possible control functions ω, it is also (locally) γ-H¨ older on bounded sets in the metric dp-var .) Proof. Given ε > 0 and R > 0 and x, y ∈ Bp-var (R) we take the corresponding with the properties as stated in Proposition 8.4. Taking control ω η = δ ε, 31/p R; ω /31/p then shows that dp-var (x, y) < η =⇒ dp-ω (x, y) < δ so that dp-ω (φ (x) , φ (y)) < ε and we conclude with (8.5).

8.1.2 Inhomogenous p-variation and H¨ older distances Our deﬁnition of homogenous (variation, H¨ older, ω-modulus) distance was based on measuring the distance of increments xs,t , ys,t ∈ GN Rd using the Carnot–Caratheodory distance. Alternatively, recalling that GN Rd ⊂ T N Rd we can use the (vector space) norm deﬁned on the latter which leads to the distance of increments given by |xs,t − ys,t |T N (Rd ) =

max |π k (xs,t − ys,t )| .

k =1,...,N

Observe that, for N > 1, this distance is not homogenous with respect to dilation on GN Rd .

Variation and H¨ older spaces on free groups

170

Deﬁnition 8.6 (inhomogenous older distance) variation and H¨ Given x, y ∈ C [0, T ] , GN Rd we deﬁne (i) for k = 1, . . . , N, (k ) ρp-var;[0,T ]

(x, y) =

π k xt

sup (t i )⊂[0,T ]

i ,t i + 1

− yt i ,t i + 1

p/k

k /p

i

and ρp-var;[0,T ] (x, y) =

(k )

max ρp-var;[0,T ] (x, y) ;

k =1,...,N

(ii) for any control function ω on [0, T ], for k = 1, . . . , N, (k )

ρp-ω ;[0,T ] (x, y) =

sup

|π k (xs,t − ys,t )|

0≤s< t≤T

k /p

ω (s, t)

and ρp-ω ;[0,T ] (x, y) =

(k )

max ρp-ω ;[0,T ] (x, y) ;

k =1,...,N

(iii) for k = 1, . . . , N, (k )

ρ1/p-H¨o l;[0,T ] (x, y) =

sup

|π k (xs,t − ys,t )| |t − s|

0≤s< t≤T

k /p

and ρ1/p-H¨o l;[0,T ] (x, y) =

(k )

max ρ1/p-H¨o l;[0,T ] (x, y) .

k =1,...,N

Some remarks are in order. By taking y ≡ o we have a notion of an inhomogenous (variation, H¨ older, ω-modulus) “norm”, but this will play no role in the sequel. Let us also remark that xp-var;[0,T ] = dp-var;[0,T ] (x, o) < ∞ iﬀ ρp-var;[0,T ] (x, o) < ∞ so that C p-var [0, T ] , GN Rd is precisely the set of paths x with ﬁnite ρp-var;[0,T ] distance between x and o. The map (x, y) → ρp-var;[0,T ] (x, y) is obviously non-negative, symmetric and one easily sees that the triangle inequality is satisﬁed. Then, precisely as in the previous section, adding |x0 − y0 |T N (Rd ) gives rise to a genuine metric on C p-var [0, T ] , GN Rd ; if attention is restricted to a path with pinned starting point, ρp-var;[0,T ]

8.1 p-Variation and 1/p-H¨ older topology

171

is already a metric. Of course, all this applies mutatis mutandis in a 1/pH¨older resp. p-ω context and ρ1/p-H¨o l resp. ρp-ω gives rise to metrics on the C 1/p-H¨o l resp. C p-ω spaces. Super-additivity of controls leads easily to 1/p N /p , ρp-var;[0,T ] x1 , x2 ≤ ρp-ω ;[0,T ] x1 , x2 max ω (0, T ) , ω (0, T ) (8.6) which should be compared with (8.5).

Proposition 8.7 For all x1 , x2 ∈ C p-var [0, T ] , GN Rd there exists a control ω with ω (0, T ) = 1 and a constant C = C (p, N ) such that ρp-ω ;[0,T ] (xi , o)

≤ Cρp-var;[0,T ] (xi , o), i = 1, 2;

≤ Cρp-var;[0,T ] (x1 , x2 ). Proof. Given x, y ∈ C p-var [0, T ] , GN Rd , we deﬁne for convenience, for k = 1, . . . , N , ρp-ω ;[0,T ] (x1 , x2 )

(k )

) ω (k x,y (s, t) = ρp-var;[s,t] (x, y)

p/k

.

We then deﬁne ω ¯ (k ) (s, t)

=

1 (k ) (k ) ω 1 2 (s, t) /ω x 1 ,x 2 (0, T ) 3 x ,x 1 (k ) (k ) + ω x 1 ,o (s, t) /ω x 1 ,o (0, T ) 3 1 (k ) (k ) + ω x 2 ,o (s, t) /ω x 2 ,o (0, T ) , 3

¯ (k ) (0, T ) = 1, and ﬁnally so that ω ¯ (k ) is a control with ω N 1 (k ) ω ¯ (s, t) . N

ω (s, t) =

k =1

To see that this deﬁnition of ω does the job, observe that for all k = 1, . . . , N and all 0 ≤ s < t ≤ T 1 π k xs,t − x2s,t p/k ≤ ω (k1) 2 (s, t) x ,x (k )

≤

3¯ ω (k ) (s, t) × ω x 1 ,x 2 (0, T )

≤

3N ω (s, t) × ω x 1 ,x 2 (0, T )

(k )

from which we see that k /p 1 π k xs,t − x2s,t ≤ (3N )k /p ω (s, t)k /p × ω (k1) 2 (0, T ) x ,x ! "k /p (k ) k /p k /p ≤ (3N ) ω (s, t) × max ω x 1 ,x 2 (0, T ) k =1,...,N

≤

(3N )

N /p

k /p

ω (s, t)

ρp-var;[0,T ] x1 , x2

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Variation and H¨ older spaces on free groups

which says precisely that N /p ρp-ω ;[0,T ] x1 , x2 ≤ (3N ) ρp-var;[0,T ] x1 , x2 . The same argument applies to x1 , o and x2 , o and the proof is ﬁnished. We now show that a map uniformly continuous on bounded sets in the ρp-ω metric, for all ω, is uniformly continuous on bounded sets in the ρp-var metric. To this end, for paths x ∈ C [0, T ] , GM Rd we deﬁne the (inhomogenous) balls β p-ω (R) = x : ρp-ω ;[0,T ] (x, o) < R , x : ρp-var;[0,T ] (x, o) < R . β p-var (R) = Corollary 8.8 Consider a map φ : C p-var [0, T ] , GM Rd → C p-var [0, T ] , GN (Re ) such that for any control function ω, any ε > 0 and R > 0 there exists δ = δ (ε, R; ω) such that x, y ∈ β p-ω (R) , ρp-ω (x, y) < δ =⇒ ρp-ω (φ (x) , φ (y)) < ε. Then, for any ε > 0 and R > 0 there exists η = η (ε, R) such that x, y ∈ β p-var (R) , ρp-var (x, y) < η =⇒ ρp-var (φ (x) , φ (y)) < ε. In fact, we can choose η = δ (ε, CR; ω) /C with ω deﬁned as in Proposition 8.7 and C = C (N, p). (This shows that if φ is (locally) γ-H¨ older on bounded older on bounded sets in the ρp-ω metric, for all ω, it is also (locally) γ-H¨ sets in the ρp-var metric.) Proof. Obvious from Proposition 8.7.

8.1.3 Homogenous vs inhomogenous distances Proposition 8.9 Let ωbe a control function on [0, T ]. For all paths x, y in C p-ω [0, T ] , GN Rd 1/N 1− 1 (8.7) max 1, xp-ωN dp-ω (x, y) ≤ C max ρp-ω (x, y) , ρp-ω (x, y) and

N −1 ρp-ω (x, y) ≤ C max dp-ω (x, y) max 1, xp-ω , dp-ω (x, y)N

(8.8)

where C = C (N ). The corresponding H¨ older estimates are obtained by taking ω (s, t) = t − s.

8.1 p-Variation and 1/p-H¨ older topology

Proof. First we see that2 dp-ω ;[0,T ] (x, y)

=

ρp-ω ;[0,T ] (x, y)

=

sup 0≤s< t≤T

sup 0≤s< t≤T

173

d δ 1 1 / p xs,t , δ 1 1 / p ys,t , ω (s ,t ) ω (s ,t ) δ 1 xs,t − δ 1 1 / p ys,t 1/p ω (s ,t )

ω (s ,t )

T N (Rd )

so that these deﬁnitions indeed only diﬀer in how to measure distance in GN Rd . A quantitative comparison of these distances was given in Proposition 7.49, an application of which ﬁnishes the proof. For a concise formulation of the next theorem, let us set d˜p-ω ;[0,T ] (x, y) := dp-ω ;[0,T ] (x, y) + d (x0 , y0 ) ρ ˜p-ω ;[0,T ] (x, y) := ρp-ω ;[0,T ] (x, y) + ρ (x0 , y0 ) and similarly for d1/p-H¨o l , dp-var and ρ1/p-H¨o l , ρp-var where we have already started to omit [0, T ] in the notation when no confusion is possible. We have Theorem 8.10 Let ω be an arbitrary control function on [0, T ]. Each identity map ˜p-ω Id : C p-ω [0, T ] , GN Rd , d˜p-ω C p-ω [0, T ] , GN Rd , ρ 1 1 ˜p-H¨o l ) Id : (C p -H¨o l [0, T ] , GN Rd , d˜p-H¨o l ) (C p -H¨o l [0, T ] , GN Rd , ρ d d p-var N p-var N ˜ [0, T ] , G R , dp-var ) (C [0, T ] , G R ,ρ ˜ ) Id : (C p-var

is Lipschitz on bounded sets in → direction and 1/N -H¨ older on bounded sets in ← direction. Proof. The relevant estimates between “d (x0 , y0 ) and ρ (x0 , y0 )” follow directly from Proposition 7.49 and we focus on the path-space distance without tilde: the case of the identity map from C p-ω [0, T ] , GN Rd into itself (equipped with homogenous resp. inhomogenous distance dp-ω resp. ρp-ω ) is covered directly by Proposition 8.9, and the case of 1/p-H¨older paths is a special case, namely ω (s, t) = t − s. We thus turn to p-variation. → direction: Let x1 , x2 ∈ Bp-var (R), the “homogenous” p-variation ball of radius R as deﬁned before Corollary 8.5, and let ω denote the corresponding control constructed in Proposition 8.4. Then, with constants c1 , c2 which may depend on N, R, p we have ρp-var x1 , x2 ≤ ρp-ω x1 , x2 by (8.6) ≤ c1 dp-ω x1 , x2 by (8.8) ≤ c2 dp-var (x1 , x2 ) by the very choice of ω. ω (s, t) = 0 and x ∈ C p -ω then x s , t = o, the unit element in G N Rd ; we agree that δ 1 o = o. 2 If

0

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174

The ← direction follows the same logic, but now we rely on (8.5), (8.7) and ω as constructed in Proposition 8.7. We ﬁnish this section with a simple proposition (it will serve as a technical ingredient in our discussion of RDE smoothness later on). n d Proposition 8.11 Let A denote the canonical lift of A ∈ L R , R to d N n N p-var N [0, T ] , G (Rn ) the Hom G (R ) , G R . Then, for ﬁxed x ∈ C map A ∈ L Rn , Rd → Ax := {t ∈ [0, T ] → Axt } ∈ C p-var [0, T ] , GN Rd is continuous. Proof. We ﬁrst prove this for x ∈ C p-ω [0, T ] , GN (Rn ) . Recall that dp-ω and the inhomogenous distance ρp-ω are locally H¨older equivalent (and in particular induce the same topology). Consider A, B ∈ L Rn , Rd with operator norm bounded by some M . Controlling ρp-ω (Ax, Bx) amounts to i i i controlling (Ax) − (Bx) for i = 1, . . . , N . Every (Ax) can be written out as a contraction of A ⊗ · · · ⊗ A (i times) against the i-tensor xi . It is then easy to see that i i (Ax) − (Bx) ≤ CM |A − B| xi . This implies that A → Ax is even Lipschitz when C p-var [0, T ] , GN Rd is equipped with ρp-ω . Switching to dp-ω we still have continuity, and hence dp-var continuity.

8.2 Geodesic approximations Our interest in GN Rd -valued paths comes from the fact (cf. Section 7.2.1) that any continuous path x of ﬁnite 1-variation with values in Rd can be lifted to a path {t → SN (x)t ≡ SN (x)0,t } with values in GN Rd simply by computing all iterated (Riemann–Stieltjes) integrals up to order N . It is natural to ask some sort of converse: how can an abstract path x : [0, T ] → GN Rd be approximated by a sequence (SN (xn ))? We have Proposition 8.12 Let x ∈ Cop-var [0, T ] , GN Rd , p ≥ 1. Then there exists (xn ) ⊂ C 1-H¨o l [0, T ] , Rd , such that d∞;[0,T ] (x, SN (xn )) → 0 as n → ∞ and sup SN (xn )p-var;[0,T ] ≤ 31−1/p xp-var;[0,T ] < ∞. n

8.3 Completeness and non-separability 1/p-H¨o l

For x ∈ Co

175

[0, T ] , GN Rd we have

sup SN (xn )1/p-H¨o l;[0,T ] ≤ 31−1/p x1/p-H¨o l;[0,T ] < ∞. n

Proof. Given the fact that GN Rd is a geodesic space under Carnot– Caratheodory distance, this follows readily from the approximation results in geodesic spaces, Section 5.2.

8.3 Completeness and non-separability By Theorem 8.10 we can equip the space C p-var ([0, T ], GN Rd ) with either homogenous or inhomogenous p-variation distance and not only obtain the same topology but also the same “metric” notions of bounded sets or Cauchy-sequences. The same holds for H¨older paths, of course, and we have Theorem 8.13 (i) Let p ≥ 1. The space C p-var ([0, T ], GN Rd ) is a complete, non-separable metric space (with respect to either homogenous or inhomogenous p-variation distance). (ii) The space C 1/p-H¨o l ([0, T ], GN Rd ) is a complete, non-separable metric space (with respect to either homogenous or inhomogenous 1/p-H¨ older distance). Proof. (i) It suﬃces to consider the homogenous p-variation distance, and more precisely d˜p-var (x, y) ≡ dp-var (x, y) + d (x0 , y0 ) for x, y ∈ C p-var ([0, T ], GN Rd ). The completeness proof follows exactly from the arguments used to establish completeness of C p-var ([0, T ], Rd ). Then, if C p-var ([0, T ], GN Rd ) were separable for some N = 1, 2, . . . the same would be true for its projection to N = 1, but we know that C p-var ([0, T ], Rd ) is not separable, cf. Theorem 5.25 it is not. (ii) Similar and left to the reader.

8.4 The d0 /d∞ estimate Lemma 8.14 Let g, h ∈ GN Rd . Then there exists C = C (N, d) such that

−1 g ⊗ h ⊗ g ≤ C max h , h1/N g1−1/N .

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Variation and H¨ older spaces on free groups

n Proof. Viewing g, h as elements in T N Rd , and writing (·) ≡ π n (·) for projection to the nth tensor level, we have for n = 1, ..., N i −1 n ⊗n g −1 ⊗ hj ⊗ g k ∈ Rd g ⊗h⊗g = . i+j +k = n j>0

⊗n Since every tensor level Rd is equipped with Euclidean structure, we easily see that i n −1 g ⊗h⊗g ≤ g −1 hj g k . i+j +k =n j>0

k Using g k ≤ c1 g and similar estimates for g −1 , h, also recalling g −1 = g, we ﬁnd n n −1 n −j j g h g ⊗ h ⊗ g ≤ c2 j =1

which implies n 1/n −1 g ⊗h⊗g

1−j /n

≤

c3 max g

≤

c3

j =1,...,n

sup 1/N ≤θ ≤1

By equivalence of homogenous norms, −1 g ⊗ h ⊗ g ≤ c4 sup

1/N ≤θ ≤1

1−θ

g

g

1−θ

h

j /n

θ

h .

θ

h .

The proof is now easily ﬁnished.

8.15 (d0 /d∞ estimate) On the path-space C0 [0, 1] , GN Proposition Rd the distances d∞ and d0 ≡ d0-H¨o l are locally 1/N -H¨ older equivalent. More precisely, there exists C = C (N, d) such that 1/N d∞ (x, y) ≤ d0 (x, y) ≤ C max d∞ (x, y) , d∞ (x, y) 1−1/N . (x∞ + y∞ ) Proof. Only the second inequality requires a proof. We write gh instead of g ⊗ h. For any s < t in [0, 1], −1 −1 −1 −1 −1 x−1 s,t ys,t = xs,t ys xs xs,t xt yt xt yt yt xt .

By sub-additivity, −1 xs,t ys,t

−1 −1 −1 −1 xs,t ys xs xs,t + x−1 t yt xt yt y t xt −1 −1 −1 −1 = v ys xs v + w xt yt w ≤

8.5 Interpolation and compactness

177

with v = xs,t and w = yt−1 xt . Note that −1 −1 yt xt = xt yt = d (xt , yt ) and v , w ≤ x∞ + y∞ . The conclusion now follows from Lemma 8.14.

8.5 Interpolation and compactness This interpolation result will be used extensively. Lemma 8.16 Assume p > p ≥ 1. (i) Let x, y be two elements of C p-var [0, T ], GN Rd . Then, p/p

1−p/p

dp -var (x, y) ≤ (xp-var + yp-var ) d0 (x, y) . (8.9) (ii) Let x, y be two elements of C p-ω [0, T ], GN Rd . Then, for all p > p, dp -ω (x, y) ≤ (xp -ω + yp -ω )

p/p

d0 (x, y)

1−p/p

.

(8.10)

Proof. (i) Consider a dissection (ti ) ⊂ [0, T ]. Then (8.9) follows from p d xt i ,t i + 1 , yt i ,t i + 1

≤ ≤

p p −p d xt i ,t i + 1 , yt i ,t i + 1 d0,[0,T ] (x, y) xt ,t + yt ,t p d0,[0,T ] (x, y)p −p , i i+ 1 i i+ 1

p 1/p p 1/p followed by summation over i, the elementary ( |ai +bi | ) ≤ ( |ai | ) + p 1/p and taking the supremum over all such dissections. ( |bi | ) (ii) Left to the reader. As a consequence of the above interpolation result, Lemma 5.12, and the Arzela–Ascoli theorem, we obtain the following compactness result. Proposition 8.17 Let (xn ) be a sequence in C [0, T ] , GN Rd . (i) Assume (xn )n is equicontinuous, bounded and supn xn p-var;[0,T ] < ∞. Then xn converges (in p > p variation, along a subsequence) to some p-var [0, T ] , GN Rd . x∈C (ii) Assume (xn )n is bounded and supn ||xn ||α -H¨o l;[0,T ] < ∞. Then xn converges (in α < α H¨ o lder topology, along a subsequence) to some x ∈ C α -H¨o l [0, T ] , GN Rd . The following corollary will also be useful.

such that Corollary 8.18 (i) If (xn ), x are in C p-var [0, T ] , GN Rd supn |xn |p-var;[0,T ] < ∞ and limn →∞ d∞;[0,T ] (x, xn ) = 0, then for p > p, sup xn p -var;[s,t] − xp -var;[s,t] → 0 as n → ∞ (s,t)∈∆ T

Variation and H¨ older spaces on free groups

178

where ∆T = {(s, t) : 0 ≤ s ≤ t ≤ T }. Furthermore, {xn p -var; [·,·] : n ∈ N} is equicontinuous in the sense that for every ε > 0 there exists δ such that |t − s| < δ implies (8.11) sup xn p -var; [s,t] < ε. n

[0, T ] , GN Rd so that supn ||xn ||α -H¨o l;[0,T ] < (ii) If (x ) , x are in C ∞ and limn →∞ d∞;[0,T ] (x, xn ) = 0, then for all s < t in [0, T ], and for α < α, sup xn α -H¨o l;[s,t] − xα -H¨o l;[s,t] → 0 as n → ∞ n

α -H¨o l

(s,t)∈∆ T

and

xn α -H¨o l;[·,·] : n ∈ N

is equicontinuous, similar to part (i).

d Proof. The argument given in the proof of Corollary 5.29 for the case of R valued paths extends line-by-line to the case of GN Rd -valued paths.

8.6 Closure of lifted smooth paths We now deﬁne C 0,p-var [0, T ] , GN Rd resp. C 0,1/p-H¨o l [0, T ] , GN Rd as the closure of step-N lifted smooth paths from [0, T ] → Rd in p-variation resp. 1/p-H¨older topology. A little care is needed since, by convention, SN (x)0 = 1 ≡ o, the unit element in GN Rd . 0,p-var [0, T ] , GN Rd as the set of conDeﬁnition 8.19 (i) We deﬁne C o tinuous paths x : [0, T ] → GN Rd for which there exists a sequence of smooth Rd -valued paths xn such that dp-var (x,SN (xn )) →n →∞ 0 and C 0,p-var [0, T ] , GN Rd as the set of paths x with 0,p-var [0, T ] , GN Rd . x0,· = x−1 0 ⊗ x· ∈ Co 0,1/p-H¨o l (ii) Similarly, Co [0, T ] , GN Rd is the set of paths x for which there exists a sequence of smooth Rd -valued paths xn such that d1/p-H¨o l (x,SN (xn )) →n →∞ 0 d 0,1/p-H¨o l 0,1/p-H¨o l N [0, T ] , G R are those paths x with x0,· ∈ Co and C [0, T ] , GN Rd .

8.6 Closure of lifted smooth paths

179

Obviously, all these spaces are closed subsets of C p-var [0, T ] , GN Rd imresp. C 1/p-H¨o l [0, T ] , GN Rd and thus complete. Proposition 7.63 1-var 1-var d [0, T ] , R to C plies a fortiori continuity of S , as a map from C N o o [0, T ] , GN Rd . Clearly then C 0,1-var [0, T ] , GN Rd = SN C 0,1-var [0, T ] , Rd (8.12) d and also C 0,1-H¨o l [0, T ] , GN Rd = SN C 1 [0, T ] , R d . The reader will 0,1-var [0, T ] , R , deﬁned as a recall from Proposition 1.32 that C 1-variation closure of smooth paths, turned out to be precisely the space of absolutely continuous paths (with respect to Euclidean metric on Rd ); in Exercise 8.20 it is seen that the same is true in the case of GN Rd -valued paths. Exercise 8.20 Show that C 0,1-var [0, T ] , GN Rd is precisely the space of absolutely continuous paths (with respect to Carnot–Caratheodory metric on GN Rd ). Solution. It suﬃces to consider x ∈ Co0,1-var [0, T ] , GN Rd . Any such x (·) is of the form SN (x)· where x is an Rd -valued absolutely continuous path. Hence, for every ε > 0and s1 < t1 ≤ s2 < t2≤ · · · < sn < tn in [0, T ] there exists δ so that i |ti − si | < δ implies i |xs i ,t i | < ε and in fact (cf. Exercise 5.15) |x|1-var;[s i ,t i ] < ε. i

But then, thanks to Proposition 7.59 |d (xs i , xt i )| ≤ |x|1-var;[s i ,t i ] = |x|1-var;[s i ,t i ] < ε. i

i

i

Lemma 8.21 We ﬁx p > 1. (i) If Ω ⊂ C 1-var [0, T ] , GN Rd and if C 0,1-var [0, T ] , GN Rd is included in the d1-var-closure of Ω, then the dp-var -closure of Ω is equal to C 0,p-var [0, T ] , GN Rd . (ii) If Ω ⊂ C 1-H¨o l [0, T ] , GN Rd and if C 0,1-H¨o l [0, T ] , GN Rd is included in the d1-H¨o l -closure of Ω, then the d1/p-H¨o l -closure of Ω is equal to C 0,1/p-H¨o l [0, T ] , GN Rd . Proof. Same proof as the N = 1 case (cf. Lemma 5.30). We can now extend “Wiener’s characterization” from the Rd setting (Theorem 8.22) to the group setting. Recall in particular that xD denotes the geodesic approximation to x based on some dissection D = (ti ) of [0, T ]. D That is, xD t i = xt i for all i and x |[t i ,t i + 1 ] is a geodesic connecting xt i and xt i + 1 as in Proposition 7.42. The proof of the Rd case then extends without any changes and we have

Variation and H¨ older spaces on free groups

180

8.22 (Wiener’s characterization) Let x ∈ C p-var [0, T ], GN Theorem Rd , with p > 1. The following d statements are equivalent. 0,p-var N (i.1) x ∈ C ([0, T ], G R ). p (i.2a) limδ →0 supD =(t i ),|D |< δ i xp-var;[t i ,t i + 1 ] = 0. p (i.2b) limδ →0 supD =(t i ),|D |< δ i d xt i , xt i + 1 = 0. (i.3) lim|D |→0 dp-var xD , x = 0. Secondly, let x ∈ C 1/p-H¨o l [0, T ], GN Rd , with p > 1. The following statements are equivalent. (ii.1) x ∈ C 0,1/p-H¨o l [0, T ], Rd . (ii.2a) limδ →0 sup|t−s|< δ x1/p-H¨o l;[s,t] = 0. (ii.2b) limδ →0 sup|t−s|< δ d(xs , xt )/|t − s|1/p = 0. (ii.3) lim|D |→0 d1/p-H¨o l xD , x = 0. Exercise 8.23 Let p ≥ 1. Show that C 0,p-var [0, T ] , GN Rd is precisely the space of paths which are absolutely continuous of order p. Solution. The case p = 1 was dealt with in Exercise 8.20. For p > 1 we simply combine the result of Exercise 5.15 with Wiener’s characterization. Corollary 8.24 For p > 1, we have the following set inclusions: -

C q -var [0, T ] , GN Rd

1≤q < p

⊂

C 0,p-var [0, T ] , GN Rd

C p-var [0, T ] , GN Rd . C q -var [0, T ] , GN Rd . ⊂

⊂

q>p

Proof. Similar to Corollary 5.33. Proposition 8.25 Let p ≥ 1. The spaces C 0,p-var [0, T ] , GN Rd , C 0,1/p-H¨o l [0, T ] , GN Rd are Polish with respect to either homogenous or inhomogenous p-variation resp. 1/p-H¨ older distance. Proof. As remarked in Section 8.3, either choice (homogenous, inhomogenous) of p-variation distance leads to the same topology and notion of Cauchy-sequence. Clearly then, C 0,p-var is complete under either distance. It remains to discuss separability. 1.35, there exists a count From Corollary 0,1-var d [0, T ] , R ; by continuity of SN , SN (Ω) is able space Ω dense in C d 0,1-var N [0, T ] , G R . We conclude using Lemma 5.30. Similar dense in C arguments for 1/p-H¨older spaces are left to the reader.

8.7 Comments

181

8.7 Comments Section 8.1 introduces the basic path space distance for GN Rd -valued paths (homogenous such as dp-var , inhomogenous such as ρp-var ). As will be discussed in detail in the next chapter, if one chooses N = [p] these distances are “rough path” distances. Noting that both dp-var and ρp-var induce the same topology, both notions are useful. Typical rough path continuity statements are locally Lipschitz continuous in the inhomogenous distance (which is also the distance put forward by Lyons, e.g. Lyons and Qian [120] in the references cited therein). The homogenous distance, on the other hand, comes in handy when establishing large deviation results via exponentially good approximations (as seen in Lemma 13.40 for instance); not to mention its general convenience, which often allows us to write arguments in the same way as for paths on Euclidean space. The geodesic approximation result of Section 8.2 appeared in Friz and Victoir [63]; here it is derived as a special case of a general approximation result on geodesic spaces. Section 8.3 follows Friz and Victoir [63]. The d0 /d∞ estimate in Section 8.4 is taken from Friz and Victoir [61]; Sections 8.5 and 8.6 follow Friz and Victoir [63].

9 Geometric rough path spaces of ﬁnite p-variation for any N ∈ N We have studied GN Rd -valued paths and p ≥ 1. If one thinks of GN Rd as the correct state-space that allows us not only to keep track of the spatial position in Rd but also to keep track of the accumulated area (and higher indeﬁnite iterated integrals up to order N ) then it should not be surprising that N and p stand in some canonical relation. To wit, if p = 1, knowledge of the Rd -valued path allows us to compute iterated integrals by Riemann–Stieltjes theory and there is no need to include area and other iterated integrals in the state-space. The same remark applies, more generally, to p ∈ [1, 2) using iterated Young integration and in this case we should take N = [p] = 1. When p ≥ 2, this is not possible and knowledge of higher indeﬁnite iterated integrals up to order N = [p] must be an a priori information, i.e. assumed to be known.1 However, we shall establish that integrals of order greater than [p] are still canonically determined. More precisely, we shall see that for N ≥ [p] there exists a canonical bijection2 SN : Cop-var [0, T ] , G[p] Rd → Cop-var [0, T ] , GN Rd such that for all x ∈ Cop-var [0, T ] , G[p] Rd we have xp-var;[0,T ] ≤ SN (x)p-var;[0,T ] ≤ CN xp-var;[0,T ] .

The analogous 1/p-H¨ older estimate also holds, and is a consequence of the p-variation estimate. Indeed, by reparametrization, [0, T ] may be replaced by [s, t] so that the H¨older statement follows trivially from xp-var;[s,t] ≤ 1/p

x1/p-H¨o l |t − s| . This gives a ﬁrst hint on the importance of these so-called (weak) geometeric rough whose regularity (p-variation) is in relation to their paths state-space G[p] Rd . 1 In a typical probabilistic situation (cf. Part III of this book), N = 2 or 3 and the required iterated integrals will be constructed via some stochastic integration procedure. 2 Recall that o in C p -va r indicates that all paths start at the unit element of G [p ] . o

9.1 The Lyons-lift map x → SN (x)

183

9.1 The Lyons-lift map x → SN (x) 9.1.1 Quantitative bound on SN We start with two simple technical lemmas. Lemma 9.1 Let g1 , g2 ∈ T N +1 Rd . Then, g1 ⊗ g2 − (g1 + g2 ) = π 0,N (g1 ) ⊗ π 0,N (g2 ) − π 0,N (g1 + g2 ) . In particular, if h1 , h2 ∈ T N +1 Rd are such that π 0,N (g1 ) = π 0,N (h1 ) and π 0,N (g2 ) = π 0,N (h2 ) , we have g1 ⊗ g2 − (g1 + g2 ) = h1 ⊗ h2 − (h1 + h2 ) . Proof. Simple algebra.

1 2 1-var [s, u] , Rd , such that Lemma ufor some N ≥ 1, 1 9.2 Let x 2,x ∈ C u SN x s,u = SN x s,u . Then, if ≥ max s dx1 , s dx2 , we have for some constant C depending only on N, (9.1) SN +1 x1 s,u − SN +1 x2 s,u ≤ C N +1 . Proof. By assumption, SN +1 x1 s,u − SN +1 x2 s,u = π N +1 SN +1 x1 s,u − SN +1 x2 s,u and (9.1) follows from π N +1 SN +1 xi s,u ≤

1 (N + 1)!

u

N +1

i dx

, i = 1, 2.

s

Alternative proof of (9.1) which introduces the useful idea of “reducing two paths to one path”: Without loss of generality, assume (s, u) = (0, 1) and observe that SN x1 0,1 = SN x2 0,1 implies −1 SN +1 x1 0,1 − SN +1 x2 0,1 = SN +1 x1 0,1 ⊗ SN +1 x2 0,1 − 1. ← − Deﬁne x = x1 x2 , i.e. as the concatenation of x1 (·) and x2 (1 − ·) and assume x is (re-)parametrized on [0, 1]. It follows that SN +1 x1 0,1 − SN +1 x2 0,1 = SN +1 (x)0,1 − 1, with SN (x)0,1 = 1 and |x|1-var;[0,1] = x1 1-var;[0,1] + x2 1-var;[0,1] ≤ 2. But then from SN (x)0,1 = 1 SN +1 (x)0,1 − 1 = π N +1 SN +1 (x)0,1 ≤ and (9.1) follows.

1 N +1 (2) (N + 1)!

Geometric rough path spaces

184

We are now ready for the crucial quantitative estimate on SN (x)p-var for N ≥ p. Proposition 9.3 Let x ∈ C 1-var [0, T ] , Rd . Then, for all N ≥ [p] , there exists a constant C depending only on N and p (and not depending on the 1-variation norm of x nor its p-variation) such that for all s < t in [0, T ] , (9.2) SN (x)p-var;[s,t] ≤ C S[p] (x)p-var;[s,t] . The constant C can be chosen to be right-continuous with respect to p. Proof. It is enough to show that for all N ≥ [p] , SN +1 (x)p-var;[s,t] ≤ c1 (p) SN (x)p-var;[s,t] ,

(9.3)

where p → c1 (p) is right continuous. Explicit dependency on p in our constant will be written in this proof. p

We deﬁne x = SN (x), y = SN +1 (x) and ω (s, t) = xp−var;[s,t] . By s,t Theorem 7.32 we can ﬁnd, for all s < td in [0, T ], a “geodesic” path x : d N [s, t] → R associated with xs,t ∈ G R which is the shortest (Lipschitz) path which has step-N signature equal to xs,t . Then, deﬁne Γs,t = ys,t − SN +1 xs,t s,t . For s < t < u, we also deﬁne xs,t,u to be the concatenation of xs,t and xt,u , and observe that SN +1 (xs,t,u )s,u = SN +1 (xs,t )s,t ⊗ SN +1 (xt,u )t,u . Then, as ys,t ⊗ yt,u = ys,u , Γs,u − (Γs,t + Γt,u ) = ys,t ⊗ yt,u − (ys,t + yt,u ) −SN +1 xs,t s,t ⊗ SN +1 xt,u t,u +SN +1 xs,t s,t + SN +1 xt,u t,u +SN +1 xs,t s,t ⊗ SN +1 xt,u t,u − SN +1 (xs,u )s,u . By construction, for all k ≤ N and all s, t ∈ [0, T ] , π k (ys,t ) = π k s,t SN +1 (x )s,t , hence we can apply Lemma 9.1 to see that ys,t ⊗ yt,u − (ys,t + yt,u ) = SN +1 xs,t s,t ⊗ SN +1 xt,u t,u − SN +1 xs,t s,t + SN +1 xt,u t,u . Hence, we are left with Γs,u − (Γs,t + Γt,u ) = SN +1 xs,t,u s,u − SN +1 (xs,u )s,u ,

9.1 The Lyons-lift map x → SN (x)

185

which we bound using Lemma 9.2: |Γs,u − (Γs,t + Γt,u )|

u

≤ c1 max

s,t,u dxr ,

s

≤

c2 ω (s, u)

N +1

u

|dxs,u r | s

N +1 p

.

(9.4)

t

t Secondly, using s |dxs,t | ≤ s |dx|, Lemma 9.2 gives N +1 N +1 ˜ (s, t) . |Γs,t | = ys,t − SN +1 xs,t s,t ≤ c3 |x|1-var;[s,t] =: ω The last two inequalities allow us to apply the same (analysis) Lemma 6.2 (which we have already used for establishing the Young–L´ oeve estimate). We get, for all 0 ≤ s < t ≤ T , N +1 ys,t − SN +1 xs,t s,t ≤ c4 CN ,p ω (s, t) p where CN ,p can be taken to be 1/(1 − 21− inequality that |π N +1 (ys,t )|

≤ c4 CN ,p ω (s, t)

N +1 p

N +1 p

). This implies by the triangle

+ π N +1 SN +1 xs,t s,t

≤ c5 (1 + CN ,p ) ω (s, t)

N +1 p

,

and hence, using the equivalence of homogenous norms, that for all s, t ∈ [0, T ] , 1/p 1/p ω (s, t) . ys,t ≤ c6 1 + (1 + CN ,p ) This means that

1/p yp-var;[s,t] ≤ c6 1 + (1 + CN ,p ) xp-var;[s,t]

for all 0 ≤ s < t ≤ T and the proof is ﬁnished.

9.1.2 Deﬁnition of the map SN on Cop-var [0, T ] , G[p] Rd

d p-var [p] [0, T ] , G R . A path Deﬁnition 9.4 Let N ≥ [p] ≥ 1 and x ∈ C o d p-var N [0, T ] , G R that projects down onto x is said to be a pin Co Lyons lift of x of order N . When p is ﬁxed, we will simply speak of a Lyons lift. We now show that there exists a unique Lyons lift of order N , for all x ∈ Cop-var [0, T ] , G[p] Rd . In the terminology of the forthcoming Deﬁnition 9.15 this says precisely that a weak geometric p-rough path admits a unique Lyons lift.

Geometric rough path spaces

186

Theorem 9.5 Let N ≥ [p] ≥ 1 and x ∈ Cop-var [0, T ] , G[p] Rd . Then there exists a unique Lyons lift of order N of x. By writing SN (x) for this path, we deﬁne the map SN on Cop-var [0, T ] , G[p] Rd . Moreover, (i) the map SN : Cop-var [0, T ] , G[p] Rd → Cop-var [0, T ] , GN Rd is a bijection with inverse π 0,[p] ; (ii) we have for some constant C = C (N, p), which may be taken rightcontinuous in p, xp-var;[s,t] ≤ SN (x)p-var;[s,t] ≤ C xp-var;[s,t]

(9.5)

for all s < t in [0, T ]. d q -var [q ] [0, T ] , G R for q ≤ p, SN Remark 9.6 Observe that if x ∈ C S[p] (x) = SN (x). This justiﬁes using the same notation for all p, and in particular the same notation for p > 1 and p =1. For further convenience, we will deﬁne for N ≤ [p] and for x ∈ C p-var [0,T ] ,G[p] Rd the path SN (x) which is just the projection of x onto GN Rd . In particular, the estimate SN (x)p-var;[s,t] ≤ C xp-var;[s,t] still holds for N ≤ [p] . Proof. First step, existence: Let xn be a sequence in C 1-var [0, T ] , Rd such that sup S[p] (xn )p-var;[0,T ] < ∞ and lim d∞ S[p] (xn ) , x = 0. n →∞

n

By Proposition 9.3, for all s < t in [0, T ] and ε small enough (namely, such that [p + ε] = [p]), SN (xn )s,t ≤ cp+ ε S[p] (xn )(p+ε)-var;[s,t] . By Corollary 8.18 the right-hand side above can be made arbitrarily small, uniformly in n, provided t − s is small enough; this implies readily that SN (xn ) is equicontinuous. Boundedness is clear and so, by Arzela– Ascoli and switching toa subsequence if necessary, we have the existence of a continuous GN Rd -valued path z such that SN (xn ) → z uniformly on [0, T ] as n → ∞. From the very choice of (xn ) it then follows that the projection of z to a G[p] Rd -valued path must be equal to x. Then, for all 0 ≤ s < t ≤ T, zs,t = lim SN (xn )s,t n →∞

≤ ≤

lim SN (xn )(p+ε)-var;[s,t] cp+ε lim S[p] (xn )

n →∞

n →∞

(p+ε)-var;[s,t]

9.1 The Lyons-lift map x → SN (x)

187

where we used (9.2), Proposition 9.3, for the last estimate. On the other hand, from the ﬁrst part of Corollary 8.18, lim S[p] (xn )(p+ε)-var;[s,t] = x(p+ε)-var;[s,t] n →∞

and hence, for all 0 ≤ s < t ≤ T, zs,t ≤ cp+ ε x(p+ε)-var;[s,t] . Using “right-continuity” of p → cp (Proposition 9.3) and also right-continuity of the homogenous p-variation norm with respect to p (Lemma 5.13) we may send ε → 0 to obtain zs,t ≤ cp xp-var;[s,t] . p

Super-additivity of (s, t) → xp-var;[s,t] implies that zp-var;[s,t] ≤ cp xp-var;[s,t] ; the converse estimate xp-var;[s,t] ≤ zp-var;[s,t] is trivial since we know that z lifts x. In particular, we found a (Lyons) lift of x in Cop-var [0, T ] , GN Rd , which satisﬁes (9.5). with z ∈ Cop-var [0, T ] , GM +1 Rd Second step, uniqueness: Given z, ˜ π 0,M (z) ≡ π 0,M (˜ z) we show (by induction in M ) that M ≥ [p] implies zt deﬁnes a path in gM +1 Rd ∩ z≡˜ z. From Lemma 7.61, ht = log z−1 t ⊗˜ d ⊗(M +1) R , and for all s, t ∈ [0, T ] , M +1

|hs,t | ≤ c1 zs,t

+ c1 ˜ zs,t

p

M +1

.

p

We deﬁne the control ω (s, t) = zp-var;[s,t] + ˜ zp-var;[s,t] . The previous inequality now reads (M +1)/p

|hs,t | ≤ c2 ω (s, t)

.

In particular, h is of ﬁnite M p+1 -variation. As M p+1 < 1, we deduce that h z, which is what we wanted to is constant equal to h0 = 0, i.e. that z = ˜ show. Third step: It remains to see that, as stated in (i), SN is a bijection with inverse on Cop-var [0, T ] , G[p] N isthe identity map d π 0,[p] . Obviously, π 0,[p] ◦Sp-var R . Conversely, given x ∈ Co [0, T ] , GN Rd it is clear from (9.5) that SN ◦ π 0,[p] (x) has ﬁnite p-variation. By uniqueness, we see that SNN p-var [0, T ] , G ◦ π (x) = x so that S ◦ π acts as identity map on C N 0,[p] o d0,[p] R . This completes the proof.

188

Geometric rough path spaces

Exercise 9.7 Let N ≥ [p] ≥ 1 and x∈ Cop-var [0, T ] , G[p] Rd . Prove N d such that for all sequences (xn ) ⊂ that there existsdz ∈ Co [0, T ] , G R 1-var [0, T ] , R such that C d∞;[0,T ] S[p] (xn ) , x → 0 and sup S[p] (xn ) p-var;[0,T ] < ∞, n

we have SN (xn ) → z uniformly on [0, T ] as n → ∞. (This exercise shows that we could have deﬁned the Lyons lift as a limit, similar to our deﬁnition of the Young integral and the forthcoming deﬁnition of the solution to a rough diﬀerential equation.)

9.1.3 Modulus of continuity for the map SN We shall now establish that the Lyons-lifting map is locally Lipschitz continuous with respect to inhomogenous rough path distances on pathspace. To this end, we need the following two lemmas. Lemma 9.8 Let N ≥ [p] ≥ 1 and x ∈ Cop-var [s, t] , G[p] Rd , and xs,t ∈

t C 1-var [s, t] , Rd with s |dxs,t | ≤ K xp-var;[s,t] such that SN (xs,t )s,t = SN (xs,t ) . Then, for some constant C depending only on K, N and p, N +1 SN +1 (x)s,t − SN +1 xs,t s,t ≤ c xp-var;[s,t] . Proof. This is fairly obvious. As SN (x)s,t = SN (xs,t )s,t , we have SN +1 (x)s,t − SN +1 xs,t s,t = π N +1 SN +1 (x)s,t − π N +1 SN +1 xs,t s,t ≤ π N +1 SN +1 (x)s,t + π N +1 SN +1 xs,t s,t . Using equivalence of homogenous norms and the quantitative estimates on the Lyons lift obtained in Theorem 9.5, we have SN +1 (x)s,t − SN +1 xs,t s,t N +1 s,t N +1 + SN +1 x s,t ≤ c1 SN +1 (x)s,t N +1 N +1 ≤ c2 xp-var;[s,t] + xs,t 1-var;[s,t] N +1

≤ c3 xp-var;[s,t] . The following result generalizes Lemma 9.2.

9.1 The Lyons-lift map x → SN (x)

189

Lemma 9.9 Let x1 , x2 , x ˜1 , x ˜2 ∈ C 1-var [s, u] , Rd such that SN x1 s,u = SN x2 s,u , 1 2 SN x ˜ s,u = SN x ˜ s,u , and assume there exist ≥ 0, ε > 0 such that u u u u 1 2 1 2 max |dx | + |dx |, |d˜ x |+ |d˜ x | s s s u s u 1 2 1 2 dxr − d˜ dxr − d˜ xr , xr max s

≤ , ≤

ε.

s

Then, for some constant C depending only on N , 1 2 ˜ s,u −SN +1 x ˜ s,u ≤ CεN +1 . SN +1 x1 s,u −SN +1 x2 s,u − SN +1 x Proof. Working as in the proof of Lemma 9.2, we see that we can assume ˜2 = 0 and (s, u) = (0, 1) . Then, scaling x1 and x ˜1 by 1 , we can x2 = x assume = 1. The lemma then follows from Proposition 7.63. We can now prove local Lipschitzness of the Lyons-lifting map SN . Theorem 9.10 Let x1 , x2 ∈ C p-var [0, T ] , G[p] Rd and ω a control such that for all 0 ≤ s < t ≤ T and i = 1, 2, i p x ≤ ω (s, t) , p-var;[s,t] 1 2 ≤ ε. ρp,ω x , x Then, for all N ≥ [p] , there exists a constant C depending only on N and p such that for all s < t in [0, T ] , ρp,ω SN x1 , SN x2 ≤ Cε. 1 2 Proof. Itis enough to show that for all 1N ≥2 [p] , if x and x are two paths p-var N d [0, T ] , G R with ρp,ω x , x ≤ ε, then, for some constant in C cN , ρp,ω SN +1 x1 , SN +1 x2 ≤ cN ε. Let x1,s,t , x2,s,t ∈ C 1-var [s, t] , Rd such that SN xi,s,t s,t = xis,t ,

and such that

s

d xi,s,t

≤

c1 ω (s, t)

d x1,s,t − x2,s,t

≤

c1 εω (s, t)

t

1/p

,

s

t

1/p

.

Geometric rough path spaces

190

This is possible thanks to Proposition 7.64, applied to δ λ x1s,t and δ λ x2s,t 1/p

with λ = 1/ω (s, t) . We deﬁne similarly xi,s,u and xi,t,u , and then xi,s,t,u to be the concatenation of xi,s,t and xi,t,u . Observe in particular that u 1/p d x1,s,t,u − x2,s,t,u ≤ 21−1/p c1 εω (s, u) . s

Following the proof of Proposition 9.3, we deﬁne for s < t, Γis,t = SN +1 xi,s,t s,t − SN +1 xi s,t , i = 1, 2, and Γs,t = Γ1s,t − Γ2s,t . It is clear from the proof of Proposition 9.3 that i c2 N +1 Γs,t ≤ ω (s, t) p , i = 1, 2, 2 and, by the triangle inequality, N +1 Γs,t ≤ c2 ω (s, t) p .

(9.6)

On the other hand, employing the same logic as in the proof of Proposition 9.3, we see that Γis,u − Γis,t + Γit,u = SN +1 xi,s,t,u s,u − SN +1 xi,s,u s,u . We can therefore use Lemma 9.9 to see that N +1 Γs,u − Γs,t + Γt,u ≤ c3 εω (s, u) p .

(9.7)

Inequalities (9.6) and (9.7) allow us to use the (analysis) Lemma 6.2, and we learn that for all 0 ≤ s < t ≤ T , N +1 Γs,t ≤ c4 εω (s, t) p . From Proposition 7.63, N +1 π N +1 SN +1 x1,s,t s,t − SN +1 x2,s,t s,t ≤ c5 εω (s, t) p . This implies by the triangle inequality that N +1 π N +1 SN +1 x1 s,t − SN +1 x2 s,t ≤ c6 εω (s, t) p , i.e. that ρp,ω SN +1 x1 , SN +1 x2 ≤ ε. From Theorem 8.10 we immediately deduce Corollary 9.11 Let N ≥ [p] . (i) The map SN : Cop-var [0, T ] , G[p] Rd → Cop-var [0, T ] , GN Rd

9.2 Spaces of geometric rough paths

191

is uniformly continuous on bounded sets, using the dp-var -metric. (ii) The map SN : Co1/p-H¨o l [0, T ] , G[p] Rd → Co1/p-H¨o l [0, T ] , GN Rd is uniformly continuous on bounded sets, using the d1/p-H¨o l -metric.

9.2 Spaces of geometric rough paths Theorem 9.12 Let p ≥ 1 and N ≥ [p] . Let Ω denote either Cop-var , 1/p-H¨o l 0,1/p-H¨o l or Co . Then Co0,p-var , Co (i) the map x ∈ Ω([0, T ], GN Rd ) → π 0,[p] (x) ∈ Ω([0, T ], G[p] Rd ) is a bijection, with inverse the map SN ; (ii) for p ≥ 2 and d ≥ 2, the map x ∈ Ω([0, T ], GN Rd ) → π 0,[p]−1 (x) ∈ Ω([0, T ], G[p]−1 Rd ) is not a bijection. Remark 9.13 The proof of part (ii) will show that: (case (ii a)) π 0,[p]−1 is not an injection, when p is not an integer. It is proven in [122] that it is a surjection when p is not an integer. (case (ii b1)) π 0,[p]−1 is not an injection when p is an integer, and Ω = p-var C or Ω = C 1/p-H¨o l . (case (ii b2)) π 0,[p]−1 is not a surjection, when p is an integer, and Ω = C 0,p-var or Ω = C 0,1/p-H¨o l ; we leave it as an exercise for the reader to prove that it is not an injection in this case. Proof. (i) The case Ω = C p-var follows from Theorem 9.5 and C 1/p-H¨o l is an obvious corollary from the C p-var case. The case Ω = C 0,p-var (resp. C 0,1/p-H¨o l ) follows from the case Ω = C p-var (resp. C 1/p-H¨o l ) by Wiener’s characterization (Theorem 8.22). (ii) (a) We ﬁrst assume that Ω = C p-var or C 1/p-H¨o l . Let h be a non-zero ⊗[p] -valued path which is 1/q-H¨older with q = N/p. As in g[p] Rd ∩ Rd the proof of Lemma 7.61, we see that if x ∈ Ω([0, T ], G[p] Rd ) and y is deﬁned by yt = xt ⊗ exp (ht ) , d [p] R ) and π 0,[p−1] (y) = π 0,[p−1] (x) . This means then y ∈ Ω([0, T ], G that π 0,[p]−1 (y) = π 0,[p]−1 (x), and as y = x, π 0,[p]−1 is not an injection from Ω([0, T ], GN Rd ) into Ω([0, T ], G[p−1] Rd ).

Geometric rough path spaces

192

(b) We now assume that Ω = C 0,p-var or C 0,1/p-H¨o l . (b1) We deal with the case N > p and again take a non-zero ⊗[p] , h ∈ C 0,1/q -H¨o l [0, T ] , g[p] Rd ∩ Rd with q = N/p. Deﬁne the path y as above, yt = xt ⊗ exp (ht ). We have already seen that y ∈ Ω([0, T ], G[p] Rd ). Using Wiener character ization we actually see that y ∈ Ω([0, T ], G[p] Rd ) (it is at this point that we need q > 1, that is N > p). Once again, π 0,[p]−1 (y) = π0,[p]−1 (x), and as y = x, π 0,[p]−1 is not an injection from Ω([0, T ], GN Rd ) into Ω([0, T ], G[p]−1 Rd ). (b2) It only remains to deal with the case Ω = C 0,p-var or C 0,1/p-H¨o l , and N = p (which implies that p ∈ {2, 3, . . . }). We aim to prove in this case that π 0,[p]−1 is not a surjection or, in other words, that there exists a path x ∈ Ω([0, T ], G[p]−1 Rd ) which admits no lift to Ω([0, T ], G[p] Rd ). To this end, assume T = 1 for simplicity of notation, and assume we have a path y ∈ Ω([0, 1], G[p] Rd ) that projects down onto x. Let ω (s, t) = p yp-var;[s,t] , which is ﬁnite by assumption. By deﬁnition of the increment of y, we have n 24 −1 y in , i +n1 . y0,1 = 2

2

i=0

˜ in , i +n1 = log x in , i +n1 , Deﬁne x ˜ in , i +n1 ∈ G[p] Rd by π 0,[p]−1 log x 2 2 2 2 2 2 ˜ in , i +n1 = 0. Observe that x ˜ in , i +n1 are not the increments and π [p] log x 2 2 2 2 ˜ as a map that associates to every of a G[p] Rd -valued path; we view x d i i+1 [p] an element in G R . Hence, as we also dyadic interval of form n , 2n 2 have π 0,[p]−1 y

i 2n

, i2+n1

y0,1 =

=x

n 24 −1

x ˜

i 2n

i 2n

,

, i2+n1

i+ 1 2n

i=0

, a short computation gives

+

n 2 −1

π [p] log y

i 2n

, i2+n1

.

i=0

Then, by equivalence of homogenous norms, there exists c > 0 such that π [p] log y 2 in

,

i+ 1 2n

[p] c y in , i +n1 2 2 i i+1 by deﬁnition of ω. ≤ cω , 2n 2n

≤

In particular, we obtain that n −1 24 x ˜ in , i +n1 ≤ ω (0, 1) + |y0,1 | < ∞. 2 2 i=0

9.2 Spaces of geometric rough paths

193

Therefore, we proved that a necessary condition for a path x ∈ Ω([0, T ], G[p]−1 Rd ) to admit a lift y ∈ Ω([0, T ], G[p] Rd ) is given by 2 n −1 4 sup x ˜ in , i +n1 < ∞. 2 2 n ≥0 i=0

We therefore aim to provide a path x such that the above expression is inﬁnite. To this end, using d ≥ 2, we let e1 , e2 be the ﬁrst two vectors in the ⊗i standard basis of Rd . Deﬁne v1 = e2 , and vi+1 = [e1 , vi ] so that vi ∈ Rd . 0, p −1 ol p -H¨

0,1/p-H¨o l

For two paths, f ∈ C0 ([0, T ] , R) and g ∈ C0 deﬁne x ∈ Ω([0, T ], G[p]−1 Rd by

([0, T ] , R) , we

x (t) = ef t e 1 + g t v [ p −1 ] = ef t e 1 ⊗ eg t v [ p −1 ] ∈ G[p]−1 Rd . We note that x is indeed in Ω = C 0,p-var , from the deﬁnition of f, g and 1 xs,t = ef s , t e 1 ⊗ eg s , t v [ p −1 ] ≤ (const) × |fs,t | + |gs,t | p −1 . Deﬁning x ˜ in , i +n1 ∈ G[p] Rd as explained above, the Campbell–Baker– 2 2 Hausdorﬀ formula yields n 24 −1

i=0

x ˜

i 2n

, i2+n1

= exp f0,1 e1 + g0,1 v[p−1] + 6 n 2 −1 In particular, supn i=0 x ˜

n 2 −1

f 2 in g

i 2n

, i2+n1

− g 2 in f

i 2n

, i2+n1

v[p]

.

i=0 i 2n

2 n −1 f 2 in g in sup 2 n ≥0

, i2+n1

, i2+n1

< ∞ if and only if − g 2 in f

i=0

which itself is equivalent to

2 n −1 f 2 in g in sup 2 n ≥0 i=0

i 2n

i + 1 < ∞, , 2n

i + 1 < ∞. , 2n

The following Exercise 9.14 shows that for any p, q ≥ 1 with 1/p + 1/q = 1 there exist paths 0,1/p-H¨o l

0,1/q -H¨o l

([0, T ] , R) , g ∈ C0 ([0, T ] , R) f ∈ C0 n 2 −1 such that supn ≥0 i=0 f 2 in g in , i +n1 = ∞. In particular, we now see that 2 2 there exists a path x ∈ Ω([0, T ], G[p−1] Rd ) which does not admit a lift to Ω([0, T ], G[p] Rd ).

Geometric rough path spaces

194

Exercise 9.14 Assume [0, 1] is a continuous increasing 1] → ∞ that−1φ, ϕ−k: [0, ∞ −1 −k 2 and 2 are convergent φ ϕ bijection such that k =1 k =1 series. We then deﬁne the functions xφ : t ∈ [0, 1] → y φ : t ∈ [0, 1] →

∞ k =1 ∞

φ−1 2−k sin 2k +1 πt , φ−1 2−k 1 − cos 2k +1 πt .

k =1

(i) Prove that if φ is such that limx→0 xp /φ (x) = 0, then xφ and y φ belong 0,1/p-H¨o l ([0, T ] , R) . to C0 (ii) Prove that n 2 −1

i=0

1 j −1 −j −1 −j 2 ϕ 2 . 2 φ 4 j =1 n

xφi y ϕi 2n

2n

,

i+ 1 2n

≥

0,1/p-H¨o l

(iii) Provide an example of functions f, g ∈ C0 ([0, 1] , R) , for 1/p + 1/q = 1, such that 2 n −1 sup f 2 in g in , i +n1 = ∞. 2 2 n ≥0

0,1/q -H¨o l

([0, 1] , R)×C0

i=0

Solution. (i) First case. We ﬁrst assume that φ−1 (x) ≤ K |x| for all s, t ∈ [0, 1] , φ x (t) − xφ (s) ∞ ≤ 2−k /p sin 2k +1 πt − sin 2k +1 πs k =1



≤ 2K  

1

log 2

t −s

k =1

k =log 2

≤ 2K π |t − s|

1 log 2 t −s

2k (1−1/p) +

k =1 1/p

≤ cK |t − s|

+∞

. Then,

 φ−1 2−k 

+∞

φ−1 2−k 2k +1 π |t − s| +

1/p

1 t −s



2−k /p 

1 k =log 2 t −s

.

1/p . Hence, for all = 0, φ−1 (x) = o |x| −k −1 2 ≤ ε2−k /p . We ε > 0, there exists n ≥ 0 such that k ≥ n implies φ write xφ = xφ,0,n + xφ,n ,∞ , Second case. Now, if limx→0

xp φ(x)

9.2 Spaces of geometric rough paths

where xφ,i,j = we have

195

φ−1 2−k sin 2k +1 πt . Clearly, as xφ,0,n is smooth, φ,0,n x (t) − xφ,0,n (s) sup = 0. lim 1/p h→0 s,t,|t−s|≤h |t − s|

j

i+1

Moreover, working as in the ﬁrst case, we have φ x (t) − xφ (s) ≤ cε |t − s|1/p . Hence, we proved that for all ε > 0, φ x (t) − xφ (s) sup ≤ cε, lim 1/p h→0 s,t,|t−s|≤h |t − s| which concludes the proof of (i). − cos 2k +1 π 2in and sin 2k +1 π 2in are equal to 0, (ii) As cos 2k +1 π i+1 2n we obtain that n 2 −1

i=0

xφi y ϕi 2n

2n

,

i+ 1 2n

=

n

n φ−1 2−k ϕ−1 2−j .Ij,k

j,k =1

where n Ij,k

=

n 2 −1

i=0

% & i+1 i − cos 2j +1 π n . sin 2πi2k −n cos 2j +1 π n 2 2

n n Trigonometric exercises show that Ij,k = 0 if j = k, and that Ij,j ≥ 2j −2 . As φ−1 and ϕ−1 are positive, we therefore obtain that n 2 −1

i=0

1 j −1 −j −1 −j 2 ϕ 2 . 2 φ 4 j =1 n

xφi y ϕi 2n

2n

, i2+n1

≥

(iii) Just take f, g = xφ , xϕ with φ (x) = xp log x1 and ϕ (x) = xq log x1 . We therefore showed that the sets Cop-var ([0, T ], G[p] Rd ), Co0,p-var ([0, T ], 1/p-H¨o l 0,1/p-H¨o l G[p] Rd ), Co ([0, T ], G[p] Rd ) and Co ([0, T ], G[p] Rd ) are quite fundamental! We therefore give their elements names: Deﬁnition 9.15 (i) A weak geometric p-rough path is a continuous path of ﬁnite p-variation with values in the free nilpotent group of step [p] over Rd , i.e. an element of C p-var ([0, T ], G[p] Rd ). (ii) A geometric p-rough path is a continuous path with values in the free nilpotent group of step [p] over Rd which is in the p-variation closureof the set of bounded variation paths, i.e. an element of C 0,p-var ([0, T ], G[p] Rd ). (iii) A weak geometric 1/p-H¨older rough path is a 1/p-H¨older path with values in the free nilpotent group of step [p] over Rd , i.e. an element of

196

Geometric rough path spaces

C 1/p-H¨o l ([0, T ], G[p] Rd ). (iv) A geometric 1/p-H¨older rough path is a continuous path with values in the free nilpotent group of step [p] over Rd which is in the 1/p-H¨older closure older paths, i.e. an element of C 0,1/p-H¨o l ([0, T ], G[p] d of the set of 1-H¨ R ). Recall from the interpolation results of the previous chapter that ⊂ C 0,p-var ([0, T ], G[p] Rd C (p+ ε)-var ([0, T ], G[p] Rd ⊂ C p-var ([0, T ], G[p] Rd and for this reason the diﬀerence between weak and genuine geometric p-rough paths is important only when we care about very precise results. Exercise 9.16 Identify the G2 R2 with the 3-dimensional Heisenberg group H ∼ = R3 . Verify that the “pure-area path” (0, 0; t) is a weak geometric 2-rough path but not a genuine 2-rough path. ⊗n . Show that t → exp (t g) ⊂ Exercise 9.17 Assume g ∈ gn Rd ∩ Rd older geometric n-rough path and compute explicitly Gn Rd is a weak H¨ SN (exp ((·) g)) ⊂ GN Rd , n ≤ N. Solution. Claim that SN (expn (· g))0,t = expN (tg) where we write expk for the exp-map in T k Rd in order to distinguish from expn and expN . To see this, note that d (expN (s g) , expN (t g)) = expN (−sg) ⊗ expN (tg) and from the Campbell–Baker–Hausdorﬀ formula, this clearly equals 1/n 1/n expN ((t − s) g) , which is bounded by a constant times (t − s) |g| . older and by uniqueness of the Lyons lift It follows that expN (· g) is 1/n-H¨ the claim is proved.

9.3 Invariance under Lipschitz maps The content of this section is not used directly in the sequel and depends on techniques of the forthcoming Section 10.6 on rough integration. The probabilistic motivation here is the fact that Φ ◦ M , the image of a semimartingale M under a C 2 -map Φ, is again a semi-martingale; a manifest consequence of Itˆo’s lemma. Having deﬁned what wemean by a weak geometric p-rough path, say x ∈ C p-var [0, T ] , G[p] Rd , it is natural to ask whether the (yet to be

9.4 Young pairing of weak geometric rough paths

197

deﬁned!) image of x under a suﬃciently smooth map Φ : Rd → Re is also a weak geometric p-rough path. The following result can then be summarized by saying that the image of a weak geometric p-rough path under a Lipγlo c map, γ > p, is indeed another weak geometric p-rough path. Theorem 9.18 Assume d [p] R ; (i) x ∈ C p-var [0, T ] , G (ii) Φ ∈ Lipγlo c Rd , Re , γ > p. Then there exists a unique continuous (in fact, uniformly continuous on bounded sets) map Φ∗ : C p-var [0, T ] , G[p] Rd → C p-var [0, T ] , G[p] (Re ) with the property that, whenever x = S[p] (x) for some x ∈ C 1-var [0, T ] , Rd then Φ∗ x = S[p] (Φ ◦ x) . Proof. The proof relies on the forthcoming Theorem d e 10.47 in Section 10.6. R , R satisﬁes the assumpIndeed, ϕ := DΦ = (∂1 Φ, . . . , ∂d Φ) ⊂ Lipγlo−1 c tion of that theorem, so that · Φ∗ x := ϕ (xs ) dxs (rough integral) 0

is a well-deﬁned geometric p-rough path; more precisely, an element of C p-var [0, T ] , G[p] (Re ) and the rough integral Φ∗ x is a (uniformly) continuous (on bounded sets) function of the integrator x in p-variation met“push-forward” behaviour whenever ric. To see that Φ∗ x has the claimed x = S[p] (x) for some x ∈ C 1-var [0, T ] , Rd it suﬃces to note that, by the fundamental theorem of calculus, t Φ (x)0,t = ϕ (xs ) dxs (classical Riemann–Stieltjes integral). 0

By the basic consistency properties of a rough integral Φ∗ x is the precisely the step-[p] lift of the indeﬁnite Riemann–Stieltjes integral · ϕ (xs ) dxs 0

and so Φ∗ x = S[p] (Φ ◦ x) as was claimed.

9.4 Young pairing of weak geometric rough paths Throughout this section, we ﬁx p > q ≥ 1 such that q −1 +p−1 > 1. Observe that this implies q ∈ [1, 2).

Geometric rough path spaces

198

9.4.1 Motivation

Consider a path x ∈ C p-var [0, T ] , Rd . It is natural, e.g. in the context of diﬀerential equations with drift term, to consider the “space-time” path t → (x (t) , t), plainly an element of C p-var [0, T ] , Rd+1 . It can also be important to replace x by x + h where h is a suitable perturbation.3 Let us now move to a genuine geometric rough path setting and consider x ∈ C p-var [0, T ] , G[p] Rd . Recall the intuition that x contains the a priori information of up to [p] iterated integrals which are not deﬁned, in general, in Riemann–Stieltjes or Young sense; there is, however, enough regularity built into the deﬁnition of such a geometric p-rough path that higher iterated integrals, i.e. beyond level [p], are canonically deﬁned, as was seen in our discussion of the Lyons-lifting map.

Now, even if x = π 1 (x) has not suﬃcient

regularity

to form the integral x ⊗ dx, one surely can form the integral xdt or x ⊗ dh for suﬃciently regular h, e.g. when the last integral is a well-deﬁned Young integral as discussed in Section 6.2. In particular, one would hope that, given a suﬃciently regular h : [0, T ] → Rd there is a canonically deﬁned geometric rough path, say S[p] (x, h), with values in G[p] Rd ⊕ Rd which coincides with S[p] (z)

where z = (x, h) : [0, T ] → Rd ⊕ Rd whenever x = S[p] (x) for some nice path x. By the same token, given a suﬃciently regular h : [0, T ] → Rd one would hope that there is a canonically deﬁned geometric rough path, say Th (x), with values in G[p] Rd which coincides with S[p] (x + h) whenever x = S[p] (x) for some nice path x. Moreover, such constructions should be robust, for example so that x → Th (x) is continuous in (e.g. p-variation) rough path distance. When p ∈ (2, 3), so that x takes values in G2 Rd , the reader will have no diﬃculties in deriving such results by making use of the Young–L´ oeve estimates of Section 6.1. The remainder of this chapter is devoted to handling the general case.

9.4.2

The space C (p,q)-var [0, T ] , Rd ⊕ Rd

We recall from Section 7.5.6 that for ﬁxed λ ∈ R, the dilation map δ λ : GN Rd → GN Rd is the unique group homomorphism which extends scalar mutiplication a ∈ Rd → λa ∈ Rd . Similarly, for ﬁxed γ 1 , γ 2 ∈ R, the map (a, b) ∈ Rd ⊕ Rd → (γ 1 a, γ 2 b) ∈ Rd ⊕ Rd lifts to a group homomorphism δ γ 1 ,γ 2 : GN Rd ⊕ Rd → GN Rd ⊕ Rd . (Elements in GN Rd ⊕ Rd arise as the step-N signature of a path in

Rd ⊕ Rd . Scaling the ﬁrst d (resp. last d ) coordinates of the path by γ 1 3 For

example, when adding a Cameron–Martin path to Brownian motion.

9.4 Young pairing of weak geometric rough paths

199

(resp. γ 2 ) gives rise precisely to the δ γ 1 ,γ 2 -dilation of the original signature.) Observe that almost by deﬁnition of δ γ 1 ,γ 2 , we have for a path (x, h) ∈ 1-var d 1-var d [0, 1] , R × C C [0, 1] , R , δ γ 1 ,γ 2 SN (x ⊕ h)0,1 = SN (γ 1 x ⊕ γ 2 h)0,1 . is the set of continuous GN Rd Noting that C p-var [0, T ] , GN Rd valued paths x such that for some control function ω (e.g. (s, t) → xp-var;[s,t] ) one has δ 1 < ∞, sup x s,t 1/p ω (s ,t )

s< t in [0,T ]

we are led to the following deﬁnition.

Deﬁnition 9.19 We say that a continuous path x ∈ C [0, T ] , GN Rd ⊕ Rd is of ﬁnite mixed (p, q)-variation, and write x ∈ C (p,q )-var [0, T ] , GN Rd ⊕ Rd if, for some control ω, xp,q -ω ;[0,T ] := sup δ 1 1 / p , 1 1 / q (xs,t ) < ∞. ω (s ,t )

s< t in [0,T ]

ω (s ,t )

(A convention of type 0/0 = 0 is in place, to deal with s < t such that ω (s, t) = 0.) If we can take ω (s, t) = |t − s| , we say that x is p1 , 1q 1 1 . H¨ older and write x ∈ C ( p , q )-H¨o l [0, T ] , GN Rd ⊕ Rd (p,q )-var As usual, we write Co [0, T ] , GN Rd ⊕ Rd , etc. if we only con sider paths started at o, the unit element in GN Rd ⊕ Rd . Deﬁnition 9.20 For a pair of controls ω 1 , ω 2 , we also deﬁne 1 1 sup (x ) xp,q -ω 1 ,ω 2 ;[0,T ] = δ s,t , ω 1 (s ,t )1/ p , ω 2 (s ,t )1/ q s< t in [0,T ]

, and, for two paths x1 , x2 ∈ C (p,q )-var [0, T ] , GN Rd ⊕ Rd 1 2 ρp,q -ω 1 ,ω 2 ;[0,T ] x , x = 1 2 1 1 1 1 sup δ xs,t − δ xs,t . , , ω 1 (s ,t )1/ p ω 2 (s ,t )1/ q ω 1 (s ,t )1/ p ω 2 (s ,t )1/ q s< t in [0,T ]

Exercise 9.21 Let x ∈ C

(p,q )-var

[0, T ] , G

N

d

R ⊕R d

and assume ω 1

and ω 2 are control functions. Show that, for all s < t in [0, T ] , 1/p 1/q . xs,t ≤ c xp,q -ω 1 ,ω 2 ;[0,T ] ω 1 (s, t) + ω 2 (s, t)

Geometric rough path spaces

200

9.4.3 Quantitative bounds on SN

For (x, h) ∈ C 1-var [0, T ] , Rd ⊕ Rd , we now aim to show that for every N ∈ {1, 2, . . . }, SN (x ⊕ h)s,t ≤ (const) × S[p] (x)p-var;[s,t] + |h|q -var;[s,t] . As in earlier chapters, the constant here depends only on the p-variation of x and the q-variation of h, allowing for a subsequent passage to the limit. The argument is similar to proving the Lyons-lift estimate SN (x)s,t ≤ (const) × S[p] (x)p-var;[s,t] , although in the latter the case N ∈ {1, . . . , [p]} is trivial. To start, we need a lemma replacing the use of geodesics. Lemma 9.22 Let N ≥ 1, and (x, h) ∈ C 1-var [s, t], Rd × C 1-var [s, t], Rd such that, for ﬁxed α, β > 0, SN +1 (αx)s,t + SN (αx ⊕ βh)s,t ≤ C1 . Then there exists a path (xs,t , hs,t ) such that (i) SN (x ⊕ h)s,t = SN xs,t ⊕ hs,t s,t SN +1 (x)s,t = SN +1 xs,t s,t ;

(9.8) (9.9)

(ii) there exists a constant CN which depends only on C1 and N such that

t

α

s,t dxu + β

s

t

s,t dhu ≤ CN .

s

Proof. Observe that if SN (x ⊕ h)s,t = SN (xs,t ⊕ hs,t )s,t , then SN (αx ⊕ βh)s,t = SN (αxs,t ⊕ βhs,t )s,t and SN +1 (αx)s,t = SN +1 (αxs,t )s,t . Hence, we can assume without loss of generality that α = β = 1. By deﬁnition of the Carnot–Caratheodory homogenous norm, there exist two paths x1,s,t , h1,s,t such that SN (x ⊕ h)s,t = SN x1,s,t ⊕ h1,s,t s,t , with

s

t

1,s,t dxu +

s

t

1,s,t dhu ≤ c1 SN (x ⊕ h)s,t .

9.4 Young pairing of weak geometric rough paths

201

−1 Then, deﬁne g = SN +1 x1,s,t s,t ⊗ SN +1 (x)s,t , and observe that

g

≤

1,s,t dxu + SN +1 (x)s,t s c3 SN (x ⊕ h)s,t + SN +1 (x)s,t

≤

c4 .

t

≤ c2

Deﬁne a path x2,s,t such that SN +1 x2,s,t s,t = g, with

t

2,s,t dxu = g ≤ c4 .

s

We have

and

SN +1 x1,s,t s,t ⊗ SN +1 x2,s,t s,t = SN +1 (x)s,t SN x1,s,t ⊕ h1,s,t ⊗ SN x2,s,t ⊕ 0 = SN (x ⊕ h)s,t .

Therefore, concatenating x1,s,t ⊕ h1,s,t and x2,s,t ⊕ 0 gives us a path that satisﬁes the required conditions of the lemma. We then need a slight generalization of Lemma 9.2. Lemma 9.23 Let x1 ⊕h1, x2 ⊕h2 be two paths in C 1-var [s, u], Rd ⊕Rd . Assume that (i) SN x1 ⊕ h1 s,u = SN x2 ⊕ h2 s,u , 1 2 (ii) SN +1 x s,u = SN +1 x s,u , u u

u

u (iii) s dx1r + s dx2r ≤ 1 and s dh1r + s dh2r ≤ 2 . Then, there exists a constant C = C (N ) such that N +1 +1−k k N 2 . SN +1 x1 ⊕ h1 s,u − SN +1 x2 ⊕ h2 s,u ≤ C 1 k =1

Proof. Working as in Lemma 9.2, we can assume without loss of generality that x2 ⊕ h2 = 0, and (s, u) = (0, 1) . Deﬁne for convenience y = x1 ⊕ h1 . We have, by deﬁnition of SN +1 and the triangle inequality SN +1 (y)s,u − 1 ≤

i 1 ,...,i N + 1

i1 iN + 1 dyr 1 . . . dyr N + 1 . s< r 1 < ...< r N + 1 < u

202

Geometric rough path spaces

Because SN +1 x1 s,u = 1, we have SN +1 (y)s,u − 1 ≤ i 1 ,...,i N + 1 {i 1 ,...,i N + 1 }∩{d+1,...,d+ d } = 0

i1 iN + 1 dyr 1 . . . dyr N + 1 . s< r 1 < ...< r N + 1 < u

i By (iii), s< r 1 < ...< r N + 1 < u dyri 11 . . . dyrNN ++ 11 is bounded by a constant times +1−k k N 2 , where k is the cardinal of {i1 , . . . , iN +1 } ∩ {d + 1, . . . , d + d }. 1 That concludes the proof. We can now generalize lemma 9.3, to give a quantitative estimate on SN (x ⊕ h)p-var for N ≥ 1. Lemma 9.24 Let (x, h) be a path in C 1-var [0, T ] , Rd ⊕ Rd , and p q ω 1 = S[p] (x)p-var;[.,.] , ω 2 = |h|q -var;[.,.] .

For all N ≥ 1, there exists a constant C = C (N, p, q) (and not depending on the 1-variation norm of x or h) such that (9.10) SN (x ⊕ h)p,q -ω 1 ,ω 2 ;[0,T ] ≤ C. Proof. Deﬁne (HN ): for all paths x ⊕ h ∈ C 1-var [0, T ] , Rd ⊕ Rd with S[p] (x) and |h|q -var;[0,T ] bounded above by 1, we have for some p-var;[0,T ] constant cN , for all s < t in [0, T ] , SN (x ⊕ h)p,q -ω 1 ,ω 2 ;[0,T ] ≤ cN . First observe that, for N = 1, we have for s < t ∈ [0, T ] , hs,t xs,t ⊕ ≤ 2, 1/q ω 1 (s, t)1/p ω 2 (s, t) i.e. (H1 ) is satisﬁed. Then, to prove that (HN ) implies for all x ⊕ h ∈ notice it is enough C 1-var [0, T ] , Rd ⊕ Rd with S[p] (x)p-var;[0,T ] and |h|q -var;[0,T ] bounded above by 1, we have SN (x ⊕ h)0,T ≤ cN +1 . Indeed, for an arbitrary x ⊕ h ∈ C 1-var [0, T ] , Rd ⊕ Rd , applying the x h would imply that , above to S [ p ] (x) p - v a r ; [ 0 , T ] |h|q - v a r ; [ 0 , T ] h x ⊕ SN ≤ cN . S[p] (x) |h| q -var;[0,T ] p-var;[0,T ]

9.4 Young pairing of weak geometric rough paths

203

By time change, the above would also hold for all s, t, when replacing [0, T ] by [s, t] . This is precisely saying that SN (x ⊕ h)p,q -ω 1 ,ω 2 ;[0,T ] ≤ C. Let us therefore ﬁx some path x ⊕ h ∈ C 1-var [0, T ] , Rd ⊕ Rd with S[p] (x) and |h|q -var;[0,T ] less than 1. We deﬁne the control ω by p-var;[0,T ] ω (s, t) =

1 q S[p] (x)p + |h| q -var;[0,T ] , p-var;[0,T ] 2

which satisﬁes ω (0, T ) ≤ 1. The induction the hypothesis tells us that h x SN ≤ c1 . ⊕ 1/p 1/q ω (s, t) ω (s, t) s,t x ≤ c2 . If If N + 1 ≤ [p] , the hypothesis tells us that SN +1 ω (s,t) 1 / p s,t x ≤ c2 . N + 1 > [p] , then Theorem 9.5 tells us that S N +1 ω (s,t) 1 / p s,t

s,t

Hence, following Lemma 9.22, we can deﬁne the path x that SN (x ⊕ h)s,t = SN xs,t ⊕ hs,t s,t SN +1 (x)s,t = SN +1 xs,t s,t and such that

1 ω (s,t) 1 / p

t s

|dxs,t u |+

⊕ hs,t , such

t 1 |dhs,t u | is bounded ω (s,t) 1 / q s t,u t,u s,u s,u

above by a

constant c2 . Deﬁne similarly the paths x , h , x , h , and deﬁne xs,t,u (resp. hs,t,u ) to be the concatenation of xs,t and xt,u (resp. hs,t and ht,u ). Then, deﬁne Γs,t = SN +1 (x ⊕ h)s,t − SN +1 xs,t ⊕ hs,t s,t . Working as in Proposition 9.3, |Γs,u − (Γs,t + Γt,u )| ≤ SN +1 xs,t,u ⊕ hs,t,u s,u − SN +1 (xs,u ⊕ hs,u )s,u , which we bound using Lemma 9.23: |Γs,u − (Γs,t + Γt,u )| ≤ c3

N +1

ω (s, u)

k =1

≤

c4 ω (s, u)

θ

N + 1 −k p

k

ω (s, u) q

204

Geometric rough path spaces

where θ = min1≤k ≤N +1 N +1−k + kq > 1. Using, as in the proof of Prop postion 9.3, that x and h are actually of bounded variation, we obtain using the (analysis) Lemma 6.2 that for all s, t, θ

|Γs,t | ≤ c5 ω (s, t) . We then deduce from the triangle inequality that π N +1 SN +1 (x ⊕ h)0,T ≤ c6 , which concludes the proof.

9.4.4 Deﬁnition of Young pairing map We deﬁne of (i) pN to be the extension of the projection onto the ﬁrst d coordinates Rd ⊕ Rd to a homomorphism from GN Rd ⊕ Rd onto GN Rd ; (ii) p N to be the extension of the projection of onto the last d coordinates d d N d d N d R ⊕R onto G R . R ⊕ R to a homomorphism from G

Deﬁnition 9.25 Let (x, h) ∈ Cop-var [0, T ] , G[p] Rd ×Coq -var [0, T ] , Rd . such that pN (z) = SN (x) A path z in Cop,q -var [0, T ] , GN Rd ⊕ Rd and p N (z) = SN (h) is said to be a (p, q)-Lyons lift or Young pairing of (x, h) of order N . We shall see in the following theorem that such a Young pairing is unique, and will denote it by SN (x, h) or SN (x ⊕ h). Theorem 9.26 Let (x, h) ∈ Cop-var [0, T ] , G[p] Rd ×Coq -var [0, T ] , Rd , and N ≥ 1. Then there exists a unique (p, q)-Lyons lift of order N of p-var (x, h). By writing SN (x, h) for this path, we deﬁne the map SN on Co d [p] q -var d [0, T ] , G R × Co [0, T ] , R . Moreover, if ω 1 = S[p] (x)p-var;[.,.] , ω 2 = |h|q -var;[.,.] , then, for some constant C = C (N, p, q), SN (x, h)p,q -ω 1 ,ω 2 ;[s,t] ≤ C. Proof. The existence of such a lift follows the same lines as the homogenous case, but using Lemma 9.24 rather than Lemma 9.3. Uniqueness follows from the following lemma. Lemma 9.27 Let y and z be two elements of Cop,q -var [0, T ], GN Rd ⊕Rd such that pN (z) = pN (y) and p N (z) = p N (y). Then we have z = y.

9.4 Young pairing of weak geometric rough paths

205

Proof. Let (HN ) be the induction hypothesis that the lemma holds true for level-N paths, i.e. as written in the statement. For N = 1, there is nothing to prove, hence (H1 ) is true. Assume now that (HN −1 ) is true, and p q let us prove that (HN ) is true. Deﬁne ω 1 = yp-var;[.,.] , ω 2 = zq -var;[.,.] .

We ﬁx e1 , . . . , ed a basis of Rd and ed+1 , . . . , ed+ d a basis of Rd , so that e1 , . . . , ed+ d is a basis of Rd ⊕ Rd . ⊗N ∩ gN Rd ⊕ Rd -valued path f by Deﬁne the Rd ⊕ Rd f (t) = log z−1 t ⊗ yt . With f (s, t) = f (t) ≡ f (s) we have exp (f (s, t)) = z−1 s,t ⊗ ys,t ; we may also write f (t) =

fi 1 ,...,i N (t) e1 ⊗ . . . ⊗ ed+ d .

1≤i 1 ,...,i N ≤d+d

0 if {i1 , . . . , iN } ∩

Hypothesis (HN −1 ) implies that fi 1 ,...,i N = {d + 1, . . . , d + d } = ∅ and from Lemma 7.61, |fi 1 ,...,i N (s, t)| ≤ c1 ω 1 (s, t)

N −a p

α

ω 2 (s, t) q ,

where a is the cardinal of the set {i1 , . . . , iN } ∩ {d + 1, . . . , d + d } . Set ω = ω 1 + ω 2 so that |fi 1 ,...,i N (s, t)| ≤ c1 [2ω (s, t)]

N −a p

+ aq

.

But for all i1 , . . . , iN such that the cardinal of {i1 , . . . , iN }∩{d+1, . . . , d+d } is greater than or equal to 1, N p−a + aq > 1, which implies that for the respective component fi 1 ,...,i N is of ﬁnite r-variation for some r < 1, i.e. it is equal to fi 1 ,...,i N (0) = 0.

9.4.5 Modulus of continuity for the map SN We go quickly in this section as there are no new ideas required. First, we need to generalize Lemmas 9.22 and 9.23 to handle the “diﬀerence of paths”. Lemma 9.28 Let N ≥ 1, and xi , hi i=1,2 be two paths in C 1-var [s, t], Rd × C 1-var [s, t], Rd such that, for ﬁxed α, β > 0, SN +1 (αx)s,t + SN (αx ⊕ βh)s,t ≤ C.

Geometric rough path spaces

206

Then there exist paths xi,s,t , hi,s,t : i = 1, 2 such that (i) we have SN xi ⊕ hi s,t = SN xi,s,t ⊕ hi,s,t s,t SN +1 xi s,t = SN +1 xi,s,t s,t ; (ii) there exists a constant CN depending only on C and N such that t t i,s,t i,s,t dxu + β dhu ≤ CN ; α s

s

(iii) with ε = SN +1 αx1 s,t − SN +1 αx2 s,t + SN αx1 ⊕ βh1 s,t − SN αx2 ⊕ βh2 s,t , we have

α

t

1,s,t dxu − dx2,s,t +β u

s

t

1,s,t dhu − dh2,s,t ≤ CN ε. u

s

Proof. As in the proof of Lemma 9.22,we can assume α = β = 1. Using Proposition 7.64, there exist two paths xi,1,s,t , hi,1,s,t i=1,2 such that SN xi ⊕ hi s,t = SN xi,1,s,t ⊕ hi,1,s,t s,t , with

t

s

i,1,s,t dxu +

t

i,1,s,t dhu ≤ c1 ,

s

t 1,1,s,t 1,1,s,t dxu dhu − dxu2,1,s,t + − dhu2,1,s,t s s i i ≤ c1 SN x ⊕ h s,t − SN xi ⊕ hi s,t . t

−1 Then, deﬁne g i = SN +1 xi,1,s,t s,t ⊗SN +1 xi s,t . Observe that g i ≤ c2 . Then, from Lemma 9.1 1 g − g 2 = SN +1 x1,1,s,t s,t −SN +1 x2,1,s,t s,t − SN +1 x1 s,t −SN +1 x2 s,t ≤ SN +1 x1 s,t − SN +1 x2 s,t + SN +1 x1,1,s,t s,t − SN +1 x2,1,s,t s,t ≤ c3 ε using Proposition 7.63. Using Proposition 7.64 one more time, we deﬁne the paths xi,2,s,t i=1,2 by SN +1 xi,2,s,t s,t = g i ,

9.4 Young pairing of weak geometric rough paths

with

i,2,s,t dxu

=

i g ≤ c2

1,2,s,t dxu − dxu2,2,s,t

=

1 g − g 2 ≤ c3 ε.

t

s

t

207

s

Concatenating the paths xi,1,s,t ⊕ hi,1,s,t and xi,2,s,t ⊕ 0 gives us two paths that satisfy the required conditions of the lemma. We leave the proof of the next lemma, extending Lemma 9.23, to the reader. ˜i ˜i , h be four pairs in C 1-var Lemma 9.29 Let xi , hi i=1,2 , x i=1,2 [s, u] , Rd ⊕ Rd . Assume that ˜1 ˜2 ˜1 ⊕ h ˜2 ⊕ h (i) SN x1 ⊕ h1 s,u =SN x2 ⊕h2 s,u and SN x =SN x ; s,u s,u 1 2 1 2 ˜ s,u = SN +1 x ˜ s,u ; (ii) SN +1 x s,u = SN +1 x s,u and SN +1 x (iii) u u u u 1 2 1 2 dxr + dxr ≤ 1 and dhr + dhr ≤ 2 , s s s s u u u u 1 2 ˜1 ˜2 d˜ d˜ xr + xr ≤ 1 and dhr + dhr ≤ 2 ; s

(iv)

s

u

s

1 dxr − d˜ x1r +

u

s

2 dxr − d˜ x2r ≤ ε1 and

s s u u 1 2 1 ˜ + ˜ 2 ≤ ε2 . dh − d h dhr − dh r r r s

s

Then, SN +1 x1 s,u − SN +1 x2 s,u − SN +1 x3 s,u − SN +1 x4 s,u ≤ Cε

N +1

+1−k k N 2 . 1

k =1

Similar to the proof of Theorem 9.10 we are also led to

Theorem 9.30 Let 1 ≤ q ≤ p so that 1/p + 1/q > 1. Assume xi , hi i=1,2 are two pairs of elements in Cop-var [0, T ] , G[p] Rd × Coq -var [0, T ] , Rd , and ω a control such that for all s, t ∈ [0, T ] , for i = 1, 2, i p q x + hi q -var;[s,t] ≤ ω (s, t) p-var;[s,t] ≤ ε. ρp-ω x1 , x2 + ρq -ω h1 , h2

Geometric rough path spaces

208

Then, for all N ≥ 1, there exists a constant C depending only on N , p, and q such that ρp,q -ω SN x1 , h1 , SN x2 , h2 ≤ Cε. Corollary 9.31 Let ω be a control, 1 ≤ q ≤ q , 1 ≤ p ≤ p , 1/p + 1/q > 1.Then, for ﬁxed R > 0, the maps4 xp-ω ≤ R , dp -ω × |h|q -ω ≤ R , dq -ω , dp -ω → C p-ω [0, T ] , G[p] Rd ⊕ Rd (x, h)

→

and xp-var ≤ R , dp -var

× →

(x, h) → and

xp-var ≤ R , d∞

× →

(x, h) →

SN (x, h)

|h|q -var ≤ R , dq -var , dp -var C p-var [0, T ] , G[p] Rd ⊕ Rd SN (x, h)

|h|q -var ≤ R , d∞ , d∞ C p-var [0, T ] , G[p] Rd ⊕ Rd SN (x, h)

are uniformly continuous. Proof. A consequence of the previous theorem and an interpolation argument. Remark 9.32 As a typical application, we see that the Young pairing (x, h) → SN (x, h) is also continuous in the sense of “uniform convergence with uniform bounds”. Indeed, take any sequence of paths (xn , hn ) ∈ C 1-var [0, T ] , Rd such that < ∞ sup S[p] (xn )p-var;[0,T ] + |hn |q -var;[0,T ] n lim d∞ S[p] (xn ) , x + d∞ S[p] (hn ) , h = 0. n →∞

Then, by Theorem 9.26 and the last part of the previous corollary above it, it follows that sup S[p] (xn ⊕ hn )p-var;[0,T ] < ∞, n lim d∞ S[p] (xn ⊕ hn ) , S[p] (x ⊕ h) = 0. n →∞

4d q -ω

h 1 , h 2 = h 1 − h 2 q -ω .

9.4 Young pairing of weak geometric rough paths

209

9.4.6 Translation of rough paths

In Section 7.5.6, we deﬁned the map plus from GN Rd ⊕ Rd to GN Rd d to be the unique homomorphism such that d for d all x, y ∈ R , N plus (exp (x ⊕ y)) = exp (x + y) . If x is a G R ⊕ R -valued path, we N d can therefore deﬁne d the G R -valued path plus (x) : t ∈ [0, T ] → N plus (xt ) ∈ G R . When x is the weak geometric rough path equal to S[p] (y ⊕ h) , where y is a weak geometric p-rough path and h a weak geometric q-rough path, plus (x) is then a canonical notion of addition of two paths. Theorem 9.33 Let (x, h) ∈ C p-var [0, T ] , G[p] Rd × C q -var [0, T ] , Rd . The translation of x by h, denoted Th (x) ∈ C p-var [0, T ] , G[p] Rd , is deﬁned by Th (x)t = plus S[p] (x ⊕ h)t . (i) We have for some constant C1 depending only on p and q, Th (x)p-var;[0,T ] ≤ C1 xp-var;[0,T ] + |h|q -var;[0,T ] .

(9.11)

(ii) Let xi , hi i=1,2 ∈ C p-var [0, T ] , G[p] Rd × C q -var [0, T ] , Rd , and ω a control. If we have for all s, t ∈ [0, T ], i p q x + hi ≤ ω (s, t) , p-var;[s,t]

q -var;[s,t]

then 1/q −1/p ρp-ω Th 1 x1 , Th 2 x2 ≤ C2 ρp-ω x1 , x2 + ω (0, T ) ρq -ω h1 , h2 for some constant C2 depending only on p and q. i variation and Remark 9.34 If xi =S[p] xi where xi is of bounded i if h i i is also of bounded variation, then Th i (x) = S[p] h + x , i.e. Th i x is just the canonical lift of the sum of the paths xi and hi . i Proof. We ﬁrst prove the quantitative bound on Th i x p-var;[0,T ] Th i xi s,t

= ≤

plus S[p] xi ⊕ hi s,t c1 S[p] xi ⊕ hi s,t .

From Exercise 9.21, deﬁning p ω 1 = S[p] (x)p-var;[.,.] ,

q

ω 2 = |h|q -var;[.,.] ,

we have S[p] xi ⊕ hi s,t ≤ c2 S[p] xi ⊕ hi p,q -ω

1 ,ω 2

1/p

ω 1 (s, t)

1/q

+ ω 2 (s, t)

.

Geometric rough path spaces

210

From Theorem 9.26, S[p] xi ⊕ hi p,q -ω ,ω is bounded, which proves 1 2 (9.11). Then, for s, t ∈ [0, T ] , deﬁning εs,t = δ 1 1 / p Th 1 x1 s,t − δ 1 1 / p Th 2 ω (s ,t ) ω (s ,t ) 2 x s,t and using in the third line Proposition 7.65, we have |εs,t | = 1 2 1 2 δ 1 ω ( s , t ) 1 / p plus S[p] x ⊕ h s,t − δ ω ( s , t1) 1 / p plus S[p] x ⊕ h s,t 1 2 1 2 = plus δ 1 1 / p S[p] x ⊕ h s,t − plus δ 1 1 / p S[p] x ⊕ h s,t ω (s ,t ) ω (s ,t ) 1 1 2 2 ≤ c1 δ 1 1 / p S[p] x ⊕ h s,t − δ 1 1 / p S[p] x ⊕ h s,t ω (s ,t ) ω (s ,t ) 1/q −1/p 1 h c1 δ 1 1 / p , 1 1 / q S[p] x1 ⊕ ω (s, t) ω (s ,t ) ω (s ,t ) s,t ≤ . 1/q −1/p 2 2 −δ 1 1 / p , 1 1 / q S[p] x ⊕ ω (s, t) h ω (s ,t )

ω (s ,t )

s,t

Using Theorem 9.30, we then obtain |εs,t |

≤ c2 ρp-ω ≤ c2 ρp-ω

1/q −1/p 1 1/q −1/p 2 x1 , x2 + ρq -ω ω (s, t) h , ω (s, t) h 1 2 1/q −1/p x , x + ω (s, t) ρq -ω h1 , h2 .

Hence, as q < p, taking supremum over all s, t ∈ [0, T ] , we have 1/q −1/p ρp-ω Th 1 x1 , Th 2 x2 ≤ c2 ρp-ω x1 , x2 + ω (0, T ) ρq -ω h1 , h2 . As a corollary, interpolation provides us with the following uniform continuity on bounded sets result. Corollary 9.35 The rough path translation (x, h) → Th (x) as a map from Cop-var [0, T ] , G[p] Rd × Coq -var [0, T ] , Rd → Cop-var [0, T ] , G[p] Rd is uniformly continuous on bounded sets, using the dp-var -metric. This is also true as a map from × Co1/p-H¨o l [0, T ] , Rd Co1/p-H¨o l [0, T ] , G[p] Rd → Co1/p-H¨o l [0, T ] , G[p] Rd . Exercise 9.36 Assume T−x n (x) → 0. Show that this is, in general, not equivalent to S2 (xn ) → x and neither implies the other.

9.5 Comments

211

Exercise 9.37 This exercise will demonstrate (again!) the power of pvariation estimates in the sense that they immediately imply non-trivial estimates in terms of H¨ older and Besov norm. Recall from Example 5.16 that for δ ∈ (1/2, 1] and q = 1/δ, δ −1/2

|h|q -var;[s,t] ≤ (const) × |h|W δ , 2 -var;[s,t] |t − s|

.

Assume x ∈ C α -H¨o l [0, T ] , G[1/α ] Rd , h ∈ W δ ,2 [0, T ] , Rd with α ∈ (1/4, 1/2) and δ := α + 1/2. Show that Th (x)α -H¨o l;[s,t] ≤ (const) × xα -H¨o l;[s,t] + |h|W δ , 2 ;[s,t] . Explain the restriction α > 1/4. Solution. Set q = 1/δ = 1/ (α + 1/2) and p = 1/α so that h ∈ C q -var and x ∈ C p-var . To apply the above corollary we need 1/p + 1/q > 1 ⇐⇒ α + (α + 1/2) > 1, which explains the restriction α > 1/4. The actual estimate then immediately follows from Th (x)s,t

≤

c xp-var;[s,t] + c |h|q -var;[s,t]

≤

c xα -H¨o l;[s,t] |t − s| + c |h|W δ , 2 ;[s,t] |t − s| .

α

α

9.5 Comments The main results of this chapter can be found in Lyons [116], see also Lyons and Qian [120] and Lyons et al. [123], although some of our proofs are new. Let us note that the estimate for the Lyons lift (Theorem 9.5) can be made independent of N , a consequence of Lyons’ “neo-classical inequality”, see Lyons [116] and, for a sharpened version, forthcoming work by Hara and Hino. The necessity to distinguish between geometric rough paths and weak geometric rough paths was recognized in Friz and Victoir [63]. Exercise 9.37 is taken from Friz and Victoir [64].

10 Rough diﬀerential equations Our construction of Young’s integral was based on estimates for classical Riemann–Stieltjes integrals with constants depending only on the p- and q-variation of integrand and integrator respectively, followed by a limit argument. The same approach works for ordinary diﬀerential equations: in this chapter we establish estimates for ordinary diﬀerential equations with constants only depending on a suitable p-variation bound of the driving signal. A limiting procedure then leads us naturally to “rough diﬀerential equations”.

10.1 Preliminaries As was pointed out in Section 7.1, for a ﬁxed starting time s, a natural step-N approximation for the solution of the ODE dy = V (y) dx =

d

Vi (y) dxi , ys ∈ Re ,

i=1

is given by

yt ≈ ys + E(V ) y, SN (x)s,t

(10.1)

where Deﬁnition 10.1 (Euler scheme) Let N ∈ N. Given (N − 1) times con- tinuously diﬀerentiable vector ﬁelds V = (V1 , . . . , Vd ) on Re , g ∈ T (N ) Rd and y ∈ Re we call E(V ) (y, g) :=

N

Vi 1 . . . Vi k I (y) gk ,i 1 ,...,i k

k =1 i 1 ,...,i k ∈{1,...,d}

the (increment of the) step-N Euler scheme.

When g = SN (x)s,t , the (step-N ) signature of a path segment x|[s,t] , we call E(V ) ys , SN (x)s,t the (increment of the) step-N Euler scheme for dy = V (y) dx over the time interval [s, t].

10.1 Preliminaries

213

We now prove a simple error estimate for the step-N scheme. To this end, it is convenient to assume Lipschitz regularity of the vector ﬁelds in the sense of E. Stein. To prepare for the following deﬁnition, given a real γ > 0, we agree that %γ& is the largest integer strictly smaller than γ so that γ = %γ& + {γ} with %γ& ∈ N and {γ} ∈ (0, 1]. Deﬁnition 10.2 (Lipschitz map) A map V : E → F between two normed spaces E, F is called γ-Lipschitz (in the sense of E. Stein), in symbols V ∈ Lipγ (E, F ) or simply V ∈ Lipγ (E) if E = F , if V is %γ& times continuously diﬀerentiable and such that there exists a constant 0 ≤ M < ∞ such that the supremum norm of its kth derivatives, k = 0, . . . , %γ&, and the {γ}-H¨older norm of its %γ&th derivative are bounded by M . The smallest M satisfying the above conditions is the γ-Lipschitz norm of V , denoted |V |Lip γ . (It should be noted that LipN maps have (N − 1) bounded derivatives, with the (N − 1)th derivative being Lipschitz, but need not be N times continuously diﬀerentiable.) This deﬁnition applies in particular to a collection of vector ﬁelds V = (V1 , . . . , Vd ) on Re , which we can view as a map d e d Re into L R , R , equipped y → {a = a1 , . . . , ad → i=1 V (y) ai } from with operator norm. Saying that V ∈ Lipγ Re , L Rd , Re is equivalent to V1 , . . . , Vd ∈ Lipγ (Re ), but it is usually the γ-Lipschitz norm of V which comes up naturally in estimates. We are now ready to state a ﬁrst error estimate for the Euler approximation in (10.1). Proposition 10.3 (Euler ODE estimate) Let γ > 1, V = (Vi )1≤i≤d be a collection of vector ﬁelds in Lipγ −1 (Re ) and x ∈ C 1-var [s, t] , Rd . Then there exists a constant C = C (γ) such that γ t |dxr | . π (V ) (s, ys ; x)s,t − E(V ) ys , Sγ (x)s,t ≤ C |V |Lip γ −1 s

(10.2) Proof. At the cost of replacing x by |V |Lip γ −1 x and Vi by

1 |V |L i p γ −1

Vi , we

can and will assume that |V |Lip γ −1 = 1. We set n := %γ& and ﬁrst show that ys,t − E(V ) ys , Sn (x)s,t = [Vi 1 · · · Vi n I (yr 1 ) − Vi 1 · · · Vi n I (ys )] dxir11 · · · dxirnn . i 1 ,...,i n ∈{1,...,d}

s< r 1 < ···< r n < t

Rough diﬀerential equations

214

To this end, consider a smooth function f and note that for any k ≤ n − 1, Vi 1 · · · Vi k f ∈ C 1 . By iterated use of the change-of-variable formula (cf. Exercise 3.17), n −1 Vi 1 · · · Vi k f (ys ) dxir11 · · · dxirkk f (yt ) = f (ys ) +

+

k =1 i 1 ,...,i k ∈{1,...,d}

i 1 ,...,i k ∈{1,...,d}

s< r 1 < ···< r k < t

s< r 1 < ···< r n < t

Vi 1 · · · Vi n f (yr 1 ) dxir11 · · · dxirnn

and the claim follows from specializing to f = I, the identity function. Clearly, t t |ys,t | = V (y) dx ≤ c1 |dxr | . s

s

Lipγ −1 -regularity of the vector ﬁelds implies that Vi 1 · · · Vi n I (·) is H¨older continuous with exponent {γ} ≡ γ − n. Hence, for all r ∈ [s, t], t {γ } |dxr | |Vi 1 · · · Vi n I (yr ) − Vi 1 · · · Vi n I (ys )| ≤ c2 s

and after integration, using γ = n + {γ}, i1 in [V · · · V I (y ) − V · · · V I (y )] dx · · · dx i1 in r1 i1 in s r1 rn s< r 1 < ···< r n < t t γ |dxr | . ≤ c3 s

Summation over the indices ﬁnishes the estimate. Remark 10.4 The proof also showed that, keeping the notation n = %γ&, π (V ) (0, y0 ; x)0,T − E(V ) y0 , Sγ (x)0,T = [Vi 1 · · · Vi n I (yr 1 ) − Vi 1 · · · Vi n I (y0 )] dxir11 · · · dxirnn i 1 ,...,i k ∈{1,...,d}

=

0< r 1 < ···< r n < T

[Vi 1 · · · Vi n I (yr ) − Vi 1 · · · Vi n I (y0 )]

i 1 ,...,i k ∈{1,...,d}

0< r < T

dxir1

r < r 2 ···< r n < T

dxir22 · · · dxirnn

n ;i , . . . , i n

≡d (x r , T 1

)

(we underlined integration variables here) [V n (yr ) − V n (y0 )] d xnr,T . ≡ 0< r < T

10.2 Davie’s estimate

215

10.2 Davie’s estimate The main result in this section will require a quantitative understanding of (A) the diﬀerence of ODE solutions started at the same point, with different driving signals (but with common iterated integrals up to a given order); (B) the diﬀerence of ODE solutions started at diﬀerent points but with identical driving signals. This is the content of the following two lemmas. Lemma 10.5 (Lemma A) Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ −1 (Re ), with γ > 1; (ii) s < u are some elements in [0, T ]; (iii) ys ∈ Re (thought of as a “time-s” initial condition); (iv) x and x ˜ are some paths in C 1-var [s, u] , Rd such that Sγ (x)s,u = x)s,u ; Sγ (˜ u

u x| . (v) ≥ 0 is a bound on |V |Lip γ −1 s |dx| + s |d˜ Then we have, for some constant C = C (γ) , π (V ) (s, ys ; x)s,u − π (V ) (s, ys ; x ˜)s,u ≤ Cγ . Proof. We do not give the most straightforward proof (which would be to insert the Euler approximation of order %γ& and use the triangle inequality), but provide a (still simple) proof that will be more instructive later on. By reparametrization of time, we can assume (s, u) = (0, 1) . Deﬁne the concatenation of x ˜ (1 − ·) and x (·), in symbols ← − z := x ˜ x, reparametrized so that z : [0, 1] → Rd . Then, π (V ) (0, y0 ; x)0,1 −π (V ) (0, y0 ; x ˜)0,1 = π (V ) (0, y0 ; x)1 − π (V ) (0, y0 ; x ˜)1 = π (V ) (0, π (V ) (0, y0 ; x ˜)1 ;z)1 −π (V )(0,y0 ;˜ x)1 = π (V ) (0, π (V ) (0, y0 ; x ˜)1 ; z)0,1 . By assumption (iv) and Chen’s theorem, −1

Sγ (z)0,1 = Sγ (˜ x)0,1 ⊗ Sγ (x)0,1 = 1. Hence, E(V ) ·, Sγ (z)0,1 ≡ 0 and the proof is ﬁnished with the ODE

Rough diﬀerential equations

216

Euler estimates from Proposition 10.3, π (V ) (0, ·; z)0,1 = π (V ) (0, ·; z)0,1 − E(V ) ·, Sγ (z) 0,1 % &γ 1 |dz| ≤ c1 |V |Lip γ −1 0

=

c1

|V |Lip γ −1

1

|dx| + 0

1

γ |d˜ x| ≤ c1 γ .

0

Lemma 10.6 (Lemma B) Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lip1 (Re ); (ii) t < u are some elements in [0, T ]; initial conditions); (iii) yt , y˜t ∈ Re (thought of as “time-t” (iv) x is a path in C 1-var [t, u] , Rd ;

u (v) ≥ 0 is a bound on |V |Lip 1 t |dx|. Then, if π (V ) (t, ·; x) denotes the unique solution to dy = V (y) dx from some time-t initial condition, we have π (V ) (t, yt ; x)t,u − π (V ) (t, y˜t ; x)t,u ≤ |yt − y˜t | . exp () . In particular, the ﬂow associated with dy=V (y) dx is Lipschitz continuous. Proof. This was seen in Theorem 3.8. Equipped with these two simple lemmas, and the technical Lemma 10.59 in Appendix 10.8, we are now ready to provide the crucial p-variation estimate of an ODE solution in terms of the p-variation of the driving signal. Lemma 10.7 (Davie’s lemma) Let γ > p ≥ 1. Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ −1 (Re ); (ii) x is a path in C 1-var [0, T ] , Rd , and x := S[p] (x) is its canonical lift to a G[p] Rd -valued path; (iii) y0 ∈ Re is an initial condition. Then there exists a constant C1 depending on p, γ (and not depending on the 1-variation norm of x) such that for all s < t in [0, T ] , p p π (V )(0, y0 ; x) |V | ≤ C γ −1 xp-var;[s,t] ∨ |V| γ −1 xp-var;[s,t] . 1 Lip Lip p-var;[s,t] (10.3) Moreover, if xs,t ∈ C 1-var [s, t] , Rd is a path such that t s,t s,t dx ≤ K x Sγ x s,t = Sγ (x)s,t and (10.4) p-var;[s,t] s

for some K ≥ 1 then, for any time-s initial condition ys ∈ Re , γ π (V ) (s, ys ; x)s,t − π (V ) s, ys ; xs,t s,t ≤ C2 K |V |Lip γ −1 xp-var;[s,t] (10.5) where C2 depends on p and γ.

10.2 Davie’s estimate

217

Remark 10.8 In case of non-uniqueness we abuse notation in the sense that π (V ) (0, y0 ; x) resp. π (V ) (s, ys ; x) in the above estimates stands for any choice of ODE solution to dy = V (y) dx with the indicated initial conditions at times 0, s respectively. Remark 10.9 From Proposition 10.3, inequality (10.5) is equivalent to the Euler estimate γ π (V ) (s, ys ; x)s,t −E(V ) s, ys ; Sγ xs,t s,t ≤C3 |V |Lip γ −1 xp-var;[s,t] . (10.6)

Remark 10.10 A ﬁnite variation path (and even Lipschitz continuous) with the properties (10.4) always exists. Indeed, it suﬃces to take xs,t as geodesic associated with the element g = Sγ (x)s,t ∈ Gγ Rd , parametrized on the interval [s, t]. The length of this curve is precisely equal to g, the Carnot–Caratheodory norm of g ∈ Gγ Rd , and so s

t

s,t dx

= Sγ (x)s,t ≤ Sγ (x)p-var;[s,t] ≤ C(γ ,p) S[p] (x) p-var;[s,t]

where in the last step we used the estimates for the Lyons-lift map, Proposition 9.3, applicable as γ > p and so %γ& ≥ [p]. Let us also note that t t s,t dx ≤ K |dx| (10.7) s

since

t s

s

|dxs,t | ≤ K xp-var;[s,t] ≤ K x1-var;[s,t] = K |x|1-var;[s,t] .

Proof. The case p < γ < 2 is discussed in Exercises 10.12, 10.13. We assume here γ ≥ 2, so that dy = V (y) dx with time-s initial condition ys has a unique solution, denoted as usual by π (s, ys ; x). We deﬁne x = S[p] (x), and p ω (s, t) = K |V |Lip γ −1 xp-var;[s,t] .

t Thanks to s |dxs,t | ≤ K xp-var;[s,t] and an elementary ODE estimate (Theorem 3.4), we have 1 (10.8) π (V ) s, ys ; xs,t s,t ≤ c1 ω (s, t) p . Then, for all s < t in [0, T ], we deﬁne Γs,t = yt − π s, ys ; xs,t t = ys,t − π s, ys ; xs,t s,t .

Rough diﬀerential equations

218

Then, for ﬁxed s < t < u in [0, T ], we have Γs,u − Γs,t − Γt,u = −π (s, ys ; xs,u )s,u + π s, ys ; xs,t s,t + π t, yt ; xt,u t,u . Deﬁne xs,t,u to be the concatenation of xs,t and xt,u and set, for better readability, A := π s, ys ; xs,t,u s,u − π (s, ys ; xs,u )s,u B := π t, yt ; xt,u t,u − π t, π s, ys ; xs,t t ; xt,u t,u = π t, yt ; xt,u t,u − π t, yt − Γs,t ; xt,u t,u which then allows us to write Γs,u − Γs,t − Γt,u = A + B.

(10.9)

The term A is estimated by – nomen est omen – Lemma A, noting that t u u s,t,u s,t t,u dx = dx + dx ≤ 2K x p-var;[s,u ] . s

s

t

Similarly, Lemma B was tailor-made to estimate B and we are led to γ /p 1/p + c2 π s, ys ; xs,t t − yt ω (t, u) |Γs,u − Γs,t − Γt,u | ≤ c1 ω (s, u) 1/p exp c2 ω (t, u) γ /p 1/p 1/p . + c2 |Γs,t | ω (t, u) exp c2 ω (t, u) ≤ c1 ω (s, u) The elementary inequality 1 + c2 ω (t, u)

1/p

1/p 1/p ≤ exp 2c2 ω (s, u) , exp c2 ω (t, u)

combined with the triangle inequality, then gives γ 1/p + |Γt,u | + c1 ω (s, u) p . |Γs,u | ≤ |Γs,t | exp 2c2 ω (s, u) On the other hand, using again Lemma A, |Γs,t | = ys,t − π s, ys ; xs,t s,t % & t t s,t γ dx |dx| + |V |Lip γ −1 ≤ c3 |V |Lip γ −1 % ≤

s

c3 |V |Lip γ −1 (K + 1)

=: ω ˜ (s,t)

&γ

t

|dx| s

s

thanks to (10.7).

(10.10)

10.2 Davie’s estimate

219

Obviously, ω ˜ is a control function whose ﬁniteness depends crucially on the a priori assumption that x has ﬁnite 1-variation and we summarize the previous estimate in writing: γ

ω (s, t) ) where γ > 1. |Γs,t | = O (˜

(10.11)

The two estimates (10.10), (10.11) are precisely what is needed to apply (the elementary analysis) Lemma 10.59 (found in the appendix of this chapter): it follows that, for all s < t in [0, T ], γ /p 1/p exp c4 ω (s, t) |Γs,t | ≤ c4 ω (s, t) and we emphasize that c4 does not depend on ω ˜ and in particular not on the 1-variation of x. From (10.8) and the triangle inequality, we therefore have for all s < t in [0, T ] , 1/p γ /p 1/p exp c4 ω (s, t) |ys,t | ≤ c1 ω (s, t) + c4 ω (s, t) and if attention is restricted to s, t such that ω (s, t) ≤ 1, we obviously have 1/p

|ys,t | ≤ (c1 + c4 ec 4 ) ω (s, t)

.

But then it follows from Proposition 5.10 that for all s < t in [0, T ], 1/p |y|p-var;[s,t] ≤ c5 ω (s, t) ∨ ω (s, t) . That also leads to

|Γs,t | = ys,t − π s, ys ; xs,t s,t 1/p 1/p ≤ c5 ω (s, t) ∨ ω (s, t) + c1 ω (s, t)

and hence

γ /p 1/p 1/p |Γs,t | ≤ min c4 ω (s, t) , c5 ω (s, t) ∨ ω (s, t) exp c4 ω (s, t) 1/p + c1 ω (s, t) γ /p

≤ c6 ω (s, t)

.

The proof is now ﬁnished. Exercise 10.11 Prove that we can take the constants C1 and C2 in Lemma 10.7 to be continuous in p, for p ∈ [1, γ). The following exercise deals with the case p < γ < 2 in Lemma 10.7.

Rough diﬀerential equations

220

Exercise 10.12 (i) for 1 < γ < 2, and a, b > 0, prove that aγ −1 b ≤ (γ − 1) abγ −1 + (2 − γ) bγ .

(10.12)

(ii) Under the assumption of Lemma 10.7 for p < γ < 2, prove that if Γs,t = ys,t − V (ys ) xs,t and ω (s, t) = |V |Lip γ −1 |x|p-var,[s,t] , we have for s < t < u, |Γs,u − Γs,t − Γt,u | ≤ c (γ − 1) |Γs,t | ω (t, u)

γ −1 p

γ /p

+ c (2 − γ) ω (t, u)

.

(iii) Prove Lemma 10.7 in the case γ < 2. Solution. (i) Write x = a/b, and dividing by bγ , we see that (10.12) is equivalent to xγ −1 ≤ (γ − 1) x + (2 − γ) for x > 0, which is checked by basic calculus. (ii) For s < t < u, |Γs,u − Γs,t − Γt,u | =

|[V (yt ) − V (ys )] xt,u | γ −1

1/p

≤

c |Γs,t |

≤

c (γ − 1) |Γs,t | ω (t, u) using (i).

ω (t, u)

γ −1 p

γ /p

+ c (2 − γ) ω (t, u)

(iii) Exactly the same argument as in the proof of Lemma 10.7.

Exercise 10.13 (i) Under the assumption of Lemma 10.7 for p < γ < 2, prove using Young estimates that for all s, t |y|p-var;[s,t] ≤ c1 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] . (ii) Using proposition 5.10, prove inequality (10.3) of Lemma 10.7. Solution. From Young’s inequality, t |ys,t | = V (y) dx s γ −1 ≤ c1 |x|p-var;[s,t] |V |∞ + |V |Lip γ −1 |y|p-var;[s,t] γ −1 ≤ c1 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] ≤ c2 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] 1/p p ≤ c3 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] .

10.3 RDE solutions

221 p

The p-variation of x is controlled by ω x;p ≡ |x|p-var;[·,·] . Using similar notap tion for y we see that |ys,t | is estimated by a constant times ω x;p +ω x;p ω y ;p which is a control (cf. Exercise 1.9) and from the basic super-additivity property of controls, 1/p p |y|p-var;[s,t] ≤ c3 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] ≤ c3 |V |Lip γ −1 |x|p-var;[s,t] 1 + |y|p-var;[s,t] . (ii) For s, t such that 2c2 |V |Lip γ −1 |x|p-var;[s,t] < 1, we obtain |y|p-var;[s,t]

c2 |V |Lip γ −1 |x|p-var;[s,t]

≤

≤

2c2 |V |Lip γ −1 |x|p-var;[s,t] .

1 − c2 |V |Lip γ −1 |x|p-var;[s,t]

We then obtain estimates on |y|p-var;[s,t] for s, t such that 2c2 |V |Lip γ −1 |x|p-var;[s,t] ≥ 1 using Proposition 5.10.

10.3 RDE solutions Davie’s lemma gives us uniform estimates for ODE solutions which depend only on the rough path regularity (e.g. p-variation or 1/p-H¨ older) of the canonical lift of a “nice” driving signal x ∈ C 1-var [0, T ] , Rd . It should therefore come as no surprise that a careful passage to the limit will yield a sensible notion of diﬀerential equations driven by a “generalized” driving signal, given as a limit of nice driving signals (in p-variation or 1/p-H¨older rough path sense . . . ). This class of generalized driving signals is precisely the class of weak geometric p-rough paths introduced in the previouschap ter. Indeed, we saw in Section 8.2 that for any x ∈ C p-var [0, T ] , G[p] Rd , there exist (xn ) ⊂ C 1-var [0, T ] , Rd which approximate x uniformly with uniform p-variation bounds, lim d0;[0,T ] S[p] (xn ) , x = 0 and sup S[p] (xn )p-var;[0,T ] < ∞. n →∞

n

(10.13)

10.3.1 Passage to the limit with uniform estimates Our aim is now to make precise the meaning of the rough diﬀerential equation (RDE) (10.14) dy = V (y) dx, y (0) = y0 ∈ Re where V = (Vi )1≤i≤d is a family of suﬃciently nice vector ﬁelds and x : [0, T ] → G[p] Rd is a weak geometric p-rough path. The following

Rough diﬀerential equations

222

is essentially an existence result for such RDEs; since our precise deﬁnition is very much motivated by this result, the precise deﬁnition of an RDE solution (together with remarks on alternative deﬁnitions) is postponed till the next section. Theorem 10.14 (existence) Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ −1 (Re ), where γ > p; (ii) (xn ) is a sequence in C 1-var [0, T ] , Rd , and x is a weak geometric p-rough path such that < ∞; lim d0;[0,T ] S[p] (xn ) , x and sup S[p] (xn ) n →∞

p-var;[0,T ]

n

(iii) y0n ∈ Re is a sequence converging to some y0 . y0n ; xn ) converges in uniform Then, at least along a subsequence, π (V ) (0, topology to some limit, say y ∈ C [0, T ] , Rd . Any such limit point satisﬁes the following estimates: there exists a constant C depending on p, γ such that for all s < t in [0, T ], p p |y|p-var;[s,t] ≤ C |V |Lip γ −1 xp-var;[s,t] ∨ |V |Lip γ −1 xp-var;[s,t] . (10.15) Moreover, if xs,t : [s, t] → Rd is any continuous bounded variation path such that t s,t s,t dx ≤ K x Sγ x s,t = Sγ (x)s,t and p-var;[s,t] s

for some constant K ≥ 1 then, again for all s < t in [0, T ], γ ys,t − π (V ) s, ys ; xs,t s,t ≤ C K |V |Lip γ −1 xp-var;[s,t] ,

(10.16)

where C depends on p, γ. Proof. Let ε > 0 be small enough such that p + ε < γ, and let us deﬁne for convenience the function φp (x) = x ∨ xp . By Davie’s Lemma (i.e. Lemma 10.7) for all s, t ∈ [0, T ], there exists c1 = c1,p+ε such that π (V ) (0, y0n ; xn )s,t ≤ c1,p+ε φp+ |V |Lip γ −1 S[p] (xn )(p+ε)-var;[s,t] , (10.17) where c1 is independent of n. From Corollary 5.29, S[p] (xn ) (p+ε)-var;[.,.]

is equicontinuous in the sense that for all ε > 0, there exists δ such that for all s, t with |t − s| < δ, S[p] (xn ) < ε. (p+ε)-var;[s,t]

10.3 RDE solutions

223

This implies that π (V ) (0, y0n ; xn ) is equicontinuous and hence converges (along a subsequence) to a path y. We therefore obtain that |ys,t | ≤ c1,p+ε φp+ |V |Lip γ −1 lim S[p] (xn )(p+ε)-var;[s,t] n →∞ ≤ c1,p+ε φp+ |V |Lip γ −1 x(p+ε)-var;[s,t] . By Exercise 10.11, limε→0 c1,p+ ε = c1,p , and by Lemma 5.13, limε→0 x(p+ ε)-var;[s,t] = xp-var;[s,t] . Hence, for all s, t ∈ [0, T ] , |ys,t | ≤ c1,p φp |V |Lip γ −1 xp-var;[s,t] . The right-hand side of the last expression raised to power p deﬁnes a control, hence we obtain (10.15). From Remark 10.9, proving inequality (10.16) is equivalent to the proof of the following inequality: γ ys,t − E(V ) s, ys ; Sγ xs,t s,t ≤ C |V |Lip γ −1 xp-var;[s,t] . By Davie’s Lemma (i.e. Lemma 10.7), for all s, t ∈ [0, T ], π (V ) (0, y0n ; xn )s,t − E(V ) s, ys ; Sγ xs,t n s,t γ ≤ c1,p+ ε |V |Lip γ −1 S[p] (xn )(p+ε)-var;[s,t] , where c1 is independent of n. Letting n tend to inﬁnity (along the subsequence that allows us to have convergence of π (V ) (0, y0n ; xn ) to y), we obtain γ ys,t − E(V ) s, ys ; Sγ (x)s,t ≤ c1,p+ε |V |Lip γ −1 x(p+ε)-var;[s,t] . Letting ε converge to 0 ﬁnishes the proof. Let us point out explicitly the error estimate for the Euler scheme, which was established in the ﬁnal step of the previous proof. Corollary 10.15 (Euler RDE estimates) Under the assumptions of Theorem 10.14 (in particular x ∈ C p-var [0, T ] , G[p] Rd , V ∈ Lipγ −1 (Re ) , γ > p) we have γ ys,t − E(V ) ys , Sγ (x)s,t ≤ C |V |Lip γ −1 xp-var;[s,t] where C only depends on p, γ. If x is 1/p-H¨ older, γ γ γ /p . ys,t − E(V ) ys , Sγ (x)s,t ≤ C |V |Lip γ −1 x1/p-H¨o l;[0,T ] × |t − s| Remark 10.16 Note %γ& ≥ [p] so that Sγ (x) is the Lyons lift of x. For γ close enough to p, Sγ (x) ≡ x.

224

Rough diﬀerential equations

10.3.2 Deﬁnition of RDE solution and existence Perhaps the simplest way to turn Theorem 10.14 into a sensible deﬁnition of what we mean by dy = V (y) dx, (10.18) started at y0 ∈ Re , is the following: Deﬁnition 10.17 Let x ∈ C p-var [0, T ] , G[p] Rd be a weak geometric p-rough path. We say that y ∈ C ([0, T ] , Re ) is a solution to the rough diﬀerential equation (short: RDE solution) driven by x along the collection of Re -vector ﬁelds V = (Vi )i=1,...,d and started at y0 if there exists a sequence (xn )n in C 1-var [0, T ] , Rd such that (10.13) holds, and ODE solutions yn ∈ π (V ) (0, y0 ; xn ) such that yn → y uniformly on [0, T ] as n → ∞ . The (formal) equation (10.18) is referred to as a rough diﬀerential equation (short: RDE). This deﬁnition generalizes immediately to other time intervals such as [s, t], and we deﬁne π (V ) (s, ys ; x) ⊂ C ([s, t] , Re ) to be the set of all solup-var ([s, t] , tionsto the above RDE starting at ys at time s driven by x ∈ C [p] d R . In case of uniqueness, π (V ) (s, ys ; x) is the solution of the RDE. G Let us note that Theorem 10.14 is now indeed an existence result for RDE solutions. That said, there are some possible variations on the theme of the RDE deﬁnition on which we wish to comment. Remark 10.18 (RDE, Davie deﬁnition) Theorem 10.14 and Corollary 10.15 allow us to pick other “deﬁning” properties of RDE solutions. For instance, A. M. Davie [37] deﬁnes y to be an RDE solution if there exists a control function ω ˜ and a function θ (δ) = o (δ) as δ → 0 such that for all s < t in [0, T ], ys,t − E(V ) (ys , xs,t ) ≤ θ (˜ ω (s, t)) . (10.19) (Note that (10.19) contains an implicit regularity assumption on V so that the Euler scheme E is well-deﬁned.) Applying Corollary 10.15 (take γ small enough so that %γ& ≥ [p] , hence Sγ (x) ≡ x, then θ (δ) = δ γ /p and p ω ˜ (s, t) = (const) × xp-var;[s,t] ) shows that any RDE solution in the sense of Deﬁnition 10.17 is also a solution in Davie’s sense. With either deﬁnition, let us note that (10.19) leads immediately to the statement that y satisﬁes some sort of compensated Riemann–Stieltjes integral equation, E(V ) yt i , xt i ,t i + 1 , yt − y0 = lim n →∞

t i ∈D n

for any sequence of dissection (Dn ) of [0, t] with mesh tending to zero.

10.3 RDE solutions

225

Remark 10.19 (RDE, Lyons deﬁnition) As one expects, RDE solutions can also be deﬁned as the solution to a “rough” integral equation and this is Lyons’ original approach [116, 120, 123]. To this end, one ﬁrst needs a notion of rough integration (cf. Section 10.6) which allows, for suﬃciently smooth ϕ = (ϕ1 , . . . , ϕd ), deﬁned on Rd , the deﬁnition of an (indeﬁnite) rough integral · ϕ (z) dz with z = π 1 (z) 0 such that, in the case when z = S[p] (z) for some z ∈ C 1-var [0, T ] , Rd , it coincides with S[p] (ξ) where ξ is the classical Riemann–Stieltjes integral

· ϕ (z) dz. 0 Note that (10.18) cannot be rewritten as an integral equation of the above form (for y is not part of the integrating signal x). Nonetheless, the “enhanced” diﬀerential equation (in which the input signal is carried along to the output) dx = dx dy = V (y) dx can be written in the desired form · ϕ (z) dz z0,· = 0

provided we set z = (x, y) and

1 0 . V (y) 0 The above integral equation indeed makes sense as a rough integral equation (with implicit regularity assumption on V so the rough integral is wellp-var ([0, T ] , deﬁned) d replacing z by a genuine geometric p-rough path z ∈ C [p] e R ⊕R and solutions can be constructed, for instance, by a Picard– G iteration [116, 120, 123]. An Re -valued solution is then recovered by projection z → π 1 (z) = z = (x, y) → y and again one can see that an RDE solution in the sense of Deﬁnition 10.17 is also a solution in this sense. ϕ (z) =

10.3.3 Local existence As in ODE theory, if the vector ﬁelds have only locally the necessary regularity for existence, we get local solutions. Exercise 10.20 We keep the notation of Theorem 10.14. Fix s < t in [0, T ] and assume that for some open set Ω, we have for all u ∈ [s, t] , yu ∈ Ω. Prove that p p |y|p-var;[s,t] ≤ C |V |Lip γ −1 (Ω) xp-var;[s,t] ∨ |V |Lip γ −1 (Ω) xp-var;[s,t] where C = C (p, γ).

226

Rough diﬀerential equations

As a consequence of this result, we obtain the following: Theorem 10.21 (local existence) Asssume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds Lipγloc−1 (Re ) , with γ > p; (ii) x : [0, T ] → G[p] Rd is a weak geometric p-rough path; (iii) y0 ∈ Re is an initial condition. Then either there exists a (global) solution y : [0, T ] → Re to dy = V (y) dx with initial condition y0 , or there exists τ ∈ [0, T ] and a (local) solution y : [0, τ ) → Re such that y is a solution on [0, t] for any t ∈ (0, τ ) and lim |y (t)| = +∞.

tτ

Proof. It simpliﬁes the argument to have unique ODE solutions, we thus assume γ ≥ 2. (Otherwise p < γ < 2 and some care, similar to the proof of Theorem 3.6, is needed; we leave this extension to the reader.) Without loss of generality, y0 = 0. Pick (xn ) ⊂ C 1-var [0, T ] , Rd so that S[p] (xn ) → x uniformly with supn S[p] (xn )p-var;[0,T ] < ∞. Replace V by compactly supported Lipγ −1 -vector ﬁelds V n which coincide with V on the ball {y : |y| ≤ n}. From the preceding existence theorem, π (V ( 1 ) ) (0, y0 ; xn ) → y (1) ∈ π (V 1 ) (0, y0 ; x) , where strictly speaking we have replaced xn by a subsequence along which this convergence holds. If (1) ≤1 y ∞;[0,T ]

then we can replace V1 by V and hence ﬁnd a global solution, in which case we are done. Otherwise, (1) τ 1 = inf t ≥ 0 : yt ≥ 1 ∈ [0, T ] and we switch to another subsequence so that π (V ( 2 ) ) (0, y0 ; xn ) → y (2) ; observing that y (1) ≡ y (2) on [0, τ 1 ]. Again, if y (2) ∞;[0,T ] ≤ 2 we ﬁnd a global solution, otherwise we deﬁne another τ 2 ≥ τ 1 by τ 2 = inf t ≥ 0 : y (2) ≥ 2 ∈ [0, T ] and so on. Iterating this either yields a global solution or a family of RDE solutions y (n ) ∈ π (V n ) (0, y0 ; x) , consistent in the sense that y (n ) ≡ y (n +1) on [0, τ n ] . Moreover, by deﬁnition of τ n we see that |y (τ n )| = n → ∞ as n → ∞ and the proof is ﬁnished.

10.3 RDE solutions

227

10.3.4 Uniqueness and continuity For ordinary diﬀerential equations, we saw that existence is guaranteed for continuous vector ﬁelds, while uniqueness requires Lipschitz vector ﬁelds; that is, one additional degree of smoothness. In essence, this remains true for RDEs driven by p-rough paths: we saw that RDE solutions exist for Lipγ −1 -vector ﬁelds when γ > p and we will now see that uniqueness holds, still assuming γ > p, for Lipγ -vector ﬁelds (or Lipγloc since uniqueness is a local issue!). We will show uniqueness by establishing Lipschitz regularity of the RDE ﬂows. Later, we will see that uniqueness also holds for γ = p (in this case, we will only prove uniform continuity on bounded sets of the RDE ﬂow rather than Lipschitzness), and that uniqueness still holds when we relax the rough path regularity of the driving signal in a way that gives uniqueness under optimal regularity assumptions for RDEs driven by Brownian motion and L´evy’s area. The reader may ﬁnd it useful to quickly revise the proof of Davie’s lemma, which is similar to arguments in this section. In particular, the following Lemmas A and B are essentially straightforward generalizations of what we called Lemmas A and B in Section 10.2. We start with the elementary yet useful f , Re ) with β ∈ [1, 2], and a, b, a ˜, ˜b : Lemma 10.22 Let g, g˜ ∈ Lipβ Re , L(R e 1-var f [0, 1], R . Then, with ˜∈C [0, 1] → R , and x, x ∆≡

1

(g (ar ) − g (br )) dxr − 0

1

g˜ (˜ ar ) − g˜ ˜br d˜ xr ,

0

we have1 |∆|

≤

|g|

L ip β

1

˜r − ˜br . |dxr | (ar − br ) − a

0

β −1 + |a − b|∞;[0,1] + ˜ a − ˜b |x|1-var;[0,1] ∞;[0,1] × |g| β b − ˜b + |g − g˜|Lip β −1 L ip ∞;[0,1] a − ˜b |x − x ˜|1-var;[0,1] . + |˜ g | β . ˜

L ip

1 In

∞;[0,1]

the case β = 1, |g − g˜|L ip β −1 has to be replaced by 2 |g − g˜|∞ .

Rough diﬀerential equations

228

Proof. Step 1: Fix a, b, a ˜, ˜b ∈ Re . When β > 1, g ∈ C 1 and we can write g (a) − g (˜ a) − g (b) − g ˜b as

1 0 1

= 0

g (ta + (1 − t) a ˜) (a − a ˜) dt −

1

g tb + (1 − t) ˜b b − ˜b dt

0

g (ta + (1 − t) a ˜) a − a ˜ − b − ˜b dt

1

+

!

" g (ta + (1 − t) a b − ˜b dt ˜) − g tb + (1 − t) ˜b

0

to obtain, using |g |∞ ≤ |g|

L ip β

for β > 1, ˜b (a − b) − a ˜ − L ip β β −1 a − ˜b + |g| β |a − b| + ˜ b − ˜b .

a) − g ˜b ≤ |g| (g (a) − g (b)) − g (˜

L ip

In fact, this argument remains valid for β = 1 since g ∈ Lip1 implies absolute continuity of t → g (ta + (1 − t) a ˜). β −1 ˜ a − ˜b a)−˜ g b − g (˜ a)−g ˜b ≤ |g − g˜|Lip β −1 ˜ Step 2: Obviously g˜ (˜ so that, by the triangle inequality, ˜ − ˜b a) − g˜ ˜b ≤ |g| β (a − b) − a (g (a) − g (b)) − g˜ (˜ L ip β −1 |g| β b − ˜b + |g − g˜|Lip β −1 . + |a − b| + ˜ a − ˜b L ip

Step 3: At last, we write ∆ = ∆1 + ∆2 where ∆1

1

!

1

!

0

∆2

" g (ar ) − g (br ) − g˜ (˜ dxr , ar ) − g˜ ˜br

= =

" g˜ (˜ ar ) − g˜ ˜br d (xr − x ˜r ) .

0

Using the elementary . . . dxr ≤ |. . . | |dxr| ≤ |. . . |∞ |x|1-var we bound ˜ g (˜ ar ) −˜ g ˜br ≤ |˜ g | β . ˜ a−b ∆1 using the step-2 estimate, while ˜ ∞

L ip

∞;[0,1]

allows us to bound ∆2 . Together, they imply the claimed estimate. We now turn to Lemma A and note the assumption of Lipγ -regularity, γ ≥ 1, in contrast to Lemma A which was formulated for Lipγ −1 vector ﬁelds, γ > 1. Lemma A) Assume that 10.23 (Lemma (i) Vi1 1≤i≤d and Vi2 1≤i≤d are two collections of vector ﬁelds in Lipγ (Re ) vector ﬁelds, with γ ≥ 1;

10.3 RDE solutions

229

(ii) s < u are two elements of [0, T ]; (iii) ys1 , ys2 ∈ Re (thought of as “time-s” initial conditions); ˜1 and x2 , x ˜2 are driving signals in C 1-var [s, u] , Rd such that (iv) x1 , x S[γ ] x1 s,u S[γ ] x2 s,u

= =

1 ˜ s,u , S[γ ] x 2 ˜ s,u ; S[γ ] x

(v) ≥ 0, δ > 0 are such that u u u u 1 1 2 2 max |dx | + |d˜ x |, |dx | + |d˜ x | s s s u s u 1 1 2 2 dx − dx , d˜ x − d˜ x max s

≤ , ≤

δ;

s

(vi) υ ≥ 0 is a bound on V 1 Lip γ and V 2 Lip γ . Then, for some constant C depending only on γ, we have2 ˜1 s,u − π (V 2 ) s, ys2 ; x2 s,u π (V 1 ) s, ys1 ; x1 s,u − π (V 1 ) s, ys1 ; x −π (V 2 ) s, ys2 ; x ˜2 s,u γ ≤ C ys1 − ys2 + υ1 V 1 − V 2 Lip γ −1 . (υ) exp (Cυ) [γ ]

+ Cδυ. (υ)

exp (Cυ) .

Proof. First, as in the proof of Lemma A, we take (s, u) = (0, 1) and observe ˜i 0,1=π (V i ) 0, π (V i ) 0, y0i; x ˜i 1 ; z i 0,1; i=1, 2, π (V i ) 0, y0i ; xi 0,1−π (V i ) 0, y0i ; x ← − ← − where z 1 = x ˜1 x1 and z 2 = x ˜2 x2 are reparametrized in the same way to a path from [0, 1] to Rd . By assumption (iv) and Chen’s theorem z 1 has trivial step-[γ] signature, i.e. 1 −1 ˜ 0,1 ⊗ S[γ ] x1 0,1 = 1, S[γ ] z 1 0,1 = S[γ ] x

(10.20)

1 and similarly for z 2 . Next, by assumption (v) we have 0 dzr1 − dzr2 ≤ 2δ. Using the Lipschitzness of the ﬂow of ODEs, in the quantitative form of Theorem 3.8, we see that it is enough to prove the above lemma with ˜1 = 0. We can thus assume z 1 = x1 and z 2 = x2 . To simplify the x ˜2 = x notation, deﬁne yui = π (V i ) 0, y0i ; xi u , i = 1, 2, 2 For

γ = 1, V 1 − V 2 L ip γ −1 is replaced by 2 V 1 − V 2 ∞ .

Rough diﬀerential equations

230

and, for N := [γ], = xi,N u ,1

⊗N dxir 1 ⊗ · · · ⊗ dxir N ∈ Rd ,

u < r 1 < ···< r n < 1

Observe that

1

max

i=1,2

i = 1, 2.

i,N dxu ,1 ≤ N .

0

From (10.20) and Remark 10.4 concerning the remainder representation of an Euler approximation,3 we have 1 i,N i i i V yu −V i,N y0i dxi,N =y0,1 −E(V i ) y0 , SN xi 0,1 = y0,1 u ,1 , i=1, 2, 0

and so we have 1 2 y0,1 − y0,1

1

= 0

1,N 1 V yu − V 1,N y01 dx1,N u ,1 ,

1

−

2,N 2 V yu − V 2,N y02 dx2,N u ,1 .

0

Note that V i,N ∈ Lipβ where β := γ − N + 1 ∈ [1, 2) since N = [γ]. 1 1 2 2 Using Lemma 10.22 to the paths u → yu , y0 , yu , y0 , we then obtain, using i,N γ −N + 1 ≤ c1 υ N , the bound maxi=1,2 V Lip 1 2 y0,1 − y0,1

1 1,N 2 ≤ c1 υ N y0,. − y0,. x .,1 ∞;[0,1]

1-var;[0,1]

γ −1 2 1,N 1 y0,. x + c1 y0,. + .,1 ∞;[0,1] ∞;[0,1] 1-var;[0,1] 1,N N 1 2 2,N γ −N × υ y0 − y0 + V −V Lip 1,N 2,N 2 . + c1 υ N . y0,. x.,1 − x.,1 ∞;[0,1] 1-var;[0,1]

We ﬁrst observe, by Theorem 3.18, 1 2 y0,1 − y0,1 ≤ c2 y01 − y02 υ + δυ + V 1 − V 2 ∞ ec 2 υ ∞;[0,1] and the ODE estimate of Theorem 3.4 gives 1 2 y0,. + y0,. ≤ c3 υ. ∞;[0,1] ∞;[0,1] V = (V 1 , . . . , V d ) we think of V N ≡ V j 1 . . . V j N j , . . . , j ∈{1 , . . . , d } as an element 1 N ⊗N of the (Euclidean) space Rd which contracts naturally with elements of the form ⊗N . x iu,,N1 ∈ Rd 3 If

10.3 RDE solutions

231

Moreover, we easily see that 1,N V − V 2,N Lip γ −N ≤ c4 υ N −1 V 1 − V 2 Lip γ −1 2,N and, from Proposition 7.63, we have x1,N − x ≤ c5 δN −1 . We .,1 .,1 1-var;[0,1] N also have x.,1 1-var;[0,1] ≤ N of course. Putting all these inequalities together gives the desired estimate. ¯ detail how all Exercise 10.24 In the ﬁnal step of the proof of Lemma A, the estimates are put together. Solution. Assume V 1 = V 2 at ﬁrst. Using 1 2 y0,. − y0,. ≤ c2 y01 − y02 υ + δυ ec 2 υ ∞;[0,1] we get 1 2 y0,1 − y0,1

υ N y01 − y02 υ + δυ ecυ N γ −1 N N 1 + (υ) υ y0 − y02 + υ N . υ.δN −1 ≡ y01 − y02 ∆1 + δ∆2

with ∆1

= ≤

γ −1

υ N υecυ N + (υ) N υ N " ! N +1 N +γ −1 ecυ . (const) × (υ) + (υ)

From N = [γ] it is clear that min (N + 1, N + γ − 1) ≥ γ and so γ

∆1 ≤ (const) × (υ) ecυ . Similarly, N

∆2 = υ N +1 N ecυ + υ N +1 N υ (υ) ecυ . When V 1 = V 2 we have 1 1 2 y0,1 − y0,1 ≤ y0 − y02 ∆1 + δ∆2 + V 1 − V 2 Lip γ −1 ∆3 with ∆3

=

γ −1 N

γ −1

≤ (υ) N ecυ 1 γ −1 1 cυ γ (υ) e = (υ) ecυ . υ (υ)

Rough diﬀerential equations

232

Lemma B) Assume that 10.25 (Lemma (i) Vi1 1≤i≤d and Vi2 1≤i≤d are two collections of vector ﬁelds in Lipγ (Re ) with γ ≥ 1; (ii) t < u are some elements of [0, T ]; (iii) yt1 , yt2 , y˜t1 , y˜t2 ∈ Re (thought of as “time-t” initial conditions); (iv) x1 , x2 are two driving signals in C 1-var [t, u] , Rd ; (v) ≥ 0, δ > 0 are such that

u

u

|dx1 |,

max t

|dx2 | ≤ and

t

u

1 dxr − dx2r ≤ δ;

t

(vi) υ is a bound on V 1 Lip γ and V 2 Lip γ . Then we have, for some constant C = C (γ), π (V 1 ) t, yt1 ; x1 t,u − π (V 1 ) t, y˜t1 ; x1 t,u − π (V 2 ) t, yt2 ; x2 t,u − π (V 2 ) t, y˜t2 ; x2 t,u ≤ Cυ exp (Cυ) yt1 − y˜t1 − yt2 − y˜t2 1 yt − y˜t1 + yt2 − y˜t2 m in(2,γ )−1 + Cυ exp (Cυ) × y˜t1 − y˜t2 + υ1 V 1 − V 2 Lip γ −1 + υδ + Cδυ exp (Cυ) yt2 − y˜t2 . Proof. At the price of replacing γ by min (2, γ) ∈ [1, 2], we can and will assume that γ ∈ [1, 2]. Deﬁne for r ∈ [t, u] , yri = π (V i ) t, yti ; xi r

and

y˜ri = π (V i ) t, y˜ti ; xi r

with i = 1, 2.

We deﬁne 1 2 1 2 − yt,r − y˜t,r − y˜t,r er = yt,r

with r ∈ [t, u],

and have to estimate eu . Fom Lemma 10.22, applied with γ ∈ [1, 2], we see for all r ∈ [t, u], er

r r 1 1 1 1 2 2 2 2 1 2 V ys − V y˜s dxs − V ys − V y˜s dxs = t t  r 1 υ t ys − y˜s1 − ys2 − y˜s2 . dx1s   γ −1    + y 1 − y˜1 + y 2 − y˜2 ∞;[t,r ] ∞;[t,r ] ≤ 1 1 1 y˜ − y˜2 x V − V 2 γ −1  υ × +   1-var;[t,r ] Lip ∞;[t,r ]   + υ y 2 − y˜2 ∞;[t,r ] x1 − x2 1-var;[t,r ] .

10.3 RDE solutions

From Theorem 3.15 we have i y − y˜i ≤ ∞;[t,r ] 1 2 y˜ − y˜ ≤ ∞;[t,r ]

233

c1 yti − y˜ti exp (c1 υ) , i = 1, 2, c1 y˜t1 − y˜t2 exp (c1 υ) + c1 V 1 − V 2 Lip γ −1 exp (c1 υ) + c1 υδ exp (c1 υ) .

Hence, we obtain er

≤

υ

r

es dx1s + υ yt1 − y˜t1 − yt2 − y˜t2

t

γ −1 υ exp (c2 υ) + c2 yt1 − y˜t1 + yt2 − y˜t2 1 y˜t − y˜t2 + 1 V 1 − V 2 γ −1 + υδ Lip υ 2 + c1 yt − y˜t2 υδ exp (c1 υ) . The proof is then ﬁnished by an application of (Gronwall’s) Lemma 3.2. Equipped with these two lemmas, we can prove (under some regularity assumptions) that the map x → π (V ) (0, y0 ; x) is well-deﬁned and locally Lipschitz continuous in all its parameters (vector ﬁelds, initial condition, and driving signal). Theorem 10.26 (Davie) 4 Assume that 1 1 (i) V = Vi 1≤i≤d and V 2 = Vi2 1≤i≤d are two collections of Lipγ -vector ﬁelds on Re for γ > p ≥ 1; (ii) ω is a ﬁxed control;5 (iii) x1 , x2 are two weak geometric p-rough paths in C p-var [0, T ] , G[p] Rd , with xi p-ω ≤ 1; conditions; (iv) y01 , y02 ∈ Re thought initial of as time-0 (v) υ is a bound on V 1 Lip γ and V 2 Lip γ . Then there exists a unique RDE solution starting at y0i along V i driven by xi , denoted by y i ≡ π (V i ) 0, y0i ; xi , for i = 1, 2. Moreover, there exists C = C (γ, p) such that6 ! " ρp,ω ;[0,T ] y 1 , y 2 ≤ C υ y01 − y02 + V 1 − V 2 Lip γ −1 + υρp,ω ;[0,T ] x1 , x2 · exp (Cυ p ω (0, T )) . 4 The present theorem stands in a similar relation to Davie’s lemma as Lemma A ¯ to ¯ to B. A, or Lemma B i p 5 In view of (iii) one can take ω (s, t) = i = 1 , 2 x p -va r;[s , t ] . 6 Here: ρ 1 2 1 2 = y − y p , ω ;[0 , T ] . p , ω ;[0 , T ] y , y

Rough diﬀerential equations

234

Proof. The present regularity assumptions on the vector ﬁelds, Lipγ with γ > p, are more than enough to guarantee existence of RDE solutions. More precisely, let us pick solutions y i ∈ π (V i ) 0, y0i ; xi , i = 1, 2. We may assume, without loss of generality, p < γ < [p] + 1 so that [γ] = [p] and also set ε := ρp,ω ;[0,T ] x1 , x2 . For all s < t in [0, T ] we can ﬁnd paths x1,s,t and x2,s,t such that S[p] xi,s,t s,t = xis,t , i = 1, 2, and such that, for a constant c1 = c1 (p),

i,s,t dxr

≤

c1 ω (s, t)

1,s,t dxr − dx2,s,t r

≤

c1 εω (s, t)

t

1/p

,

s

t

1/p

;

s

indeed, this is possible thanks to Proposition 7.64 applied to gi := δ 1 1 / p xis,t ∈ G[p] Rd , i = 1, 2, ω (s ,t )

noting that g1 , g2 ≤ 1 and |g1 − g2 |T [ p ] (Rd ) ≤ ε. After these preliminaries, let us now ﬁx s < t < u in [0, T ] and deﬁne xi,s,t,u := xi,s,t xi,t,u , the concatenation of xi,s,t and xi,t,u . Following Davie’s lemma, we deﬁne i − π (V i ) s, ysi ; xi,s,t s,t , i = 1, 2, Γis,t = ys,t and set Γs,t := Γ1s,t − Γ2s,t . From the estimates in (the existence) Theorem 10.14, using the fact that the vector ﬁelds are [p]-Lipschitz, we have [p]+ 1 i 1 Γs,t ≤ c1 υω (s, t)1/p , i = 1, 2, 2 and hence

(10.21)

[p]+ 1 Γs,t ≤ c1 υω (s, t)1/p .

We now proceed similarly as in the proof of Davie’s lemma, cf. (10.9). Namely, for i = 1, 2, we deﬁne Ai ≡ π (V i ) s, ysi ; xi,s,t,u s,u − π (V i ) s, ysi ; xi,s,u s,u ,

10.3 RDE solutions

235

noting that S[p] xi,s,t,u = S[p] xi,s,u , and B i ≡ π (V i ) t, yti ; xi,t,u t,u − π (V i ) t, π (V i ) s, ysi ; xi,s,t t ; xi,t,u t,u = π (V i ) t, yti ; xi,t,u t,u − π (V i ) t, yti − Γis,t ; xi,t,u t,u . ¯ := B 1 − B 2 so that Γs,u − Γs,t − Γt,u = A¯ + B ¯ We also set A¯ := A1 − A2 , B and hence ¯ . Γs,u − Γs,t − Γt,u ≤ A¯ + B 1/p

We now apply Lemmas A and B with parameters := c1 ω (s, u) , δ := ε. Lemma A was tailor-made to give the estimate ! "γ A¯ ≤ c2 ys1 −ys2 + 1 V 1 −V 2 γ −1 υω (s, u)1/p exp c2 υω (s, u)1/p Lip υ "[p]+ 1 ! 1/p 1/p . + c2 ε υω (s, u) exp c2 υω (s, u) Re-observing that [p]+ 1 1/p max π (V i ) s, ysi ; xs,t t − yti ≤ c1 υω (s, t) ,

i=1,2

(10.22)

Lemma B tells us that ¯ = B 1 − B 2 B 1 1/p ≤ c3 Γs,t υω (s, u) p exp c3 υω (s, u) 1 1/p + c3 yt1 − yt2 + V 1 − V 2 Lip γ −1 + ευω (s, t) υ ! "1+ ([p]+1)(m in(2,γ )−1) 1/p 1/p exp c3 υω (s, u) · υω (s, u) "[p]+ 2 ! 1/p 1/p . + c3 ε υω (s, u) exp c3 υω (s, u) Observe that 1+([p] + 1) (min (2, γ) − 1) ≥ min ([p] + 2, 1 + γ (γ − 1)) ≥ γ, and obviously [p] + 2 ≥ γ. Putting things together, we obtain Γs,u ≤ Γs,t exp c4 υω (s, u)1/p + Γt,u (10.23) 1 +c4 max yr1 − yr2 + ε + V 1 − V 2 Lip γ −1 υ r ∈{s,t} "γ ! 1/p 1/p . · υω (s, u) exp c4 υω (s, u) We also have, from Theorem 3.18, that 1 y − y 2 s,t − Γs,t = π (V 1 ) s, ys1 ; x1,s,t s,t − π (V 2 ) s, ys2 ; x2,s,t s,t

236

Rough diﬀerential equations

is bounded above by 1 1/p 1/p . c5 ys1 − ys2 + V 1 − V 2 Lip γ −1 + ε υω (s, t) exp c5 υω (s, t) υ (10.24) Thanks to estimates (10.23), (10.24) and (10.21), we can apply Lemma Γs,t and t → yt1 − yt2 with the ε parameter in that 10.63 to (s, t) → lemma set to υ1 V 1 − V 2 Lip γ −1 + ε. We therefore see that 1 2 1 1 1 2 2 y −y V y + ≤c −y −V +ε exp (c6 υ p ω (0, T )) , 6 0 0 ∞;[0,T ] Lip γ −1 υ and that for all s < t in [0, T ] , with θ = γ/p > 1, Γs,t ≤c7 y01 −y02 +ε+ 1 V 1 −V 2 γ −1 υ pθ ω (s, u)θ exp (c7 υ p ω (0, T )) . Lip υ These estimates plus (10.24) easily give that, for all s < t in [0, T ] , 1 1/p 2 ys,t −ys,t ≤c8 υ ys1 −ys2 + V 1−V 2 Lip γ −1 +υε ω (s, t) exp (c8 υ p ω (0, T )) and this implies the claimed Lipschitz estimate. Obtaining uniqueness of the RDE is then easy: take two solutions in π (V 1 ) 0, y01 , x1 . The above estimate tells us the supremum distance between these two solutions is 0. We now discuss some corollaries. First observe that a locally Lipschitz estimate in ρ1/p-H¨o l;[0,T ] -metric follows immediately from setting ω (s, t) proportional to (t − s). The corresponding result for ρp-var;[0,T ] -metric is the content of Corollary Assume that 10.27 (i) V 1 = Vi1 1≤i≤d and V 2 = Vi2 1≤i≤d are two collections of Lipγ -vector ﬁelds on Re for γ > p ≥ 1; (ii) x1 , x2 are two weak geometric p-rough paths in C p-var [0, T ] , G[p] Rd ; (iii) y01 , y02 ∈ Re thought initial conditions; of as time-0 (iv) υ is a bound on V 1 Lip γ and V 2 Lip γ and p a bound on x1 p-var;[0,T ] and x2 p-var;[0,T ] . Then, if y i = π (V i ) 0, y0i ; xi , we have for some constant C depending only on γ and p, ρp-var;[0,T ] y 1 , y 2 % 1 y0 − y02 + 1 V 1 − V 2 γ −1 ≤ Cvp Lip v & + ρp-var;[0,T ] δ 1/ p x1 , δ 1/ p x2 exp Cυ p pp .

10.3 RDE solutions

237

Proof. Follows from Theorem 10.26 with control ω given by the sum of the controls constructed in Propositions 8.4 and 8.7. (Note that for p = 1 we essentially obtain the ODE estimate from Theorem 3.18.) Thanks to Theorem 8.10 we can “locally uniformly” switch from the inhomogenous path-space metrics (ρp,ω ,ρp-var ) to the homogenous ones (dp,ω , dp-var ). In fact, we can state the following. Corollary 10.28 Let V = (Vi )1≤i≤d be a collection of Lipγ -vector ﬁelds on Re for γ > p ≥ 1. If ω is a control, p ≥ p and R > 0, the maps7 → (C p-ω ([0, T ] , Re ) , dp ,ω ) Re × xp,ω ≤ R , dp ,ω

and Re ×

(y0 , x)

→

π (V ) (0, y0 ; x)

xp-var ≤ R , dp -var

→

(C p-var ([0, T ] , Re ) , dp -var )

(y0 , x)

→

π (V ) (0, y0 ; x)

xp-var ≤ R , d∞

→

(C p-ω ([0, T ] , Re ) , d∞ )

(y0 , x)

→

π (V ) (0, y0 ; x)

and Re ×

are also uniformly continuous. Proof. For p = p, this follows from the above Theorem 10.26/Corollary 10.27 combined with the remark that inhomogenous “ρ” path-space metrics are “H¨older equivalent” on bounded sets to the homogenous ones (Theorem 8.10). Recalling the d0 /d∞ -estimate (Proposition 8.15) we only have to consider the metrics dp -var for p > p and d0 and since d0 x1 , x2 ≤ dp -var x1 , x2 it suﬃces to consider the d0 case. and But this is easy: take p˜ ∈ (p, γ) 1 2 1 2 consider two paths x and x in xp,ω ≤ R such that d0 x , x < ε. From interpolation, 1 2 p/p dp-ω x , x ≤ ε1−p/p (2R) ˜ and noting xp-ω ≤ cR where c may depend on ω (0, T ) it only remains to ˜ use (uniform) continuity (on bounded sets) of the Itˆ o–Lyons map, applied with p˜ instead of p. 7 All

sets refering to x are subsets of C p -va r [0, T ] , G [p ] Rd .

Rough diﬀerential equations

238

10.3.5 Convergence of the Euler scheme Consider an RDE of the form dy = V (y) dx.

(10.25)

As usual, x denotes a geometric p-rough path and we assume suﬃcient regularity on the collection of vector ﬁelds V (i.e. that they are Lipγ −1 , γ > p) to ensure existence of a solution y. An Euler scheme of order ≥ [p] is a natural way to approximate such solutions, at least locally on some (small) time interval [s, t], and we have already seen the error estimate (cf. Corollary 10.15) ys,t − E(V ) (ys , xs,t ) ω (s, t)γ /p with ω (s, t) = xp p-var;[s,t] which was closely related to our existence proof of RDE solutions. We may rewrite the above as yt = ys + ys,t ≈ ys + E(V ) (ys , xs,t ) . Iteration of this leads to an approximate solution over the entire time horizon [0, T ]. We formalize this in Deﬁnition 10.29 (Euler scheme for RDEs) Given g ∈ GN Rd and V = (Vi )1≤i≤d , a collection of vector ﬁelds in C N−1 (Re ), we write Eg y := y + E(V ) (y, g) . p-var Then, ([0, T ] , given D = {0 = t0 < t1 < · · · < tn = T } and x ∈ Co [p] d R and a ﬁxed integer N ≥ p we deﬁne the “step-N Euler apG proximation” to (10.25) at time tk ∈ D by8

ytEuler;D := Et k ←−t 0 y0 := E k

S N (x) t

k −1 , t k

◦ · · · ◦ ES N (x) t 1 , t 0 y0 .

Theorem 10.30 Assume that (i) x ∈ C p-var [0, T ] , G[p] Rd ; (ii) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ (Re ); γ > p; (iii) D = {0 = t0 < t1 < · · · < tn = T } is a ﬁxed dissection of [0, T ] and write #D = n. Set N := %γ& ≥ [p] , and deﬁne the control p

p

ω (s, t) = |V |Lip γ xp-var;[s,t] . Then there exists C = C (p, γ) so that, with y = π (V ) (0, y0 ; x), #D θ Euler;D ω (tk −1 , tk ) , θ = (N + 1) /p > 1. yT − yT ≤ CeC ω (0,T ) k =1

8x

is a geometric p-rough path and S N (x) its unique step-N Lyons lift.

10.3 RDE solutions

239

If x ∈ C 1/p-H¨o l [0, T ] , G[p] Rd and tk +1 − tk ≡ |D| for all k then Euler;D yT − yT

N +1

θ −1

N +1

≤

CT |V |Lip γ x1/p-H¨o l;[0,T ] |D|

∼

|D|

θ −1

.

Proof. With D = (tk ) let z k := π (V ) (tk , Et k ←−t 0 y0 ; x) ∈ C ([tk , T ] , Re ) be the unique RDE solution to dy = V (y) dx, with time tk initial condition given by Et k ←−t 0 y0 . Note that z 0 = yT and z n = ET ←−t 0 y0 = yTEuler;D . Hence n k Euler;D zT − z k −1 ≤ yT − yT T k =1

and since zTk

=

zTk −1

= =

π (V ) tk , Et k −1 ←−t 0 y0 + E(V ) Et k −1 ←−t 0 y0 , SN (x)t k −1 ,t k ; x , T t k −1 ←−t 0 π (V ) tk −1 , E y0 ; x T π (V ) tk , π (V ) tk −1 , Et k −1 ←−t 0 y0 ; x t ; x , k

T

we can use Lipschitzness of RDE ﬂows (implied a fortiori by theorem 10.26; here c1 = c1 (γ, p)) to estimate k zT − z k −1 ≤ c1 ztk − ztk −1 exp (c1 ω (0, T )) . T k k On the other hand, k zt − ztk −1 = E(V ) Et k −1 ←−t 0 y0 , SN (x) t k −1 ,t k k k t k −1 ←−t 0 y0 ; x t ,t − π (V ) tk −1 , E k −1 k ≤ π (V ) (tk −1 , ·; x)t k −1 ,t k − E(V ) ·, SN (x)t k −1 ,t k

∞

and using Euler RDE estimates this is bounded from above by (N +1)/p

c2 ω (tk −1 , tk )

;

indeed, V ∈ Lipγ =⇒ V ∈ Lip(N +1)−1 with N + 1 = %γ& + 1 ≥ γ > p and we apply corollary 10.15 with N + 1 (instead of γ). A similar result holds for “geodesic” approximations in the sense of the following deﬁnition. Deﬁnition 10.31 (geodesic scheme forRDEs) Given D = {0 = t0 < t1 < · · · < tn = T }, x ∈ Cop-var [0, T ] , G[p] Rd , V = (Vi )1≤i≤d a collection of Lipschitz vector ﬁelds on Re , and a ﬁxed integer N ≥ p we deﬁne

Rough diﬀerential equations

240

the “step-N geodesic approximation” to (10.25) via any xD ∈ C 1-var [0, T ] , Rd such that SN xD t ,t = xt k −1 ,t k for all k = 1, . . . , n; k −1 k D sup SN x p-var;[0,T ] ≤ K xp-var;[0,T ] . D ∈D[0,T ]

The “step-N geodesic approximation” to (10.25) at time t ∈ [0, T ] is then simply deﬁned as ytgeo;D := π (V ) 0, y0 ; xD t . Remark 10.32 Such xD always exist and can be constructed as concatenations of geodesics associated to SN (x)t k −1 ,t k ∈ GN Rd . From SN (x)p-var;[0,T ] ≤ Cp,N xp-var;[0,T ] we can then take K = 31−1/p Cp,N . Proposition 10.33 Under the assumptions of Theorem 10.30, there exists C = C (p, γ, K) such that we have the error estimate (for the step-N geodesic approximation), #D θ geo;D C ω (0,T ) ≤ Ce − y ω (tk −1 , tk ) , θ = (N + 1) /p > 1. y T T k =1

Proof. With D = (tk ) let z k := π tk , yt k , xD ∈ C ([tk , T ] , Re ) be the unique ODE solution to dy = V (y) dxD , with time tk initial condition given by yt k = π (V ) (0, y0 ; x)t k . Observe that D −1 # k +1 geo;D z − zTk . ≤ yT − yT T k =0

Since zTk +1

= π tk +1 , yt k + 1 , xD T and zTk = π tk +1 , π tk , yt k , xD t

k+1

, xD

T

we can use Lipschitzness of RDE ﬂows (implied a fortiori by Theorem 10.26) p p to see that, with ω (s, t) = |V |Lip γ xp-var;[s,t] as earlier, k +1 k D z t ≤ c − z exp (c ω (t , T )) y − π , y , x 1 1 k +1 t k t (V ) T k+1 k T tk + 1 = c1 exp (c1 ω (tk +1 , T )) yt k ,t k + 1 − π (V ) tk , yt k , xD t ,t k k+1

with c1 = c1 (p, γ, K) , uniformly over D simply because SN xD ≤ K xp-var;[t k + 1 ,T ] for all k. From the geodesic p-var;[t k + 1 ,T ] error estimate it is clear that, with c2 = c2 (p, γ), k +1 θ z − zTk ≤ c2 exp (c2 ω (0, T )) × ω (tk , tk +1 ) T for θ > 1. In fact, the same argument as in Theorem 10.30 shows that θ can be taken to be (N + 1) /p. The proof is now easily ﬁnished.

10.4 Full RDE solutions

241

10.4 Full RDE solutions The RDEs we considered in the previous subsection map weak geometric p-rough paths to Re -valued paths of bounded p-variation. We shall now see that one can construct a “full” solution as a weak geometric p-rough path in its own right. This will allow us to use a solution to a ﬁrst RDE to be the driving signal for a second RDE. Relatedly, RDE solutions can then be used (as integrators) in rough integrals, cf. Section 10.6 below. This is not only for “functorial” beauty of the theory! It is precisely this reasoning that will enable us later to deal with various derivatives of RDEs such as the Jacobian of an RDE ﬂow. Let us also remark that in Lyons’ orginal work [116], existence and uniqueness was established by Picard iteration and so it was a necessity to work with “full” RDE solutions throughout.

10.4.1 Deﬁnition

Deﬁnition 10.34 Let x ∈ C p-var [0, T ] , G[p] Rd be a weak geometric p-rough path. We say that y ∈ C [0, T ] , G[p] (Re ) is a solution to the full rough diﬀerential equation (short: a full RDE solution) driven by x [p] e along the vector ﬁelds (Vi )i and started at y0 ∈ G (R ) if there exists n 1-var d [0, T ] , R such that (10.13) holds, and ODE a sequence (x )n in C solutions yn ∈ π (V ) (0, π 1 (y0 ) ; xn ) such that y0 ⊗ S[p] (y n ) converges uniformly to y when n → ∞. The (formal) equation dy = V (y) dx is referred to as a full rough diﬀeren tial equation (short: full RDE). This deﬁnition generalizes immediately to time intervals [s, T ] and we deﬁne π (V ) (s, ys ; x) ⊂ C [s, T ] , G[p] (Re ) to be the set of all solutions to the above full RDE starting at ys at time s, and in case of uniqueness, π (V ) (s, ys ; x) is the solution of the full RDE.9 A key remark about full RDEs (driven by x along vector ﬁelds V ) is that they are just RDEs (in the sense of the previous section) driven by x but along diﬀerent vector ﬁelds as made precise in the next theorem. We have Theorem 10.35 Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ −1 (Re ), where γ > p; (ii) x : [0, T ] → G[p] Rd is a weak geometric p-rough path; ∼ R1+e+···+ e [ p ] is an initial condition; (iii) y0 ∈ G[p] (Re ) ⊂ T [p] (Re ) = (iv) y is a solution of the full RDE driven by x, along (V ) , started at y0 . 9 Make sure to distinguish between Re -valued RDE solutions denoted by π (. . . ) and G [p ] (Re )-valued full RDE solutions denoted by the bold greek letter π (. . . ) .

Rough diﬀerential equations

242

∼ R1+e+···+ e [ p ] is a solution to Then, u → zu := yu ∈ G[p] (Re ) ⊂ T [p] (Re ) = the RDE dzu = W (z) dxu [p ] e [p] driven by x along Lipγlo−1 (Re ) ∼ = R1+e+···+ e c (R )-vector ﬁelds on T given by Wi (z) = z ⊗ Vi (π 1 (z)) , i = 1, . . . , d.

Proof. By deﬁnition of full RDEs and RDEs, we can assume that x =S[p] (x) where x is of bounded variation. Then, if y = π (V ) (0, y0 ; x) , dyu

=

d y0 ⊗ S[p] (y)0,u

=

y0 ⊗ S[p] (y)0,u ⊗ dy0,u

=

yu ⊗ V (yu ) dxu .

10.4.2 Existence Theorem 10.35 is useful as it immediately implies a local existence theorem for full RDEs. For global existence we need to rule out explosion and we do this via the following quantitative estimates for full RDEs. Theorem 10.36 Assume that (i) V = (Vi )1≤i≤d is a collection of vector ﬁelds in Lipγ −1 (Re ), where γ > p; (ii) (xn ) is a sequence in C 1-var [0, T ] , Rd , and x is a weak geometric p-rough path such that lim d0;[0,T ] S[p] (xn ) , x = 0 and sup S[p] (xn )p-var;[0,T ] < ∞; n →∞

n

(iii) y0n ∈ G[p] (Re ) is a sequence converging to some y0 . Then, at least along a subsequence, y0n ⊗ S[p] π (V ) (0, π 1 (y0n ) ; xn ) converges in uniform topology. Moreover, there exists a constant C1 depending on p, γ, such that for any such limit point y, we have p p yp-var;[s,t] ≤ C1 |V |Lip γ −1 xp-var;[s,t] ∨ |V |Lip γ −1 xp-var;[s,t] . (10.26) Then, if xs,t : [s, t] → Rd is any continuous bounded variation path such that t s,t s,t dxu ≤ K x Sγ x s,t = Sγ (x)s,t and p-var;[s,t] s

10.4 Full RDE solutions

243

for some constant K ≥ 1, we have for all s < t in [0, T ] and all k ∈ {1, . . . , %γ&} , (10.27) π k Sγ (y)s,t − Sγ π (V ) s, π 1 (ys ) ; xs,t s,t pk γ +k −1 ∨ K |V |Lip γ −1 xp-var;[s,t] ≤ C2 K |V |Lip γ −1 xp-var;[s,t] γ +k −1

where C2 depends on p and γ. (For t − s small, the term (. . . ) inates.)

dom-

Proof. Observe ﬁrst that it is enough to prove the quantitative estimates for x = S[p] (x) , wherex is a bounded variation path, so that y = y0 ⊗ S[p] π (V ) (0, π 1 (y0 ) ; x) . Deﬁne the control ω by10 1/p

ω (s, t)

= K |V |Lip γ −1 xp-var;[s,t]

and consider the hypothesis (HN ) : 1/p

∃c1,N > 0 : ∀ s < t in [0, T ] : SN (y)p-var;[s,t] ≤ c1,N ω (s, t)

∨ ω (s, t) .

We aim to prove by induction that (HN ) holds for all N = 1, . . . , [p]. For N = 1, (HN ) follows from (Davie’s) Lemma 10.7. We now assume (HN ) for a ﬁxed N < [p] , and aim to prove (HN +1 ) . Fix s and t in 1/p [0, T ] , deﬁne α := ω (s, t) ∨ ω (s, t) and observe that SN (y)p-var;[s,t] ≤ 1/p ≤ c1,N . c1,N ω (s, t) ∨ ω (s, t) = c1,N α is equivalent to SN y α

For u ∈ [0, t − s], deﬁne zu = α.SN +1

1 y α

p-var;[s,t]

s,s+u

so that (we could write dxs+ u instead of dxs+u in the next line) dzu = Wα ,y s (zu ) dxs+ u ,

z0 = (α, 0, . . . , 0) ∈ T N +1 (Re )

(10.28)

with vector ﬁelds Wα ,y s on T N +1 (Re ) given by (cf. notation of Theorem 10.35) 

Wα ,y s

1 0 For

 (ys + π 1 (z))  for z ∈ T N +1 (Re ) . ... (z) =  1 α z⊗Vd (ys + π 1 (z)) 1 α z⊗V1

the proof of the ﬁrst estimate, (10.26), we take K = 1.

Rough diﬀerential equations

244

Observe that Wα ,y s (z) only depends on z through π 0,N (z); that is, it does not depend on π N +1 (z) . Then, for k = 0, . . . , N, 1 |π k (zu )| = sup π k ◦ SN y sup α α s,s+ u u ∈[0,t−s] u ∈[0,t−s] k 1 y ≤ c2,N sup SN α s,s+ u u ∈[0,t−s] ≤

c3,N from the induction hypothesis. Deﬁne Ω = z ∈ T N +1 (Re ) , α1 |π 0,N (z)| < c3,N + 1 and observe that {zu : u ∈ [0, t − s]} ⊂ Ω. On the other hand, for z ∈ Ω, ! 1 1 π 0,N (z) ⊗Vi (. . . ) ≤ (c3,N + 1) |Vi (. . . )| z⊗Vi (. . . ) = α α so that |Wα ,y s |Lip γ −1 (Ω) ≤ c5,N |V |Lip γ −1 . We can then ﬁnd compactly ˜ α ,y which coincide with Wα ,y on Ω, and such supported vector ﬁelds W s s that ˜ Wα ,y s γ −1 ≤ c4 |Wα ,y s |Lip γ −1 (Ω) ≤ c5,N |V |Lip γ −1 . Lip

Moreover, since z|[0,t−s] remains in Ω, we see that z actually solves ˜ α .y (zu ) dxs+u , dzu = W s

z0 = (α, 0, . . . , 0) .

From Lemma 10.7, we have ˜ ˜ p p |zt−s −α| ≤ c6,N Wy s γ −1 xp-var;[s,t] ∨ Wy s γ −1 xp-var;[s,t] Lip (Ω) (Ω) Lip p p ≤ c7,N |V |Lip γ −1 xp-var;[s,t] ∨ |V |Lip γ −1 xp-var;[s,t] =c7,N α. This reads α δ α1 SN +1 (y)s,t − 1 ≤ c7,N α and, using Proposition 7.45, we have δ α1 SN +1 (y)s,t ≤ c7,N , which is equivalent to 1/p SN +1 (y)s,t ≤ c7,N α = c1,N +1 ω (s, t) ∨ ω (s, t) . This ﬁnishes the induction step and thus the proof of (10.26).

10.4 Full RDE solutions

245

For the proof of (10.27) we proceed similarly. We ﬁrst write α.Sγ α1 y |[0,t−s] as solution to a diﬀerential equation of the form (10.28), but now with vector ﬁelds Wα ,y s on T γ (Re ). Applying the “geodesic approximation” error estimate from (Davie’s) Lemma 10.7 we see that 1 1 γ /p s,t y π (V ) s, ys ; x − αSγ , ≤ c1 ω (s, t) α.Sγ α s,t α s,t which means that for all k ∈ {1, . . . , %γ&} , γ /p . π k Sγ (y)s,t − Sγ π (V ) s, ys ; xs,t s,t ≤ c1 αk −1 ω (s, t) (10.29) 1/p Recalling that, by deﬁnition, α = ω (s, t) ∨ ω (s, t) leads to an estimate of the form (10.27) with right-hand side given by a constant times !

1/p

ω (s, t)

"γ +k −1

"γ +p(k −1) ! 1/p ∨ ω (s, t) .

"γ +k −1 ! 1/p Obviously, the term ω (s, t) dominates for ω (s, t) ≤ 1. For ω (s, t) ≥ 1 it is in fact better to estimate each term on the left-hand side of (10.29) separately, using in particular k k π k Sγ (y)s,t ≤ Sγ (y)s,t ≤ yp-var;[s,t] , thanks to estimates for the Lyons lift and (10.26). Remark 10.37 Observe that the (ﬁrst part of the) above proof shows that 1/p 1/p = |V |Lip γ −1 for any ﬁxed s < t, and α ≥ ω (s, t) ∨ ω (s, t) with ω (s, t) xp-var;[s,t] r ∈ [0, t − s] → zr = α δ 1/α y s,s+ r ∈ T [p] Rd ˜ α ,y (zr ) dxs+ r started from z0 = (α, 0, . . . , 0) along comsatisﬁes dzr = W s ˜ α ,y on T [p] (Re ) which satisfy pactly supported vector ﬁelds W s ˜ Wα ,y s γ −1 ≤ c5,N |V |Lip γ −1 Lip

1

˜ α ,y (z) ≡ z⊗Vj (ys + π 1 (z)) and W on z ∈ T [p] (Re ) , α1 |z| < c s α j =1,...,d for some c dependent on p.

10.4.3 Uniqueness and continuity Theorem 10.26 states uniqueness of RDE solutions, but ignores full RDEs. We observe that full RDE solutions are RDE solutions driven by diﬀerent

246

Rough diﬀerential equations

vector ﬁelds. In particular, if we put ourselves under the same conditions as Theorem 10.26, we automatically obtain uniqueness of full RDEs. What is less obvious is that we have a similar Lipschitz bound for the (full) Itˆ o– Lyons map. Just as in the existence discussion, this estimate on full RDE solutions is actually a consequence of our earlier estimate on RDE solutions. Theorem 10.38 Assume that (i) V 1 = Vi1 1≤i≤d and V 2 = Vi2 1≤i≤d are two collections of Lipγ -vector ﬁelds on Re for γ > p ≥ 1; (ii) ω is a ﬁxed control; (iii) x1 , x2 ∈ C p-var [0, T ] , G[p] Rd , with xi p−ω ≤ 1; of as initial conditions; (iv) y01 , y02 ∈ G[p] (Re ) thought time-0 (v) υ is a bound on V 1 Lip γ and V 2 Lip γ . For i = 1, 2 we set yi = π (V i ) 0, y0i ; xi ; that is, the full RDE solutions driven by xi , starting at y0 , along the vector ﬁelds V1i , . . . , Vdi . Then we have the following Lipschitz estimate on the (full) Itˆ o–Lyons map:11 ρp,ω y1 , y2 ! " ≤ C υ y01 − y02 + V 1 − V 2 Lip γ −1 + υρp,ω x1 , x2 exp (Cυ p ω (0, T )) , where C = C (γ, p) and y0i = π 1 y0i ∈ Re . 1/p

Proof. Fix s < t in [0, T ] , and deﬁne α = c2 ω (s, t) i exp (c2 ω (0, T )) ∈ R. . As noticed in Deﬁne for i = 1, 2 and r ∈ [0, t − s] , zri = α δ 1/α ys,s+r Remark 10.37, z i is the solution of the RDE ˜ i i zri dxis+r , dzri = W z0i = α, α ,y s ˜i ˜ αi ,y (z) ≡ 1 z⊗V i (ys + π 1 (z)) ≤ cυ, and W on where W α ,y s j α s j =1,...,d γ Lip i 1 [p] e i z ∈ T (R ) : α |z| < c for some c dependent on p. Writing yt = π 1 yt it is also easy to see that 2 2 ˜2 ˜ 1 1 V ys + . − V 1 ys1 + . γ −1 ≤ c Wα ,y s2 −W 1 α ,y s Lip Lip γ −1 2 ≤ c1 V − V 1 Lip γ −1 + c1 V 1 ys2 + . − V 1 ys1 + . Lip γ −1 ≤ c2 V 2 − V 1 Lip γ −1 + ys2 − ys1 .υ (10.30) 1 1 By deﬁnition of a full RDE solution y, any increment y s , t depends on the starting point y 0 only through y 0 = π 1 (y 0 ). This explains why we don’t have y 01 − y 02 = 1 y − y 2 [ p ] e on the right-hand side. 0 0 T (R )

10.4 Full RDE solutions

247

where we used crucially V 1 ∈ Lipγ in the last estimate. Therefore, we obtain from Theorem 10.26 that, with ε := ρp,ω ;[0,T ] x1 , x2 , 1 1 2 zt−s − zt−s ≤ c3 υ ys1 − ys2 + V 1 − V 2 Lip γ −1 + ευ ω (s, t) p exp (c3 υ p ω (0, T )) . But this says precisely that for all k = 1, . . . , [p] and s < t in [0, T ] , 1 k 2 ≤ c3 υ ys1 − ys2 + V 1 − V 2 γ −1 + ευ ω (s, t) p π k ys,t − ys,t Lip exp (c3 υ p ω (0, T )) .

(10.31)

∈T (R ) on the [We insist that one does not have the norm of right-hand side above, but ys1 − ys2 coming from (10.30).] But from Theo1/p rem 10.26, noting that ω (0, s) exp (Cυ p ω (0, T )) υ −1 exp (c4 υ p ω(0, T )), we have ! " υ ys1 − ys2 ≤ c4 υ y01 − y02 + V 1 − V 2 Lip γ −1 + ευ exp (c4 υ p ω (0, T )) . ys1

− ys2

N +1

e

Plugging this inequality into (10.31) gives us the desired result. The following corollaries can then be proved with almost identical arguments as in the RDE case. Corollary Assume that 10.39 (i) V 1 = Vi1 1≤i≤d and V 2 = Vi2 1≤i≤d are two collections of Lipγ -vector ﬁelds on Re for γ > p ≥ 1; (ii) x1 , x2 ∈ C p-var [0, T ] , G[p] Rd ; (iii) y01 , y02 ∈ G[p] (R e ) thought of time-0 initial conditions; (iv) υ is a bound on V 1 Lip γ and V 2 Lip γ and p a bound on x1 p-var;[0,T ] and x2 p-var;[0,T ] . Then, if yi = π (V i ) 0, y0i ; xi , we have % 1 y0 − y02 + 1 V 1 − V 2 γ −1 ρp-var;[0,T ] y1 , y2 ≤ Cvp Lip v " + ρp-var;[0,T ] δ 1 /p x1 , δ 1 /p x2 exp Cυ p pp , where C = C (γ, p) and y0i = π 1 y0i ∈ Re . For simplicity of the statement, we once again remove the dependency in the vector ﬁelds. Corollary 10.40 Let V = (Vi )1≤i≤d be a collection of Lipγ -vector ﬁelds on Re for γ > p ≥ 1. If ω is a control, p ≥ p and R > 0, the maps → C p-ω [0, T ] , G[p] (Re ) , dp ,ω Re × xp,ω ≤ R , dp ,ω (y0 , x)

→

π (V ) (0, y0 ; x)

Rough diﬀerential equations

248

and Re ×

and Re ×

xp-var ≤ R , dp -var

→

(y0 , x)

→

xp-var ≤ R , d∞

→

(y0 , x)

→

C p-var [0, T ] , G[p] (Re ) , dp -var

π (V ) (0, y0 ; x)

C p-var [0, T ] , G[p] (Re ) , d∞

π (V ) (0, y0 ; x)

are also uniformly continuous.

10.5 RDEs under minimal regularity of coeﬃcients We now show the uniqueness of solutions to RDEs driven by geometric prough paths along Lipp -vector ﬁelds. In fact, the driving signal only needs to be of ﬁnite ψ p,1 -variation where we recall from Section 5.4 that ψ p,1 (t) = tp / (ln∗ ln∗ 1/t) where ln∗ ≡ max (1, ln) . For instance, with probability one, (enhanced) Brownian sample paths have ﬁnite ψ 2,1 -variation but are not geometric 2-rough paths (i.e. don’t have ﬁnite 2-variation), as will be discussed in the later Sections 13.2 and 13.9. The main interest in the reﬁned regularity assumption is that it shows that RDE solutions driven by (enhanced) Brownian motion have unique solutions under Lip2 -regularity assumptions. Theorem Assume thatp ≥ 1, and that 10.41 (i) V 1 = Vj1 1≤j ≤d and V 2 = Vj2 1≤j ≤d are two collections of Lipp -vector ﬁelds on Re ; (ii) x1 , x2 ∈ C ψ p , 1 -var [0, T ] , G[p] Rd ; e (iii) y01 , y02 ∈ G[p] of as time-0 initial conditions; (Ri ) thought i (iv) y ∈ π (V i ) 0, y0 ; xi for i = 1, 2 (that is, they are full RDE solutions i driven by xi , starting at y0i , along i the vector ﬁelds V ); i (v) assume V Lip p ≤ υ and x ψ -var;[0,T ] ≤ R for i = 1, 2. p ,1 Then, π (V i ) 0, y0i ; xi is a singleton; that is, there exists a unique full RDE solution yi = π (V i ) 0, y0i ; xi starting at y0i driven by xi along V i . Moreover, for all ε > 0, there exists µ = µ (ε; p, υ, R) > 0 such that12 1 y0 − y02 + V 1 − V 2 p −1 + d∞ x1 , x2 < µ Lip implies d∞ y1 , y2 < ε. 1 2 For

p = 1, V 1 − V 2 L ip p −1 is replaced by 2 V 1 − V 2 ∞ .

10.5 RDEs under minimal regularity of coeﬃcients

249

Remark 10.42 The theorem applies in particular to geometric p-rough paths, i.e. when x1 , x2 ∈ C p-var [0, T ] , G[p] Rd . It is also clear that under Lipp -regularity, existence of (full) is not an issue here so that RDE solutions the set of RDE solutions π (V i ) 0, y0i ; xi is not empty. Proof. Since we only deal with supremum distance d∞ and without quantitative estimates, it is enough to prove the above result for RDE rather than full RDE. Running constants c1 , c2 , . . . may depend on p, υ, R which 1 2 y , taken , y are kept ﬁxed in this proof. Let ε(µ) be the supremum of d ∞ over all RDE solutions yi ∈ π (V i ) 0, y0i ; xi , i = 1, 2 such that V i , xi satisfy (v) and such that 1 y0 − y02 + V 1 − V 2 p −1 + d∞ x1 , x2 < µ; Lip we show that ε = ε (µ) → 0 as µ → 0. Construction of xis,t : For i = 1, 2, we deﬁne ω i (s, t) =

sup



D ∈D([s,t]) t ∈D i

ψ p,1 

i xt i ,t i + 1 xi ψ p , 1 -var;[0,T ]

 ;

Proposition 5.39, ω i is a control with ω i (0, T ) ≤ 1. Deﬁne ω = from 1 2 ω +ω . 1/p Using the fact that ψ −1 with δ (z) = p,1 (·) ≤ c1 ψ 1/p,−1/p (·) = c1 δ (·) ∗ ∗ 1 z ln ln z (see Lemma 5.48), it follows (again using Proposition 5.39) that for all s < t in [0, T ], i xs,t ≤ c2 δ (ω (s, t))1/p . 1/p 1/p As δ (z) ≤ z 1/p for all p > p, we see that xi p -var;[s,t] ≤ c3 ω (s, t) . By interpolation, we obtain that for all p > p , there exists g1 , a continuous function 0 at 0 such that, for all s < t in [0, T ] and k = 1, . . . , [p] , 1 π k xs,t − x2s,t ≤ g1 (µ)ω (s, t)k /p . This immediately implies that for α < 1 (we can take it as close to 1 as we want, and in particular we take it greater than {p}) and for all s < t in [0, T ] , and for k = 1, . . . , [p] , 1 k + α −1 π k xs,t − x2s,t ≤ c3 g1 (µ) ω (s, t) p As δ δ (ω (s,t)) −1 / p xis,t ≤ c2 , and

≤

c3 g1 (µ) δ (ω (s, t))

k + α −1 p

.

(α −1)/p , δ δ (ω (s,t)) −1 / p x1s,t − δ δ (ω (s,t)) −1 / p x2s,t ≤ c3 g1 (µ) δ (ω (s, t))

Rough diﬀerential equations

250

Proposition 7.64 provides us with two paths x1,s,t and x2,s,t such that S[p] xi,s,t s,t = xis,t , and

t

≤ d xi,s,t r

1/p

c4 δ (ω (s, t))

,

s

t

≤ d x1,s,t − x2,s,t r r

α /p

c4 g1 (µ) δ (ω (s, t))

.

s

We deﬁne similarly xi,s,u and xi,t,u , and then xi,s,t,u to be the concatenation of xi,s,t and xi,t,u . Estimates on Γ: Following closely the pattern of proof of Theorem 10.26, we set for s < t, i − π s, ysi ; xi,s,t s,t , i = 1, 2, Γis,t = ys,t and Γs,t := Γ1s,t − Γ2s,t . Theorem 10.14 gives us for p ∈ (p, [p] + 1), i 1 Γs,t ≤ c5 xi [p]+

p -var;[s,t]

.

Using Proposition 5.49 and Theorem 5.43, we see that i i x ≤ c6 ψ −1 p,1 ω (s, t) p -var;[s,t] ≤

1/p

c7 δ (ω (s, t))

.

In particular, we have with θ = ([p] + 1) /p > 1, Γs,t ≤ c7 δ (ω (s, t))θ ≤ c8 δ (ω (s, t)) .

(10.32)

Deﬁne for i = 1, 2, Ai := π (V i ) s, ysi ; xi,s,t,u s,u − π (V i ) s, ysi ; xi,s,u s,u B i := π (V i ) t, yti ; xi,t,u t,u − π (V i ) t, π (V i ) s, ysi ; xi,s,t t ; xi,t,u t,u ¯ := B 1 − B 2 , we obtain Γs,u − Γs,t − Γt,u = A¯ + B ˜ so and A¯ := A1 − A2 , B that ¯ . Γs,u − Γs,t − Γt,u ≤ A¯ + B 1/p

From Lemma A, applied with parameters := c4 δ (ω (s, u)) α /p c4 g1 (µ) δ (ω (s, u)) , it follows that A¯ ≤ c9 ys1 − ys2 + V 1 − V 2 p −1 δ (ω (s, u)) Lip α /p

≤

[p ]+ α

+ c9 g1 (µ) δ (ω (s, u)) .δ (ω (s, u)) p c9 ys1 − ys2 + V 1 − V 2 Lip p −1 + g1 (µ) δ (ω (s, u)) .

, δ :=

10.5 RDEs under minimal regularity of coeﬃcients

251

On the other hand, maxi=1,2 π (V ) s, ysi ; xs,t t − yti = maxi=1,2 Γis,t and with (10.32), Γs,t ≤ c8 δ (ω (s, t))θ ≤ c8 δ (ω (s, t)) .

(10.33)

From Lemma B it then follows that 1 ¯ ≤ c3 Γs,t δ (ω (s, u)) p + c3 g1 (µ) δ (ω (s, u))1+α /p B α /p + c3 yt1 − yt2 + V 1 − V 2 Lip γ −1 + g1 (µ) δ (ω (s, u)) 1

·δ (ω (s, u)) p

(1+p(m in(2,p)−1))

.

Observe that p1 (1 + p (min (2, p) − 1)) ≥ p1 min p + 1, p2 − 2p + 1 + p ≥ 1, and obviously 1 + α/p ≥ 1. Putting things together, we obtain 1 ¯ ≤ c12 Γs,t δ (ω (s, u)) p B + c12 yt1 − yt2 + V 1 − V 2 Lip p −1 + g1 (µ) .δ (ω (s, u)) . ¯ , we obtain that Adding the inequalities on A¯ and B 1 Γs,u − Γs,t − Γt,u ≤ c13 Γs,t δ (ω (s, u)) p (10.34) + c13 max yr1 − yr2 + V 1 − V 2 Lip p −1 + g1 (µ) δ (ω (s, u)) . r ∈{s,t}

We also have, from Theorem 3.18, that π (V 1 ) s, ys1 ; x1,s,t − π (V 2 ) s, ys2 ; x2,s,t 1/p α /p ≤ c14 ys1 − ys2 + V 1 − V 2 Lip p −1 δ (ω (s, t)) + g1 (µ) ω (s, t) α /p . (10.35) ≤ c14 ys1 − ys2 + V 1 − V 2 p −1 + g1 (µ) ω (s, t) Lip

Conclusion: Thanks to estimates (10.34), (10.35) and (10.33), we can apply Proposition 10.70 (found in Appendix 10.8) to obtain our result. By interpolation, we obtain the following corollary. Corollary 10.43 Let V = (V1 , . . . , Vd ) be a collection of Lipp -vector ﬁelds on Re . Fix p > p > p > 1, a control ω, R ∈ (0, ∞] and set13 Ωp -ω (p, R) = x : xp -ω < R and xψ p , 1 -var < R . Ω (p, R) = x : xψ p , 1 -var < R . 1 3 In

general, C ψ p , 1 -va r C p

-ω .

Rough diﬀerential equations

252

Then the maps Re × (Ωp -ω (p, R) , dp -ω ) → C ψ p , 1 -var ∩ C p -ω [0, T ] , G[p] (Re ) , dp -ω (y0 , x) → π (V ) (0, y0 ; x) and Re × (Ω (p, R) , dp -var )

→

(y0 , x) →

C ψ p , 1 -var [0, T ] , G[p] (Re ) , dp -var

π (V ) (0, y0 ; x)

are uniformly continuous for R ∈ (0, ∞) and continuous for R = +∞. (This also holds when dp -var is replaced by d∞ .) Proof. Since continuity is a local property, it suﬃces to consider the case R < ∞. Observe that for all x ∈ Ω (p, R), we have π (V ) (0, y0 ; x)

ψ p , 1 -var

< c(R),

as follows from Corollary 5.44 and Proposition 5.49. Then, we know from Theorem 10.41 that Re × (Ω (p, R) , d∞ ) → C ψ p , 1 -var [0, T ] , G[p] (Re ) , d∞ (y0 , x) →

π (V ) (0, y0 ; x)

is uniformly continuous. In particular, this that for implies some ﬁxed ε, R > 0, there exists µ > 0 such that if y01 − y02 + dp -var x1 , x2 < µ, with x1 , x2 ∈ Ω (p, R), then d∞ π (V ) 0, y01 ; x1 , π (V ) 0, y02 ; x2 < ε. Using π (V ) 0, y0i ; xi ψ -var < c(R) we can use interpolation to obtain p ,1 uniform continuity of Re × (Ω (p, R) , dp -var )

→

(y0 , x) →

C ψ p , 1 -var [0, T ] , G[p] (Re ) , dp -var

π (V ) (0, y0 ; x) .

The modulus ω case follows from the same argument, except that for the interpolation step, we need to assume that the p -ω norm of xi , i = 1, 2 is bounded by R (since, for a given ω, we cannot be sure that the ψ p,1 variation of xi is controlled by this ω).

10.6 Integration along rough paths

253

10.6 Integration along rough paths With our main interest in (rough) diﬀerential equations we constructed RDEs directly as limits of ODEs. In the same spirit, we now deﬁne “rough integrals” as limits of Riemann–Stieltjes integration. Given the work already done, we can take a short-cut and derive existence, uniqueness and continuity properties quickly from the previous RDE results. Deﬁnition 10.44 (rough integrals) Let x ∈ C p-var [0, T ] , G[p] Rd be a weak geometric p-rough path, and ϕ = (ϕi )i=1,...,d a collection of maps from Rd to Re . We say that y ∈ C [0, T ] , G[p] (Re ) is a rough path integral of ϕ along x, if there exists a sequence (xn ) in C 1-var [0, T ] , Rd such that ∀n : xn0 = π 1 (x0 ) lim d0;[0,T ] S[p] (xn ) , x = 0 n →∞ sup S[p] (xn )p-var;[0,T ] < ∞ n

. ϕ (xnu ) dxnu , y = 0. lim d∞ S[p]

and

n →∞

0

We will write ϕ (x) dx for the set of rough path integrals of ϕ along x. If this set is a singleton, it will denote the rough path integral of ϕ along x.

· We ﬁrst note that a classical indeﬁnite Riemann–Stieltjes integral 0 ϕ (x) dx can be written as a (projection to the y-component of the) ODE solution to dz = dx, dy = ϕ (z) dx, (z, y) = (x0 , 0). This leads to the key remark that rough integrals can be viewed as (projections of) a solution to (full) RDEs driven by x along vector ﬁelds V = (V1 , . . . , Vd ) given by (10.36) Vi (x, y) = (ei , ϕi (x)) , where (e1 , . . . , ed ) is the standard basis of Rd . Obviously V has the same amount of Lip-regularity as ϕ = (ϕi )i=1,...,d , viewed as a map ϕ : Rd → L Rd , Re ; in fact, |V |Lip γ −1 1 + |ϕ|Lip γ −1 . Thus, if one is happy with rough integration under Lipγ -regularity, γ > p, existence, uniqueness and uniform

254

Rough diﬀerential equations

continuity on bounded sets (in fact: Lipschitz continuity on bounded sets with respect to the correct rough path metric ρp-var ) of → C p-var [0, T ] , G[p] (Re ) Lipγ × C p-var [0, T ] , G[p] Rd (ϕ, x) → ϕ (x) dx is immediate from the corresponding results on full RDEs in Section 10.4. The point of the forthcoming theorem is that for rough path integration, one gets away with Lipγ −1 regularity. As can be seen already from

the next lemma, there is no hope for local Lipschitz continuity of (ϕ, x) → ϕ (x) dx, but uniqueness and uniform continuity on bounded sets hold true. Lemma 10.45 Let (i) ϕ1j j =1,...,d , ϕ2j j =1,...,d be two collections of Lipα Rd , Re -maps, α ∈ (0, 1], so that maxi=1,2 ϕi Lip α ≤ 1; (ii) xi ∈ C 1-var [s, t] , Rd such that, for some ≥ 0 and ε ∈ [0, 1] , i x ≤ , 1-var;[s,t] 2 1 x − x ≤ ε and x2s − x1s ≤ ε. 1-var;[s,t] Then, for k ∈ {1, 2, . . . } , 2 2 1 1 2 1 ϕ x dx ϕ x dx − π k Sk π k Sk s,t s,t 2 ≤ ϕ − ϕ1 ∞ + εα k . Proof. From Proposition 2.8, we easily see that ϕ2 x2 dx2 − ϕ1 x1 dx1 ≤ ϕ2 − ϕ1 ∞ + εα . 1-var;[s,t]

The conclusion follows then from Proposition 7.63. Lemma that 10.46 (Lemma Aintegral ) Assume (i) ϕj j =1,...,d is a collection of Lipγ −1 Rd , Re -maps where γ > 1; (ii) s < u are some elements in [0, T ]; (iii) x and x ˜ are some paths in C 1-var [s, u] , Rd such that Sγ (x)s,u = x)s,u and xs = x ˜s ; Sγ (˜

u

u x|. (iv) ≥ 0 is a bound on s |dx| + s |d˜ Then we have, for some constant C = C (γ) and for all k ∈ {1, . . . , %γ&} , k ϕ (x) dx ϕ (˜ x) d˜ x −π k ◦ Sk ≤ C ϕLip γ −1 γ +k −1 . π k ◦ Sk s,u s,u

10.6 Integration along rough paths

255

· Proof. The indeﬁnite Re -valued integral s ϕ (x) dx is also a (projection of the) solution to the ODE y· = π (V ) (s, (xs , 0) ; x) with V given by (10.36). By homogeneity, we may assume ϕLip γ −1 = 1 so that |V |Lip γ −1 ≤ c1 . We leave it to the reader to provide a self-contained “Riemann–Stieltjes/ODE” proof of the claimed estimate. Somewhat shorter, if more fanciful, we can think of y as a solution (again, strictly speaking, a projection thereof) of an RDE driven by a weak geometric 1-rough path x. The claimed estimate is now a consequence of the “geodesics error-estimate” of Theorem 10.36 with p = 1 and K = 1. Indeed, we take a geodesic xs,u associated with Sγ (x)s,u such that x)s,u and Sγ (xs,u )s,u = Sγ (x)s,u = Sγ (˜ u |dxs,u | ≤ Sγ (x)1-var;[s,u ] = |x|1-var;[s,u ] . s

˜s we see that both y· = π (V ) (s, (xs , 0) ; x) and y˜· = π (V ) Since xs = x (s, (˜ xs , 0) ; x ˜) have the same geodesic approximation given by π (V )(s, (xs , 0) ; xs,u ). On the “projected” level of the integral this amounts to consid· s,u ering s ϕ (xs + xs,u s,· ) dx· ) and so, for all s < u in [0, T ] and for all k ∈ {1, . . . , %γ&} , · · s,u s,u ϕ (x) dx − Sγ ϕ xs + xs,· dx· π k Sγ s s s,u s,u γ + k −1 ≤ c2 γ + k −1 . ≤ c2 |x|1-var;[s,u ] The same estimate holds with x replaced by x ˜ and an application of the triangle inequality ﬁnishes the proof. Theorem Existence: Assume that 10.47 (i) ϕ = ϕj j =1,...,d is a collection of Lipγ −1 Rd , Re -maps where γ > p ≥ 1; (ii) x is a geometric p-rough path in C p-var [0, T ] , G[p] Rd . Then, for all s < t ∈ [0, T ] , there

exists a unique rough-path integral of ϕ along x. The indeﬁnite integral ϕ (x) dx is a geometric rough path: there exists a constant C1 depending only on p and γ such that for all s < t in [0, T ], p ϕ (x) dx x . ≤ C ϕ ∨ x γ −1 1 Lip p-var;[s,t] p-var;[s,t] p-var;[s,t]

Also, if xs,t : [s, t] → Rd is any continuous bounded variation path such that t s,t s,t s,t dxu ≤ K x xs = π 1 (xs ) , Sγ x s,t = Sγ (x)s,t and p-var;[s,t] s

Rough diﬀerential equations

256

for some constant K, we have for all s < t in [0, T ] with K xp-var;[s,t] < 1, and all k ∈ {1, . . . , %γ&} , # $ s,t s,t ϕ xu dxu ϕ (x) dx − Sγ π k s,t s,t γ +k −1 k ≤ C2 ϕLip γ −1 K xp-var;[s,t] , (10.37) where C2 depends on p and γ.

Uniqueness, continuity: There exists a unique element in ϕ (x) dx. More precisely, if ω is a ﬁxed control, max ϕi Lip γ −1 , xi p-ω ;[0,T ] < R, i=1,2

and

ε = x10 − x20 + ρp-ω ;[0,T ] x1 , x2 + ϕ2 − ϕ1 Lip γ −1 ,

then for some constant β = β (γ, p) > 0 and C = C (R, γ, p) ≥ 0, 1 1 2 2 1 2 ϕ x dx , ϕ x dx ≤ Cεβ . ρp-ω ;[0,T ] Proof. Existence, ﬁrst proof: For some ﬁxed r > 0, consider the collection of vector ﬁelds V (r ) deﬁned by (r )

Vi

(x, y) = (ei , rϕi (x)) for x, y ∈ Rd × Re ,

where ei is the ith standard basis vector of Rd . Observe that V (r ) is (γ − 1)Lipschitz, and that for a bounded variation path we have . rϕ (xu ) dxu . π (V ( r ) ) (0, (x0 , 0) ; x) = x, 0

In view of (γ − 1)-Lipschitz regularity of the vector ﬁelds we can apply Theorem 10.36; we obtain existence of a rough integral of x along ϕ, and that every element y(r ) of rϕ (x) dx satisﬁes (r ) y

p-var;[s,t]

≤ c1

(r ) V

Lip γ −1

p xp-var;[s,t] ∨ V (r )

Lip γ −1

p xp-var;[s,t] .

Now, setting y = y(1) it is easy to see that y(r ) = δ r y and V (r ) Lip γ −1 ≤ c2 1 + r |ϕ|Lip γ −1 ; thus, picking r = 1/ |ϕ|Lip γ −1 , we obtain p yp-var;[s,t] ≤ c3 |ϕ|Lip γ −1 xp-var;[s,t] ∨ xp-var;[s,t] .

10.6 Integration along rough paths

257

The error estimate (10.37) is also a consequence of the corresponding estimate in Theorem 10.36, again applied to full RDEs along vector ﬁelds V (r ) with r = 1/ |ϕ|Lip γ −1 . Existence, second proof: We provide a second existence proof, based on by now standard arguments, which is (notationally) helpful for the forthcoming uniquess proof. Similar to the proof of RDE existence, it suﬃces to establish uniform estimates for x of the form x = S[p] (x), followed by a straightforward limiting argument. Let us thus assume x = S[p] (x) and 1/p xs,t ≤ ω (s, t) for some control ω. By assumption, there exists xs,t , which we think of as an “almost” geodesic path associated with xs,t , and we deﬁne ϕ (x) dx ϕ xs,t dxs,t − Sγ . Γs,t = Sγ s,t

s,t

Then, for s < t < u, we have Γs,u = ∆1 − ∆2 + ∆3 where ∆1

=

∆2

=

Sγ

∆3

=

Sγ

⊗ Sγ

ϕ (x) dx

Sγ

s,t

ϕ xs,t dxs,t

ϕ (x) dx

⊗ Sγ

ϕ x

s,t

ϕ xs,t,u dxs,t,u

, t,u

t,u

dx

t,u

− Sγ

, t,u

ϕ (xs,u ) dxs,u

.

s,u

s,u

First, using Lemma Aintegral (Lemma 10.46), we obtain that for all N ∈ {1, . . . , %γ&} |π N (∆3 )| ≤ c1 ω (s, u) We also see that ϕ (x) dx ∆1 − ∆2 = Sγ

γ + N −1 p

.

(10.38)

⊗ Γt,u + Γs,t ⊗ Sγ

ϕ x

t,u

dx

t,u

s,t

t,u

so that Γs,u − Γs,t ⊗ Γt,u equals ϕ xs,t dxs,t ϕ xt,u dxt,u Sγ ⊗ Γt,u + Γs,t ⊗ Sγ s,t

+ ∆3 . t,u

(10.39) We now prove by induction in N ∈ {1, . . . , %γ&} that ∀s < u in [0, T ] : |π N (Γs,u )| ≤ c2 ω (s, u)

γ + N −1 p

1/p . exp c2 ω (s, u)

Rough diﬀerential equations

258

For N = 0, this is obvious as π 0 (Γs,t ) = 0. Assume now that γ + k −1 1/p ∀s < u in [0, T ] : |π k (Γs,u )| ≤ c2 ω (s, u) p exp c2 ω (s, u) holds for all k < N . Then, from (10.39), we have π N (Γs,u ) − π N (Γs,t ) − π N (Γt,u ) =

π N (∆3 ) +

N −1 k =1

+

N −1

π k ◦ Sγ

π k (Γs,t ) ⊗ π N −k (Γt,u )

ϕ x

s,t

dx

N −1

⊗ π N −k (Γt,u )

s,t s,t

k =1

+

(10.40)

π k (Γs,t ) ⊗ π N −k ◦ Sγ

ϕ xt,u dxt,u

, t,u

k =1

so that, using the induction hypothesis, (10.38) and bounds of the type s,t s,t k /p dx ϕ x ≤ c2 ω (s, t) , π k ◦ Sγ s,t we have14

1/p . exp c2 ω (s, u) (10.41) We can then classically use Lemma 10.59 to obtain that γ + N −1 1/p , |π N (Γs,u )| ≤ c4 ω (s, u) p exp c4 ω (s, u) |π N (Γs,u ) − π N (Γs,t ) − π N (Γt,u )| ≤ c3 ω (s, u)

γ + N −1 p

which concludes the induction proof. The triangle inequality then leads to, for k ∈ {1, . . . , %γ&}, k /p 1/p , ϕ (x) dx ≤ c4 ω (s, t) exp c4 ω (s, t) π k Sγ s,t which is equivalent to saying 1/p 1/p . ϕ (x) dx Sγ ≤ c5 ω (s, t) exp c5 ω (s, t) s,t 1 4 As we shall need (10.41) in the uniqueness/continuity part below, let us point out that (10.41) also follows from the “ﬁrst” existence proof, namely from the error estimate (10.37).

10.6 Integration along rough paths

259

An application of Proposition 5.10 then gives 1/p ϕ (x) dx Sγ ≤ c6 ω (s, t) ∨ ω (s, t) . s,t Continuity/uniqueness: Without loss of generality, we assume %γ& = [p] , and also, by simple scaling, that maxi=1,2 ϕi Lip γ −1 ≤ 1 (so that we can use Lemma 10.45). We set ε = x10 − x20 + ρp-ω ;[0,T ] x1 , x2 + ϕ2 − ϕ1 Lip γ −1 and agree that, in this part of the proof, constants may depend on ω (0, T ) and R. Then we deﬁne (exactly as in the proof of Theorem 10.26) two paths x1,s,t and x2,s,t such that S[p] xi,s,t s,t = xis,t , i = 1, 2, and such that

i,s,t dxr

≤

c7 ω (s, t)

1,s,t dxr − dx2,s,t r

≤

c7 εω (s, t)

t

1/p

, i = 1, 2,

s

t

1/p

.

s

We also deﬁne (as usual!) Γis,t =

ϕi xir dxir

− S[p]

i,s,t dxr ϕi xi,s,t r

s,t

, i = 1, 2, s,t

and set Γs,t := Γ1s,t − Γ2s,t . From the existence part, we have Γs,t ≤ γ /p

c8 ω (s, t)

. Deﬁne

# ∆3 =

Sγ #

−

ϕ1 x1 , s , t , u dx1 , s , t , u

− Sγ s,u

Sγ

ϕ

2

x

2,s,t,u

dx

2,s,t,u

$

s,u

− Sγ

s,u

ϕ1 x1 , s , u dx1 , s , u

ϕ

2

x

2,s,u

dx

2,s,u

$ .

s,u

Rough diﬀerential equations

260

Observe that by continuity of the Riemann–Stieltjes integral and the map Sγ , we have for all integers k ∈ {1, . . . , [p]}

≤

π k ∆3 ϕ2 x2,s,t,u dx2,s,t,u π k ◦ Sγ s,u ϕ1 x1,s,t,u dx1,s,t,u − π k ◦ Sγ

s,u

2,s,u 2,s,u 2 dx ϕ x + π k ◦ Sγ s,u ϕ1 x1,s,u dx1,s,u − π k ◦ Sγ s,u m in{γ −1,1} ≤ 2 ϕ2 − ϕ1 Lip γ −1 + ρp,ω x1 , x2 k /p

ω (s, t) ≤

using Lemma 10.45

k /p

α

ε ω (s, t)

for some α ∈ (0, 1].

(10.43)

We now prove by induction on N ≤ [p] that ∀k ∈ {1, . . . , N } : ∃β k > 0 : π k Γs,t ≤ εβ k ω (s, t) . For N = 0, it is obvious, as π 0 Γs,t = 0. Assume now the induction hypothesis is true for all k < N. From equation (10.40), we see that π N Γs,u − π N Γs,t − π N Γt,u is equal to ∆3 + D1 + D2 + D3 , where

D1

=

N −1

−1 N π k Γs,t ⊗ π N −k Γ2t,u − π k Γ1s,t ⊗ π N −k Γt,u

k =1

D2

=

N −1

Sγ

πk

k =1

+

k =1

πk

ϕ x

2,s,t

ϕ x1,s,t dx1,s,t

− Sγ N −1

k =1

Sγ

dx

2,s,t

s,t

⊗ π N −k Γ2t,u

s,t

ϕ x

1,s,t

dx

1,s,t s,t

⊗ π N −k Γt,u

10.6 Integration along rough paths

D3

=

N −1

π k Γs,t ⊗ π N −k ◦ Sγ

k =1

−

N −1 k =1

−Sγ

πk

Γ1t,u

ϕ x2,t,u dx2,t,u

t,u

⊗ π N −k

261

ϕ x

Sγ

ϕ x1,t,u dx1,t,u

2,t,u

dx

t,u

2,t,u

. t,u

We easily see from the induction hypothesis and Lemma 10.45 that N /p

|D1 | + |D2 | + |D3 | ≤ c9 εα N ω (s, t)

, for some αN ∈ (0, α].

In particular, with (10.43), we obtain that π N Γs,u − Γs,t − Γt,u ≤ c10 εα N ω (s, t)N /p . From the existence part, equation (10.41), we also have π N Γs,u −Γs,t −Γt,u ≤ π N Γ2s,u −Γ2s,t −Γ2t,u + π N Γ1s,u −Γ1s,t −Γ1t,u ≤ c11 ω (s, t)

γ + N −1 p

θ

≤ c12 ω (s, t) with θ = γ/p > 1.

These estimates show that assumption (ii) of Lemma 10.61 is satisﬁed (checking assumption (i) is easy and left to the reader); it then follows that for some β N ∈ (0, 1], π N Γs,t ≤ εβ N ω (s, t) and so the induction step is completed. Putting this last inequality and Lemma 10.45 together, we see that for k ≤ [p] , we have from the triangle inequality 2 2 1 1 k /p 2 1 ϕ x dx − ϕ x dx ≤ εm in(β N ,γ −1) ω (s, t) . π k s,t s,t The proof is now ﬁnished. As a consequence of Theorem 10.47, we have Corollary 10.48 Let ϕi : Rd → Re , i = 1, . . . , d be some (γ − 1)-Lipschitz maps where γ > p. For any ﬁxed control ω, R > 0 and p ≤ p , the maps → C p-ω [0, T ] , G[p] (Re ) , dp -ω Lipγ −1 × xp-ω ≤ R , dp ,ω (ϕ, x) → ϕ (x) dx

Rough diﬀerential equations

262

and Lipγ −1 ×

and Lipγ −1 ×

xp-var ≤ R , dp -var → C p-var [0, T ] , G[p] (Re ) , dp -var (ϕ, x) → ϕ (x) dx

xp-var ≤ R , d∞

→

(ϕ, x)

→

C p-var [0, T ] , G[p] (Re ) , d∞ ϕ (x) dx

are uniformly continuous.

d e R ,R , i = Exercise 10.49 Extend Theorem 10.47 to ϕi ∈ Lipγlo−1 c 1, . . . , d. ˜ ˜ i ∈ Lipγ −1 Rd , Re such that ϕ ≡ ϕ Solution. It suﬃces to replace ϕi by ϕ on a ball with radius |x|∞;[0,T ] + 1.

10.7 RDEs driven along linear vector ﬁelds In various applications, for instance when studying the “Jacobian of the ﬂow”, one encouters RDEs driven along linear vector ﬁelds. Since linear vector ﬁelds are unbounded, we can, at this stage, only assert uniqueness and local existence (cf. Theorem 10.21). The aim of the present section is then to establish global existence with some precise quantitative estimates. Exercise 10.50 (linear vector ﬁelds) Let us consider a collection V = (Vi )i=1,...,d of linear vector ﬁelds of the form Vi (z) = Ai z + bi for e × e matrices Ai and elements bi of Re . Prove (by induction) that V N ;i 0 ,...,i N (z) ≡ Vi 0 . . . Vi N (z) = Ai N . . . Ai 0 z + Ai N . . . Ai 1 bi 0 . Lemma 10.51 (Lemma Alinear ) Assume that (i) V = (Vi )1≤i≤d with Vi (z) = Ai z are a collection of linear vector ﬁelds, and ﬁx N ∈ N; (ii) s, t are some elements of [0, T ]; x)s,t ; (iii) x, x ˜ are some paths in C 1-var [s, t] , Rd such that SN (x)s,t = SN (˜

t

t (iv) is a bound on s |dxu | and s |d˜ xu | and υ is a bound on maxi (|Ai |) . Then, N +1 π (V ) (s, ys ; x)s,t − π (V ) (s, ys ; x ˜)s,t ≤ C [υ] exp (Cυ) .

10.7 RDEs driven along linear vector ﬁelds

263

Proof. Deﬁne for all r ∈ [s, t] , ys,r = π (V ) (s, ys ; x)s,r ; we saw in Theorem 3.7 that |ys,r | ≤ c (1 + |ys |) υ exp (cυ) . From Proposition 10.3, we have π (V ) (s, ys ; x)s,t − E(V ) ys , SN (x)s,t t i1 iN (V . . . V I (y ) − V . . . V I (y )) dx . . . dx ≤ i1 iN r i1 iN s r r s

i 1 ,...,i N ∈{1,...,d}

t i1 iN ys,r dxr . . . dxr ≤ Ai N . . . Ai 1 s

≤

c (1 + |ys |) [υ]

N +1

exp (cυ) .

˜)s,t and π (V ) (s, ys ; x)s,t share the same Euler approximation, As π (V ) (s, ys ; x the triangle inequality ﬁnishes the proof. Equipped with this result and the Lipschitzness of the ﬂow for ordinary diﬀerential equations (Lemma 12.5), we are ready to obtain a version of Lemma 10.7 for linear vector ﬁelds. Lemma 10.52 (Davielinear ) Let p ≥ 1. Assume that (i) V = (Vi )1≤i≤d is a collection of linear vector ﬁelds deﬁned by Vi (z) = Ai and elements of Re bi ; Ai z + bi , for some e × e matrices 1-var [0, T ] , Rd , and x := S[p] (x) its canonical lift to (ii) x is a path in C a G[p] Rd -valued path; (iii) y0 ∈ Re is an initial condition; (iv) υ is a bound on maxi (|Ai | + |bi |). Then there exists a constant C depending on p (but not on the 1-variation norm of x) such that for all s < t in [0, T ] , p π (V ) (0, y0 ; x)s,t ≤ C (1 + |y0 |) υ xp-var;[s,t] exp Cυ p xp-var;[0,T ] .

Proof. To simplify the notation we set y = π (V ) (0, y0 ; x) and N := [p], a control function on [0, T ] is deﬁned by p

ω (s, t) := υ p xp-var;[s,t] . For every s < t in [0, T ] we deﬁne xs,t as a geodesic path associated with xs,t = SN (x)s,t , i.e.

s,t

SN x

s,t

= SN (x)s,t

and s

t

s,t dx = x

p-var;[s,t]

Rough diﬀerential equations

264

and we note that, as in (10.7), we have t t s,t dx ≤ |dx| . s

(10.45)

s

Let us now ﬁx s < t < u in [0, T ] and deﬁne xs,t,u := xs,t xt,u , the concatenation of xi,s,t and xi,t,u . Following the by now classical pattern of proof, ﬁrst seen in (Davie’s) Lemma 10.7, we set Γs,t = π (V ) (s, ys ; x)s,t − π (V ) s, ys ; xs,t s,t and observe Γs,u − Γs,t − Γt,u = A + B where A = π s, ys ; xs,t,u s,u − π (s, ys ; xs,u )s,u B = π t, yt ; xt,u t,u − π t, yt + Γs,t ; xt,u t,u . Lemma 10.51 (Lemma Alinear ) was tailor-made to estimate A and gives (N +1)/p 1/p |A| ≤ c1 (1 + |ys |) ω (s, u) . exp c1 ω (s, u) On the other hand, from Theorem 3.8, we have 1/p 1/p |B| ≤ c2 |Γs,t | ω (t, u) exp c2 ω (t, u) and so |Γs,u | ≤ |Γs,t | e2c 2 ω (s,u )

1/p

(N +1)/p

+ |Γt,u | + c1 (1 + |ys |) ω (s, u)

Another application of Lemma Alinear with ω ˜ (s, t) := |x|1-var;[s,t] we have lim

sup

1/p

ec 1 ω (s,u ) . (10.46) combined with (10.45) shows that

r →0 s,t s.t ω ˜ (s,t)≤r

|Γs,t | = 0. r

(10.47)

Also, ODE estimates give that for all s, t ∈ [0, T ] , we have 1/p 1/p π (V ) s, ys ; xs,t s,t ≤ c3 (1 + |ys |) ω (s, t) ec 2 ω (s,t) .

(10.48)

Inequalities (10.46), (10.47) and (10.48) allow us to use (the analysis) Lemma 10.63 (from the appendix to this chapter) to see that |y|∞;[0,T ] ≤ c4 (1 + |y0 |) exp (c4 ω (0, T )) , and that for all s, t ∈ [0, T ], θ

|Γs,t | ≤ c4 (1 + |y0 |) ω (s, t) exp (c4 ω (0, T )) .

(10.49)

10.7 RDEs driven along linear vector ﬁelds

265

The triangle inequality then provides that for all s, t ∈ [0, T ], 1/p

|ys,t | ≤ c5 (1 + |y0 |) ω (s, t)

exp (c5 ω (0, T )) .

We now give the appropriate extension to (full) RDEs. Theorem 10.53 Assume that (i) V = (Vi )1≤i≤d is a collection of linear vector ﬁelds deﬁned by Vi (z) = e Ai z + bi , for some e × e matrices Ai and elements of R [p]bi; d p-var [0, T ] , G R ; (ii) x is a weak geometric rough path in C [p] e (iii) y0 ∈ G (R ) is an initial condition; (iv) υ is a bound on maxi (|Ai | + |bi |). Then there exists a unique full RDE solution π (V ) (0, y0 ; x) on [0, T ]. Moreover, there exists a constant C depending on p such that for all s < t in [0, T ] , we have p π (V ) (0, y0 ; x)s,t ≤ C (1 + |y0 |) υ xp-var;[s,t] exp Cυ p xp-var;[0,T ] . Proof. We only have to prove the above estimate for a bounded variation path, the ﬁnal result following a now classical limiting procedure. 1 Write y0 = π 1 (y0 ) , and deﬁne y˜ = 1+|y π (V ) (0, y0 ; x). Observe that 0| using the linearity of the vector ﬁelds, we have y0 y˜ = π (V ) 0, ;x 1 + |y0 | = π (V ) (0, y˜0 ; x) . From (the proof of) Lemma 10.52 it follows that p y0 |) exp c1 υ p xp-var;[0,T ] |˜ y |∞;[0,T ] ≤ c1 (1 + |˜ p ≤ 2c1 exp c1 υ p xp-var;[0,T ] ≡ R. For any vector ﬁeld V˜ such that V ≡ V˜ on Ω = {y ∈ Re : |y| < R + 1} we have π (V ) (0, y˜0 ; x) ≡ π (V˜ ) (0, y˜0 ; x) . Moreover, we can (and will) take V˜ such that ˜ V γ ≤ c1 |V |Lip γ (Ω) ≤ |V |∞;Ω + |V |∞;Ω ≤ υ (R + 2) . Lip

It then suﬃces to use the estimate of Theorem 10.36, applied to the full RDE with vector ﬁelds V˜ driven by S[p] (x), to see that p p S[p] π (V ) (0, y˜0 ; x) s,t ≤ c2 V˜ γ −1 xp-var;[s,t] ∨ V˜ γ −1xp-var;[s,t] Lip Lip p p ≤ c3 υ xp-var;[s,t] exp c3 υ xp-var;[0,T ] .

266

Rough diﬀerential equations

This implies that S[p] π (V ) (0, y0 ; x) s,t = (1 + |y0 |) S[p] π (V ) (0, y˜0 ; x) s,t p ≤ C(1+ |y0 |) υxp-var;[s,t] exp Cυ p xp-var;[0,T ] .

Remark 10.54 This estimate shows in particular that (full) solutions to linear RDEs have growth controlled by p exp (const) × xp-var;[0,T ] which has implications on the integrability of such a solution when the driving signal x is random. It is therefore interesting to know that this estimate cannot be improved and the reader can ﬁnd a construction of the relevant examples in [58]. Exercise 10.55 Assume x ∈ C p-var [0, T ] , G[p] Rd drives linear vector ﬁelds as in Theorem 10.53 above. If x is controlled by a ﬁxed control ω in the sense that ∀0 ≤ s < t ≤ T : υ xp-var;[s,t] ≤ ω (s, t)

1/p

,

the conclusion of Theorem 10.53 can be written as 1/p π (V ) (0, y0 ; x)s,t ≤ C (1 + |y0 |) ω (s, t) exp (Cω (0, T )) , valid for all s < t in [0, T ], where any dependence on p, υ has been included in the constant C. Show that an estimate of this exact form remains valid, if the assumption on x is relaxed to 1/p

∀0 ≤ s < t ≤ T : υ xp-var;[s,t] ≤ ω (s, t)

∨ ω (s, t) .

(10.50)

The importance of this exercise comes from the fact (cf. Theorem 10.36) that (10.50) is a typical estimate for solutions of full RDEs, i.e. when x itself arises as the solution to a full RDE along Lipγ -vector ﬁelds, γ > p. 1/p

1/p

= ω (s, t) ∨ ω (s, t). For Solution. Assume (10.50) and write ω ˜ (s, t) s, t : ω (s, t) ≤ 1, Theorem 10.53 then gives 1/p π (V ) (0, y0 ; x)s,t ≤ C (1 + |ys |) ω (s, t) exp (Cω (s, t)) 1/p ≤ C 1 + |y|∞;[0,T ] ω (s, t) exp (Cω (0, T )) and we are done if we can show that |y|∞;[0,T ] ≤ c (1 + |y0 |) exp (cω (0, T )) .

10.7 RDEs driven along linear vector ﬁelds

267

From equation (10.49) (now applied with ω ˜ !) this estimate follows from (the p analysis) Lemma 10.63 but since ω ˜ (0, T ) = ω (0, T ) for large ω (0, T ), this is not good enough. However, from Remark 10.64 we can do a little better and get   |y|∞;[0,T ] ≤ c (|y0 | + ε) exp c

 sup D =(t i )⊂[0,T ] such that ω (t i ,t i + 1 )≤1 for all i

 ω ˜ (ti , ti+1 ) .

i

But since ω ˜ ≡ ω, when ω ≤ 1 we can replace ω ˜ by ω, and by superadditivity, |y|∞;[0,T ] ≤ c (|y0 | + ε) exp (cω (0, T ))

as required.

Exercise 10.56 (non-explosion) Consider V = (Vi )1≤i≤d , a collection of locally Lipγ −1 -vector ﬁelds on Re for γ ∈ (p, [p] + 1), such that (i) Vi are Lipschitz continuous; older (ii) the vector ﬁelds V [p] = Vi 1 . . . Vi [ p ] i ,...,i ∈{1,...,d} are (γ − [p])-H¨ 1 [p ] continuous. Show that if x is a geometric p-rough path, and if y0 , then π (V ) (0, y0 ; x) does not explode. Provide a quantitative bound. Solution. The argument is the same as for linear RDEs. We only need to extend Lemma Alinear , i.e. we need to prove the following: if (a) s, t are some elements of [0, T] ; x)s,t ; (b) x, x ˜ are some paths in C 1-var [s, t] , Rd such that SN (x)s,t = SN (˜ [p] 1/[p]

t

t (c) is a bound on s |dxu | and s |d˜ xu | and υ is a bound on V (γ −[p])-H¨o l and supy ,z |V (y) − V (z)| / |y − z| . Then, γ π (V ) (s, ys ; x)s,t − π (V ) (s, ys ; x ˜)s,t ≤ C [υ] exp (Cυ) . To do so, deﬁne for all r ∈ [s, t] , ys,r = π (V ) (s, ys ; x)s,r ; as the vector ﬁelds Vi are Lipschitz continuous, Theorem 3.7 gives |ys,r | ≤ c1 (1 + |ys |) υ exp (cυ) .

Rough diﬀerential equations

268

From Proposition 10.3, π (V ) (s, ys ; x)s,t − E(V ) ys , SN (x)s,t t i1 i[ p ] Vi 1 . . . Vi [ p ] I (yr ) − Vi 1 . . . Vi [ p ] I (ys ) dxr . . . dxr ≤ i 1 ,...,i N ∈{1,...,d}

≤

s

t

γ −[p]

|ys,r |

υ [p]

i i dxr1 . . . dxrN

s

i 1 ,...,i N ∈{1,...,d}

γ −[p]

γ

≤ c2 (1 + |ys |) [υ] exp (cυ) γ ≤ c2 (1 + |ys |) [υ] exp (cυ) . ˜)s,t and π (V ) (s, ys ; x)s,t share the same Euler approximation, As π (V ) (s, ys ; x the triangle inequality ﬁnishes the proof. Equipped with this result, we then prove the exercise by going through the proof of Theorem 10.53.

10.8 Appendix: p-variation estimates via approximations Our discussion of the Young–L´ oeve inequality was based on some elementary analysis considerations; Lemmas 6.1 and 6.2. We now give the appropriate extensions, still elementary, upon which we base our discussion of rough diﬀerential equations. Lemma 10.57 Let θ > 1, K, ξ ≥ 0, α > 0. Assume : [0, R] → R+ satisﬁes (i) (r) = 0; lim r →0 r (ii) for all r ∈ [0, R] , r + ξrθ exp (Krα ) . (r) ≤ 2 2 Then, for all r ∈ [0, R], ξrθ exp (r) ≤ 1 − 21−θ

2K rα 1 − 2−α

.

10.8 Appendix: p-variation estimates via approximations

269

Proof. Note that it is enough to prove the ﬁnal estimate for r = R; indeed, given any other r ∈ [0, R], it suﬃces to replace the interval [0, R] by [0, r]. Assumption (ii) implies that for all r ∈ [0, R], r exp (Krα ) + ˆξrθ (r) ≤ 2 2 with ˆξ = ξ exp (KRα ). By induction, we obtain that (r) ≤2n

r 2n

exp Krα

n −1

2−k α

+ ˆξrθ

k =0

n −1

 2k (1−θ ) expKrα

k −1

 2−j α .

j =0

k =0

n −1 We bound exp Krα k =0 2−k α ≤ exp (Krα / (1 − 2−α )) and so obtain (r) exp

−Krα 1 − 2−α

n −1 r θ ˆ ≤ 2 n + ξr 2k (1−θ ) . 2 n

k =0

By assumption (i), sending n to ∞ yields ˆξrθ Krα . exp (r) ≤ 1 − 21−θ 1 − 2−α As ˆξ = ξ exp (KRα ) ≤ ξ exp (KRα / (1 − 2−α )), we then obtain ξRθ 2KRα (r) ≤ . exp 1 − 21−θ 1 − 2−α As a variation on the theme, let us give Lemma 10.58 Let θ > 1, K, ξ ≥ 0, α > 0 and β ∈ [0, 1). Assume : [0, R] → R+ satisﬁes (i) (r) = 0; lim r →0 r (ii) for all r ∈ [0, R] , r (r) ≤ 2 + ξrθ ∧ εrβ exp (Krα ) . 2 Then, for all r ∈ [0, R], for some constant C depending on θ and β, we have 1 −β θ −1 2K α . r (r) ≤ Crε θ −β ξ θ −β exp 1 − 2−α

Rough diﬀerential equations

270

Proof. Just as in the previous proof, we only prove the estimate for r = R. Deﬁning ˆξ = ξ exp (KRα ) and ˆε = ε exp (KRα ), we have for all r ∈ [0, R], r exp (Krα ) + ˆξrθ ∧ ˆεrβ . (r) ≤ 2 2 By induction, we obtain that n −1 r (r) ≤ 2n n exp Krα 2−k α 2 k =0   n −1 k −1 ˆξrθ 2k (1−θ ) ∧ ˆεrβ 2k (1−β ) exp Krα 2−j α  . + j =0

k =0

n −1 We bound exp Krα k =0 2−k α ≤ exp (Krα / (1 − 2−α )) and then let n tend to ∞ to obtain ∞ −Krα ˆξrθ 2k (1−θ ) ∧ ˆεrβ 2k (1−β ) (r) exp ≤ −α 1−2 k =0 2k (1−β ) + ˆξrθ ≤ ˆεrβ 1 0≤k ≤ θ −β ln 2

ˆ ξ ˆε

1 k > θ −β ln 2

ˆ ξ ˆε

2k (1−θ ) +ln 2 r

1 β ≤ c1 ˆεr 2 θ −β

+ ˆξrθ 2 ≤

θ −1

1 θ −β

+ln 2 r

ln 2

ln 2

ˆ ξ ˆε

ˆ ξ ˆε

+ln 2 r (1−β )

+ln 2 r (1−θ )

1 −β

c2 rˆε θ −β ˆξ θ −β ,

where c1 , c2 are constants which depend on θ and β. This estimate ﬁnishes the proof. An important consequence of Lemma 10.57 is the following estimate. Typically (e.g. in the proof of Davie’s estimate, Lemma 10.7), Γ is the difference between a path y (for which we are trying to bound its p-variation) and a “local” approximation of y which is easier to control. Lemma 10.59 Let ξ > 0, θ > 1, K ≥ 0, α > 0 and Γ : {0 ≤ s < t ≤ T } ≡ ∆T → Re be such that: (i) for some control ω ˆ, lim

sup

r →0 (s,t)∈∆ : ω ˆ (s,t)≤r T

|Γs,t | = 0; r

(10.51)

10.8 Appendix: p-variation estimates via approximations

271

(ii) for some control ω we have that, for all s < t < u in [0, T ], θ α |Γs,u | ≤ |Γs,t | + |Γt,u | + ξω (s, u) exp (Kω (s, u) ) .

(10.52)

Then, for all s < t in [0, T ] , θ

ξω (s, t) exp |Γs,t | ≤ 1 − 21−θ

2K α ω (s, t) 1 − 2−α

.

Remark 10.60 It is important to notice that the control ω ˆ