Ergodicity for Infinite Dimensional Systems (London Mathematical Society Lecture Note Series)

0, is said to be irreducible at time to > 0 if, for arbitrary non empty open set r and all x E F,

Pt.(x,F) = PtoXr(x) > 0. Again by the semigroup property if a Markovian semigroup Pt, t > 0, is a strongly Feller (resp. irreducible) semigroup at time to > 0 then it is a strongly Feller (resp. irreducible) semigroup at arbitrary time

t > to. The following result is due to R.Z. Khas'minskii. Proposition 4.1.1 If a Markovian semigroup Pt, t > 0, is a strongly Feller semigroup at time to > 0 and irreducible at time so > 0, then it is regular at time to + so.

Proof - Assume that for some xo E E and F E E, Pto+so (xo, F) > 0. Since Pto+so (xo, r) =

Pso (xo, dy)Pto (y, F),

JE for some yo E Ewe have Pto (yo, r) > 0. Since Pto (y, r) = Pto (Xr)(yo) and Pto(Xr) E Cb(E), because the semigroup Pt, t _> 0 is a strongly

Feller semigroup at time to > 0, there exists ro > 0 such that

Pto(y,F)>0forallyEB(yo,ro) Consequently, for arbitrary x E E,

Pto+so(x,F) = JE0(0) >

J (yo,ro) PO (x, dy)Pto (y, F) > 0.

We used here the fact that Pso(x, B(yo, ro)) > 0 and Pto(y, F) > 0 for all y E B(yo, ro). Thus if Pt0, +,0(xo, F) > 0 for some xo E E then Pto+so (x, r) > 0 for all x E E and the (to + so)-regularity follows.

Regular Markovian systems

4.2

43

Doob's theorem

The main result concerned with regular semigroups is due to J. L. Doob [56] and can be formulated as follows.

Theorem 4.2.1 Let Pt, t > 0, be a stochastically continuous Markovian semigroup and µ an invariant measure with respect to Pt, t > 0. If Pt, t > 0, is to-regular for some to > 0, then (i) p is strongly mixing and for arbitrary x E E and F E E

lim Pt(x, F) = µ(r).

t-++o3

(ii) p is the unique invariant probability measure for the semigroup Pt, t > 0. (iii) p is equivalent to all measures Pt(x, ), for all x E E and all t > to.

Proof Step 1 - µ is ergodic Let r E 9 be an invariant set PtXr = Xr, for all t > 0,

(4.2.1)

such that p(F) > 0; we have to show that µ(F) = 1. From (4.2.1) Pt0, (x, F) = 1, x E F, p-a.s., and for µ(F) > 0 there exists xo E I' such that Pt0, (xo, F) = 1. Since all probabilities Ptp(x, ) are equivalent, we have Pt0, (x, F) = 1, d x, E E. It follows that

p(F) = JEo() = 1. Step 2- p is the unique invariant measure If v is another invariant probability measure for Pt, t > 0, then v is ergodic by Step 1. Since the two measures µ and v are equivalent with respect to any measure P(to, x, ), x E E, they are equivalent. By Proposition 3.2.5 we have p = v.

Step 3- Pt, t > 0 has no angle variable

Chapter 4

44

Assume that for some A E R, A # 0, we have

Pt f_eaatf, t > 0.

(4.2.2)

We will show first that If is constant. Since µ is ergodic, it is enough to prove that

PtIfI = If I, µ-a.s.

(4.2.3)

By (4.2.2) it follows that

If(x)I = IPtf(x)I <- Ptlfl(x) Set now

K = {x E E : If(x)I < Ptlfl(x)}, and assume in contradiction that µ(K) > 0. Then, taking into account that µ is invariant,

4 If(x)Ii(dx) = 1K If(x)Ip(dx)

+f

C

If(x)Ii(dx)

< fh PtIfl(x)µ(dx) + I Ptifl(x)µ(dx) =

f EPtl fl (x)jt(dx) = f EI f(x)I u(dx),

a contradiction. So I f Now we show that

f(X(t)) =

is constant.

f (X(0)), t > 0, Pl''-a.s.,

(4.2.4)

where X (t), t E R, is the canonical process. We have in fact, using (4.2.3),

E'`[I f(X(t)) - e%at

f(X(0))11] = E [I f(X(t))I2] + Eµ[I f(X(O))I2]

-e-'AtE, µ[f (X (t)) f (X (0))]

- ezatEµ[f(X (t)) f (X (0))]. (4.2.5)

Since the process X (t), t E R, is stationary, we have

=1µ[If(X(0))I2],

Regular Markovian systems

45

and since it is a Markov process

E"If(X(t)) f(X(O))l = i" {E"[f(X(t)) f(X(o))I.Fol}

= i" {f(X(o)) E"[f(X (t))I .Fo]} = EE" {f(X(o)) Ptf(X (o))} = e:'tE" {If(X(o))12} . Now the conclusion (4.2.4) follows on substituting this in (4.2.5). Let r be a Borel set on the circumference {A : JAI = 11. We show now that if ]P"(f (X (t)) E r) > 0, then we have

Pt(X(0), f-1(r)) = 1, P"-a.s. In fact

P"(f(X(t)) E r) = P"(X(t) E f-1(r))

= )E" [Pt(X(0),f-1(r))] and so, by (4.2.4), P"(f(X (t)) E F) = 1P"(eiat f(X (o)) E r)

= ]"(f(X (t)) E r, e'at f(X (0)) E r)

J e'atf(X(0))EF} Pt(X(O), f-1(r))dµ. So if ]P"(f(X(t)) E r) > 0 we have Pt(X(0), f-1(r)) = 1, P"-a.s., as required.

To complete the proof of Step 3 set v = C(f(X(t))). Then v is concentrated on the circle {A E C : IAI = 1}. Let ei° belong to the support of v. Then for all n E N P

(f(x(t)) E S(c- n,c+

n> 0,

where

= {AEC: c-1 < argA
Chapter 4

46

By (4.2.3) we have

Pt (X(0), f-1 1 S l c - n,c +

P"-a.s..

Hill

Consequently we have Pt

(xr'

(S \c-

n'

n')))

1, P"-a.s.

,

for one (and then for all) x E E, by the hypothesis of equivalence of all transition probabilities. It follows that

Pt(x, f-1(c)) = 1, t > 0, x E H, and so P(t, x, f-1(c))µ(dx) = 1 p(f-1(c)) = JE and f is a.s. constant. Step 4 - For arbitrary x E E and r E E, we have

t

iim Pt(x, r) = µ(r) 00

(4.2.6)

From the absolute continuity of P(x, dy), u > to, x E E, with respect to y there exists a density p(u, x, z) such that

PP (x, I') = fp(n,x,z)(dz), V F E , u > to. On the other hand, for t > u > to and r E E Pt(x, r)

JE Pu(x, dy)Pt-u(y, r)

f EPt-u(y, r)p(u, x, y)µ(dy) ]Eµ

(Pt-u(X (u), r)p(u, x, x (u))) = 1E (/3 Uta), (4.2.7)

where a = Xr (X (0)), 0 = p(u, x, X (U)).

Regular Markovian systems

47

We have in fact

]EI(QUt(a)) = E"( Q Xr(X (t))) = ]Ea [,QIEa (Xr(X (t))ITu)] = ]Eµ [0 Pt-ux (X (u))] = El [3 Pt-u(X (u), r)] .

By Proposition 3.2.1 and Step 3, the canonical dynamical system is weakly mixing. Thus there exists a set I C [0, +oo[ of relative measure 1 such that

lim Pt(x, r) = ,t(r),

ItI- +00 tEI

for arbitrary x E E, r E 8(E). We show finally that the limit in (4.2.6) does exist without any restriction on t. Let t - +oo, then there exists s = s(t) such that 3 < s(t) < 3 and s(t) E I, t - s(t) E I. This is true for sufficiently large t. Since Pt (x, F)

=

JE P5(t)(x, dy)Pt-s(t)(y, A),

the result follows from the previous considerations and by Lemma 4.2.2 below.

Lemma 4.2.2 Let r be a Bored set in [0, 1] with A(r) > a where A is the Lebesgue measure. Then there exists s E IF such that 1 - s E F.

Proof - Let

r={1-s:sEr}.

Since A(r) = A(r) > 2 we have A (r n r) >0. Remark 4.2.3 Using different methods J. Seidler [136] and L. Stettner [147] were able to show that, under the conditions of Theorem 4.2.1, the convergence in (i) can be replaced by the convergence in the variation norm.

Part II

Invariant measures for stochastic evolution equations

49

Chapter 5

Stochastic Differential Equations Part II of these lecture notes deals with general methods of studying invariant measures with respect to Markovian semigroups Pt, t >_ 0, determined by solutions of stochastic evolution equations on a Hilbert or a Banach space. In this chapter we summarize basic results on stochastic integration and on stochastic evolution equations on infinite dimensional spaces. We recall important inequalities for stochastic integrals and stochastic convolutions and prove the existence of a regular solution to a class of evolution equations with Lipschitz or locally Lipschitz drift and diffusion coefficients. Then dissipative systems are discussed in some detail. Regular dependence of solutions on initial data are established as well.

5.1

Introduction

We deal here with equations of the form:

dX = (AX + F(X))dt + B(X)dW(t), (5.1.1)

X(0)=

,

51

Chapter 5

52

As a rule the process W(t), t > 0, will be a cylindrical Wiener process on a Hilbert space U. Moreover F and B are nonlinear transformations and A the infinitesimal generator of a strongly continuous

semigroup S(t), t > 0. Precise definitions of all the data related to problem (5.1.1), and to the solution X to (5.1.1), will be given in §5.2 and in §5.3.

If X(t,x), t >_ 0, x E E, denotes the unique solution to (5.1.1) with = x constant, then the transition semigroup Pt, given by

Ptw(x) = IE[cp(X (t, x))], t > 0, x E E, cp E Bb(E).

t > 0, is (5.1.2)

If there exists an invariant measure it for Pt, t > 0, given by (5.1.2) then the solution to (5.1.1) corresponding to a random variable such that C(e) = µ is a stationary solution to (5.1.1), and this solution has important limit properties, see §3.5. In the following chapters we shall concentrate on basic principles leading to the existence of invariant measures for Pt, t > 0, given by (5.1.2) and to other related properties. These include uniqueness, ergodicity, mixing and regularity of the density of the invariant measure with respect to a natural reference measure. The present chapter, however, contains a selection of basic results on stochastic evolution equations to which we will often refer.

5.2

Wiener and Ornstein-Uhlenbeck processes

Let U be a separable Hilbert space and it a Gaussian measure on U. The measure y is completely determined by its mean value m E U and its covariance operator Q which is a trace-class nonnegative linear operator. Such a measure will be denoted by N(m,Q). A U-valued stochastic process W(t), t > 0, defined on a probability space (St, 1', P) is called a Q-Wiener process if it has the following properties.

(i) W(.) has continuous paths and W(0) = 0. (ii) W() has independent increments.

Stochastic Differential Equations

53

(iii) For arbitrary t > 0 and h > 0

G(h-112(W(t + h) - W(t))) = N(m, Q). We will always assume that the probability space is equipped with

a right-continuous filtration {.Ft}t>o such that FO contains all sets of IP-measure zero. The Wiener process is assumed to be adapted

to {Ft}t>o and for every t > s the increments W(t) - W(s) are independent of Ft. Let Uo = Image Q1/2 be a Hilbert space equipped with the scalar product (u,V)o =

(Q-1/2u,Q-ll2v),

u,v E Uo,

where Q-1/2 is the pseudo-inverse of Q1/2, and let {ek} be an arbitrary, complete, orthonormal basis in U. Then the formula W(t) = E00 k(t)ek, k=1

t > 0,

(5.2.1)

where /3k, k = 1, ..., are independent real processes defines a QWiener process on U. The series (5.2.1) converges in U, IF-a.s. uniformly on an arbitrary finite time interval [0, T]. Note that for arbitrary g, h E Uo and t, s > 0 E(g, W(t))o(h, W(s))o = (h,g)o t A S.

This is why the process W is also called a cylindrical Wiener process

on Uo. Its covariance operator with respect to Uo is the identity operator. If W is a Wiener process on U with the covariance operator different from the identity then it is called a coloured Wiener process.

Example 5.2.1 An important example of a coloured Wiener process is a process on U = L2(Rd) used to describe "Random environment", see D. A. Dawson and H. Salehi [53]. The covariance operator is defined as the convolution operator

Qu(a) = fd q( - 77)u(rl)drl, u E U,

(5.2.2)

Chapter 5

54

where q is a continuous, positive definite function: N

_ E Rd, a1i...,AN E C,N E N.

0, 1,..., Moreover by Bochner's theorem

q(\) = fd e-(A,')ir(dx), A

E ][Yd

where a is a nonnegative, finite measure, the so-called spectral measure. The Wiener process W(t, e), t _> 0, E Rd, corresponding to

Q has the property that for fixed t, W(t, ) is a stationary Gaussian field with the correlation function EW(t, e)W (t,,1) =

Vtiq( - 1)), e, rl E Rd.

Typical examples: q(e) = e-1E1°, E Rd, for arbitrary a E ]0, 2]. The corresponding spectral measures are then isotropic a-stable distribu-

tions, and in particular for a = 2 the spectral measure is Gaussian. Note that the operator Q is never compact on U and thus never of trace class. A natural generalization of this example consists in replacing Rd by a group G with a translation invariant measure v and in considering covariance operators Q on U = L2(G, v) as generalized convolutions

Qu(e) = J q( - r))u(r))v(di),

E G, U E L2(G, v).

It is sometimes convenient to consider Wiener processes represented in a form more general than (5.2.1):

W(t) = E00akek0k(t), t > 0,

(5.2.3)

k=1

where {ak} C R and the basis {ek} consists of eigenvectors of a self-adjoint operator A.

Example 5.2.2 The sequence cos ne,

sin n

n E N,

E [0, 2r],

Stochastic Differential Equations

55

forms a complete orthonormal basis in U = L2(0, 21r). Its elements are eigenfunctions of the operator A=

,

D(A) _ {x E H2(0, 2ir) : x(O) = x(2ir), x'(0) = x'(2zr)}.

2

In this case the series (5.2.3) can be equivalently written as follows: +00

W(t,e) _

(a.,3.1 (t) cos ne + bn,Q,n(t) sin ne), t > 0, E [0, 1],

n.o (5.2.4) where {an} and {bn} are sequences of real numbers and {,@j}, {,32} are mutually independent Wiener processes. Denote by G the interval [0, 27r] treated as a group with addition calculated mod 21r. Covariance operators Q on L2(G) are of convolution type with a kernel q if and only if, by Bochner's Theorem, the positive definite function q is of the form 00

q(C) = yo +

with 7n

yn, cos of , t: E [0, 21r],

(5.2.5)

n-1

0, n E N and F_°_1 -fn < +oo.

It is easy to see that if an = bn = y;., n E N, then the covariance operator Q of (5.2.4) is of convolution type with the kernel q of the form (5.2.5). Thus, in some cases, Wiener processes defined by (5.2.3) can have a physical meaning. Similar considerations can be made if G is a d dimensional torus and A is either a Laplace-Beltrami operator on G or a Stokes operator.

Assume that U and H are separable Hilbert spaces, A is the infinitesimal generator of a Co-semigroup S(t), t > 0, on H and B is a bounded linear operator from U into H. Let W(t),t > 0, be a cylindrical Wiener process on U and an H-valued To-measurable random variable.

Chapter 5

56

An H-valued Ft-adapted stochastic process Z(t), t > 0, is said to be a weak solution to the equation

dZ = AZdt + BdW, Z(0) _ e,

(5.2.6)

if for arbitrary h E D(A*) and all t > 0, P-a.s.

(h, Z(t)) _ (h, ) + j(A*h, Z(s))ds + (B*h, W(t)).

(5.2.7)

Note that all terms in (5.2.7) have well defined meanings. One can show that there exists a solution to (5.2.6) if and only if the operators t

Qt = f S(r)BB*S*(r)dr, t > 0,

(5.2.8)

are of trace class. In this case the solution is given by the formula t

Z(t) = S(t)d + f S(t - s)BdW(s), t > 0.

(5.2.9)

For the definition of stochastic integral see next section. The process Z is called an Ornstein-Uhlenbeck process and the process WA(t) =

f

t

S(t - s)BdW(s), t > 0,

-

o

(5.2.10)

a stochastic convolution. The process Z is Gaussian and Markovian with transition semigroup Rt, t > 0, of the form

RtV(x) = f(y)H(S(t)x,Qt)(dy), V E Bb(H).

(5.2.11)

The operator BB* will often be denoted by Q:

BB* = Q.

5.2.1

Stochastic integrals and convolutions

We assume, see earlier, that W(t), t > 0, is a cylindrical Wiener process on a separable Hilbert space U, given by a formal expansion

W(t) _

/3,,,(t)e,,, n=1

t > 0,

Stochastic Differential Equations

57

where en,, n E N, is an orthonormal basis on U. Very often the space U will be identical with H. The random variable is assumed to be Fo-measurable.

By IIRIIHs or IIRII2 we denote the Hilbert-Schmidt norm of the operator R E L(U, H). The space of all Hilbert-Schmidt operators

from U into H (endowed with the Hilbert-Schmidt norm) will be denoted by L2(U, H). This is again a separable Hilbert space. The stochastic integral

I (s)dW(s), t > 0, 0t

0

is well defined, see G. Da Prato and J. Zabczyk [44], for any L2(U, H)valued adapted process such that II,t(S)I12

Clot

ds < +oo, t > 0) = 1.

Moreover

ift'(s)dW(S) ]E

2 1

< ]E Jot I I , ( s ) I I Hs ds, t > 0,

0

(5.2.12)

with the equality applying in (5.2.12) if the right hand side is finite. The following generalization of (5.2.12) will be used, see G. Da Prato and J. Zabczyk [44, Lemma 7.7, page 194]: P

sup lE IJa (a)dW(o,) < cp SE[O,t]

(jui)P/2

0

(5.2.13)

where t E [0, T] and cp = (p(p - 1)/2)p/2, valid for arbitrary p > 2, t > 0 and an arbitrary predictable process s E [0,T], with values in L2(U, H). We will also need the so-called maximal inequalities formulated in the following two theorems.

Theorem 5.2.3 (Doob's inequalities) Assume that

El' IIp(S)IIHSds < +oo.

Chapter 5

58

(i) For arbitrary p > 1 and A > 0, P

sup 1JO § (s)dW(s)A t
]P

aP

fT 4(s)dW(s)

1E

0

(ii) For arbitrary p > 1, lE

I

sup t
P) <

t o 110

p

p-1

E

f T'(s)dW(s)

P

0

Theorem 5.2.4 (Burkholder-Davis-Gundy) For arbitrary p > 0 there exists a constant cP > 0 such that fT

t lE

(sup

t
4 (s)dW (s)

P

/J

< CPIE

p/2

II (S)II Hsds

Let A be e the infinitesimal generator of a Co-semigroup S(t), t >

0, on H. If fi(t), t E [0, T], is an L2(U, H)-valued adapted process such that the stochastic integrals

f S(t t

WA (t), t E [0,T],

0

are well defined, then the process WA is called a stochastic convolution. If fi(t) = B, t E [0, T], we write for short WA instead WA . Let a E ]0, 1] and Y,, (t), t E [0, T], be the following stochastic process,

j(t - s)-'S(t - s)(D(s)dW(s), t E [0,T]. t

Ya (t) =

The following result, which will often be referred to as the factorization formula, is an easy corollary of the stochastic Fubini theorem, see e.g. G. Da Prato and J. Zabczyk [Theorem 4.18][441.

Theorem 5.2.5 Assume that for all t E [0,T]

f t(t - sWa-1 {j(s - a)-2'E (II S(t -

0,)4(Q)II2) do] 1/2 ds

< +oo. (5.2.14)

Then

J

t

S(t-s)4(s)dW(s) =

sinrar

f

t(t-s)cr-1S(t-s)Y(s)ds, t E [0,T]. (5.2.15)

Stochastic Differential Equations

59

We note that condition (5.2.14) is only sufficient for the factorization (5.2.15) to hold. Different sufficient conditions can be more useful in specific cases. We will need the following result on stochastic convolutions whose proof is a typical application of the factorization formula.

Theorem 5.2.6 Assume that (i) The operator A generates an analytic semigroup S(t), t > 0, on H such that IIS(t)II < Me-wt, t > 0, with w and M positive numbers (i). (ii) There exists a E ]0, 1/2[ such that for arbitrary T > 0

/T t-2a J0

(5.2.16)

II S(t)BII2 Sdt < +00.

Then for arbitrary y E [0, a[ there exists a version of WA which is Holder continuous with values in D((-A)-') with arbitrary exponent

smaller than a - y. (2) Proof - For any a E ]0, 1[, y E [0, a[ and p > 1, define the integral operator Ra,7w(t) =

f 0t(t - o,)a-1(-A)-YS(t - a)cp(a)di, t E [0,T],

on the space LP(O,T;H). Let us choose p > I.. By Proposition A.1.1 and Theorem 5.2.5 on the factorization, it is enough to show that the process Y.(s) =

J0

s

(s - or)`S(s - a)BdW(v), s E [0,T],

has p-integrable trajectories. By Theorem 5.2.4 ]IYa(s)I" < cp (10s II(s - o)-°S(s - o )BII2do,

p/2

, s E [O, T],

'The assumption w > 0 is not essential in Theorems 5.2.6 and 5.2.7 2(-A)" means the fractional power of -A, see e.g. A. Lunardi [108, §2.2.2]

Chapter 5

60

and therefore E

IT 0

IYa(s)Ipds < c,

T JO

(f

s(o)_2aII

ds < +oo.

The proof is complete.

With exactly the same proof the following result can be established.

Theorem 5.2.7 Assume that (i) The operator A generates an analytic semigroup S(t), t > 0, on H such that IIS(t)II < Me-Wt, t > 0, with w and M positive numbers,

(ii) there exists a E ]0,1/2[ such that for arbitrary T > 0 0T

t-2a II S(t)II Hsdt < +oo,

(5.2.17)

Jo

(iii) there exists a constant C > 0 such that P-a. s. II(b(t)II < C for all t > 0.

Then for arbitrary y E [0, a[ there exists a version of WA which is Holder continuous with values in D((-A)") with arbitrary exponent smaller than a - y.

Example 5.2.8 Let H = L2(0,1) = U, B = I, 2

A=

2,

D(A) = H2(0,1) n Hol (0, 1).

The functions =V2-sin

n E N,

E ]0, 1[,

form an orthonormal sequence of eigenfunctions of the operator A corresponding to the eigenvalues

-7 2n, 2n E Isl.

Stochastic Differential Equations

61

The stochastic convolution Z(t) = WA(t), t > 0, is in this case a weak solution of the equation

dZ(t) = AZ(t)dt + dW(t), Z(O) = 0. Note that e-2a2n2t

II S(t)II HS = n=1

The condition (5.2.16) of Theorem 5.2.6 is

JrT t- 2a

°O

e-2"2n2tdt < +oo. n=1

0

It is fulfilled if and only if a < 9. Consequently the process Z is Holder continuous in H with any exponent smaller than 4.

We will now prove a more general result concerned with the stochastic convolution on L2(O) where 0 is a bounded domain in Rd and the semigroup S(t), t > 0, is given by

S(t)x = E e-a"t (en, x) en, x E H,

(5.2.18)

n=1

and {en} is a complete orthonormal set on L2(0). Moreover we will assume that

W(t) =

Xn/3n(t)en, t > 0,

(5.2.19)

n=1

n E N, are independent Wiener processes. As far as the domain 0 is concerned we will assume that where

k(0) > 0

(5.2.20)

where the constant r,(0) is defined in Appendix B, by (B.1.1). Concerning the functions {en} we shall assume that

ek E Co(), I ek(e)I

C, I Dkek(6)I C

ak,

(5.2.21)

E 0, k E N.

(5.2.22)

Chapter 5

62

In the formulation of the theorem below II IIa stands for the Holder

semi-norm on 0:

WO - xWI

IIXIIa. = sup

f#7iEO

14 - 771°

Theorem 5.2.9 Assume that (5.2.18) and (5.2.19)-(5.2.22) hold and for some y E ]0, 1[ 00

Ak 1--Y < -boo.

(5.2.23)

k=1 ak

Then there exists a continuous version of the process with values in Co(?). aMoreover, if r > 0, y E ]0, 1[, a > 0 are numbers such that

,Q-2d

d

2r > 2r > 0' then the version has the additional property that 00

E

Ak 1--y

*/Z

k=1 ak

Proof - Fix T > 0 and define N

t

ZN(t,e) = E an J S(t 0

n=1

N

t

,\nen(b) f n=1

e-ant-8)d%n(s),

t E [0,T].

0

Set also

ZN,M = ZM - ZN, M, N E N.

It is clear that we can assume that ZN is a continuous function on [0, T] x 0 and that ZN(t, ) = 0 for t E [0, T] and belonging to the boundary 00 of 0. By a straightforward interpolation argument it follows that I ek(e) - ek(ll)I <- Cak/2I -

for ally E [0,1] and ',l E E.

glry, k E N,

Stochastic Differential Equations

63

Let M > N then for all t E [0, T], ]E

77 E 0

I ZNM(t, ) - ZN,M(t, 77)12 M

C` J: Ak ojt <

C

ek(17)I2ds

M

E kakIe-71127 ak

2 k=N+1

and therefore

2k=N+1 Lm

1k-y(C -11127

(5.2.24)

k

Moreover for 0 < s < t < T, 1; E 0 )E I ZN,M(t, ) - ZN,M(t,

M k=N+1

t

\k Js

M

77)I2

e-2(t-°)ak Iek(e)I2do

Js

k=N+1

= Il(t,S,e) + I2(t,S,S) Let c.y > 0 be a number such that for arbitrary u, v > 0, crylu - vl'Y. Then C I,(t,S, S)

C

1 C2 2

C' 21--v

M

E k=N+1 a

-(1-e- 2(t-s)ak/ll

Ak

1-7 k=N+1 ak

It - s I

Ie-u

- e-v I

Chapter 5

64 Moreover

M

I2(t, s, e)

< C2

is

E

e-2(S-a)ak (1

Ak

- e-2(t-S)ak) 2 da

k_N+ 1 M

1A

< 2 C2 k_N+1 ak k C-Y

-y

It -

81-y.

Collecting all the estimates and taking into account that the random

are Gaussian we obtain that for arbitrary r > 0

variables

there exists Cr such that )E I ZN,M(t, e) - ZN,M(t, n)I r/2

M k

E

Cr

r

(I

k =N+1

-qI2+It -sI2)

rry/4

(5.2.25)

ak

From Theorem B.1.1 we have ZM,N(u, I£) - ZM,N(v, C)I r (I2) a-2d

(Iu - vI2 + I' -

IT

C J TJ

- ZM,N(S, )I r (It - sI2 + Ie iiI2) 2 ZM,N(t, I)

J

T

I

d 0

-

or

IT jT IIIZN,MIII2d < C

0

T

ZM,N(s, ,)I r d dqdtds f f ZM,N(t, (It-812+IS-17I2)2

d0

IIIa denotes the a-Holder norm on [0,T] x Taking expectation, using (5.2.24) and assuming that

where III

-

y

d

a-2d >0, andr> 1, r

r > we obtain that there exists a finite constant C such that 2

M EIJIZN,MIII aa_2d < C E

)

Ak 1-

r (k=N+1 ak

ry

r/2

Stochastic Differential Equations

65

r

So the sequence of random variables {ZN} satisfies the Cauchy condition in the space L of random variables Z whose values are -2d Holder continuous functions on [0, T] x vanishing on [0, T] x 00.

Therefore {ZN} has a limit Z on this Banach space which is the desired version. In a similar way one proves the second part of the theorem using (5.2.23) instead of (5.2.24).

Remark 5.2.10 It follows from the proof that if y

d

2

r >

a-2d >0,andr>1, r

then

:5 C

T II r EIIIWA(., -)t

(

M Ak

aa_2d

r

5.3

1-ry

k=N+1 ak

Stochastic evolution equations

In this section we will recall basic existence and uniqueness results on stochastic evolution equations of the form

dX(t) = (AX + F(X))dt + B(X)dW(t) (5.3.1)

on a separable Hilbert space H. We will also study regular dependence of solutions on initial data and Kolmogorov's equation for transition functions.

We will recall now existence results for problem (5.3.1), under suitable Lipschitz continuous conditions on F and B. It is convenient to introduce the following assumptions:

Hypothesis 5.1 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H. (ii) F is a mapping from H into H and there exists a constant co > 0 such that I F(x)I <- co(1 + Ixl), x E H,

IF(x) - F(y)I < coi x - yI, x, Y E H.

Chapter 5

66

(iii) B is a strongly continuous mapping from H into L(U; H) 0)

such that for any t > 0 and x E H, S(t)B(x) belongs to L2(U; H), and there exists a locally square integrable mapping

K : [0,+oo[-* [0,+oo[, t -* K(t), such that 1IS(t)B(x)IlHs < K(t)(1 + IxI), t > 0, x E H,

1IS(t)B(x) - S(t)B(y)IIHS < K(t)lx - yl, t > 0, x, y E H. An 17t-adapted process X(t), t > 0, is said to be a mild solution of (5.3.1) if it satisfies the following integral equation,

X(t) = S(t)e +

t

J

S(t - s)F(X(s))ds (5.3.2)

+

It t

S(t - s)B(X (s))dW (s), t E [0, T].

0

We shall denote by xp,T the Banach space of all (equivalence classes)

of predictable H-valued processes Y(t), t > 0, such that IIYiip,T = sup (]EIY(t)Ip)1/p < +oo. tE[0,T]

We have the following result.

Theorem 5.3.1 Assume Hypothesis 5.1 and let p > 2. Then for an arbitrary Fo-measurable initial condition e such that +oo there exists a unique mild solution X of (5.3.1) in xp,T and there exists a constant CT, independent of , such that sup IEIX(t)Ip < CT(1 +

(5.3.3)

tE[0,T]

Finally, if there exists a E ]0,1/2[ such that 1

10

s-2"K2(s)ds < +oo,

(5.3.4)

where K is the function from Hypothesis 5.1-(iii), then the solution X(.) is continuous P-a.s. 'Means that for any u E U the mapping x continuous.

B(x)u from H into H is

Stochastic Differential Equations We shall denote by

67

the mild solution of (5.3.1).

The proof of Theorem 5.3.1 is similar to that in G. Da Prato and J. Zabczyk [44, Theorem 7.4, page 186]. We will sketch it here for the sake of completeness. A more general result, needed in some applications, see S. Peszat and J. Zabczyk [124], will be given in Part III.

Proof of Theorem 5.3.1 - For arbitrary E LP(ft, H) and X E fp,T define a process Y = K(6, X) by the formula

Y(t) = S(t)e +

jt

S(t - s)F(X(s))ds (5.3.5)

+

jS(t - s)B(X(s))dW(s), t E [0,T].

We will note first that, by inequality (5.2.13), we have K(e,X) E 7{p,T for arbitrary X E 1p,T. Moreover, setting MT = suptE[o,T] IIS(t)II, we have

EIY(t)IP <

l

]E

[(J t

S(t - s)F(X(s))Ids)p] I

+ E [Ift S(t - s)B(X(s))dW(s) 1P

JJJ

t

< 3P-1 Jo

EIF'(X(s))IPds p/2

+ cp I JOt

(EIIS(t - s)B(X

(s))IIHS)2/p ds]

Moreover t

EIF(X(s))IPds < 2P-'cps sup] (1 +EIX(s)IP)t, 10

O,t

Chapter 5

68 and

p/2

U'

(EIIS(t - s)B(X(s))I)

Hs)2/p ds

t

< 2p-1 (10 K2 (t - s)(1 + ]EIX(s)Ip)2/pds

/

p/2

2

_< 2p-1 (10 t K2(t - s)ds

/

p

sup (1 + ]EIX(s)Ip). sE[0,t]

It is now clear that, for some constants c1, c2, c3, sup ]EIY(t)Ip <_ c1 + c2]EISIp + c3 sup ]EIX(t)Ip. tE[0,T]

(5.3.6)

tE[0,T]

Thus Y E lp,T. In exactly the same way, if X1, X2 E ?1 p,T and Y1 = Y2 = X2), then

X1),

sup )EIY1(t) - Y2(t)I p _< C3 sup EIX1(t) - X2(t)I.

tE[0,T]

tE[0,T]

It is easy to see that if T is small enough then c3 < 1 and consequently, by the contraction principle, the equation (5.3.1) has a unique solution in fp,T. The case of general T > 0 can be treated by considering the equation in intervals [0, T], [T, 2T],... with T such that c3(T) < 1. Moreover with such a T we get from (5.3.6) for the solution of (5.3.1) that sup ]EIX(t)Ip < tE[0,T]

1

1 - C3(T)

[c1 +c2EISIp]

which is inequality (5.3.3). The case of general T > 0 can be easily obtained by iteration as well. Finally, the continuity of the solution can be proved by using factorization, see Theorem 5.2.5, in a. similar way to Theorem 5.2.6 and Theorem 5.2.7.

Remark 5.3.2 If the operator A generates an analytic semigroup, then trajectories of solutions are even Holder continuous with values in D((-A)a). Compare the proofs of Theorems 5.2.6 and 5.2.7.

Stochastic Differential Equations

69

Regular dependence on initial conditions and Kolmogorov equations

5.4

If one assumes that the coefficients F and B of equation (5.3.1) are smooth then the solutions to the equation are also smooth in a proper sense. Note that in the proof of Theorem 5.3.1 we showed that the transformation x -+ X(., x) from H into xp,T was Lipschitz continuous. We want now to prove that this mapping is, under suitable regularity assumptions, differentiable. For this we need a generalization of the local inversion theorem proved in Appendix C.

5.4.1

Differentiable dependence on initial datum

We prove here the result.

Theorem 5.4.1 Assume that the mappings A, F and B satisfy Hypothesis 5.1.

(i) If F and B have first Frechet derivatives bounded and continuous then the solution X(., x) to problem (5.3.1) is continuously differentiable in x as a mapping from H into "12,T. Moreover, for any h E H, the process (h(t) = XX(t,x)h, t E [0,T], is a mild solution of the following equation,

d(h

= (A(h + FA(X) (hdt + B,(X) . ChdW (t), (5.4.1)

(h(0) = h.

In addition there exists a constant C1,T, independent of h, such that sup IEIXx(t,x)h12 _< C1,T1h12.

(5.4.2)

tE[0,TI

(ii) Assume in addition that F and B have bounded and continuous second Frechet derivatives and that for any t > 0, and x, y, z E H, S(t)Bx,(x)(y, z) belongs to L2(U; H) and there exists a locally square integrable mapping A71 :

[0,+oo[-+ [0,+oo[, t - Kl(t),

such that II S(t)Bxx(x)(y, z)ullHs < Ki (t)jyJ IzI Jul, V x, y, z E H, u E U.

Chapter 5

70

x) to problem (5.3.1) is twice continuously difThen the solution ferentiable, and for any h, g E H, the process rlh'9(t) = Xxx(t, x)(h, g), t E [0, T], is a mild solutions of the following equation,

= (Arih,s + Fx(X) iih,s)dt + Bx(X) . 77h9dW(t),

dgh,s

+Fxx(X) ((h, (9))dt + Bxx(X) . (Ch, (9)dW(t) 71

h 9(0)

= 0. (5.4.3)

Proof - Consider the mapping .77 : H X "2,T - 'H2,T,

defined by

x,X

t

= S t x+ jtSS(t +

FXs

ds

JS(t - s)B(X(s))dW(s), t E [0,T].

It remains to check that, setting

A = H, E=H2,T, G=H4,T, F fulfils the hypotheses of Proposition C.1.3, and the conclusion follows.

5.4.2

Kolmogorov equation

Under the hypotheses of Theorem 5.4.1 one can show, see G. Da Prato and J. Zabczyk [44, Theorem 9.8], that for any x E H, X(t,x), t > 0, is a Markov process. The corresponding transition semigroup Pt, t > 0 is defined by PtV(x) = E[ca(X(t, x))], x E H, cp E Bb(H).

Stochastic Differential Equations

71

From Theorem 5.4.1 we can also deduce an important result about the Kolmogorov backward equation associated to (5.3.1) with = x :

at

v(t, x) = 2 Tr [B*(x)vx.,(t, x)B(x)] + (Ax + F(x), vx(t, x)), t > 0, x E D(A),

v(0, x)

cp(x); x E H. (5.4.4)

We need the following hypothesis, stronger than Hypothesis 5.1.

Hypothesis 5.2 (i) Hypothesis 5.1-(i)-(ii) hold. (ii) B is a mapping from H into L2(U; H), and there exists a constant cl > 0 such that II B(x)II Hs < ci(1 + IxI), X E H, II B(x) - B(y)II Hs <_ c1I x - yI , x, y E H.

A strict solution of problem (5.4.4) is a continuous function v [0, +oo[ xH --> R having continuous first and second partial derivatives with respect to x, such that u(., x) is continuously differentiable in t for all x E D(A), and fulfilling equation (5.4.4) for all x E D(A)

andt>0. The following result is proved in G. Da Prato and J. Zabczyk [44, Theorem 9.16].

Theorem 5.4.2 Assume that the mappings F and B satisfy Hypothesis 5.2. If in addition the first and the second derivatives of F and B are bounded and continuous and cp E Cb (H) then equation (5.4.4) has a unique strict solution v and it is given by the formula v(t, x) = E( 0, x E H.

5.5

(5.4.5)

Dissipative stochastic systems

There is an important class of infinite dimensional deterministic dynamical systems for which the asymptotic behaviour is well understood. They are called dissipative. After recalling their basic prop-

erties we will study their stochastic perturbation described by the

Chapter 5

72

evolution equation (5.1.1) with A, F and B having additional properties.

5.5.1

Generalities about dissipative mappings

Let E be a Banach space with the norm II -1I. Let us first recall some

properties of the subdifferential of the norm. The subdifferential aIIxII of II II at x is defined as follows,

aIIxII = {x* E E* : llx + yII - IIxII >_ (y, x*), V Y E E},

where E* is the dual of E. One can easily prove that the set aIIxII is convex, closed, nonempty and given by {x* E E* : (x,x*) = IIxII, llx*lI =1} if x

0,

{x* E E* : llx*ll < 1} if x = 0. Moreover:

D+IIxII . y = max{(y, x*) : x* E allxll},

D-IIxII . y = min{(y,x*) : x* E allxll}.

We will use the following chain rule, see for instance G. Da Prato and J. Zabczyk [44, Proposition D.4].

Proposition 5.5.1 Let u : [0,T] -+ E,t , u(t), differentiable in to E [0,T]. Then the function y = Ilu(.)II is differentiable on the right and on the left at to and we have, d+ 7

dt (to)

= D+II u(to)ll . u'(to)

= max{(u'(to), x*) : x* E allu(to)ll}, d- -y

dt

(to) = D-Ilu(to)II . u'(to) = min{(u'(to),x*); x* E allu(to)ll}.

Stochastic Differential Equations

73

A mapping f : D(f) C E -p E is said to be dissipative if and only if for any x, y E D(f) there exists z* E all x - yll such that

(f(x) - f(y),z*) < 0. We will need the following simple result.

Proposition 5.5.2 Let x, y E E, then the following assertions are equivalent.

(i)

llx 11

< ll x + ayll, d a > 0;

(ii) 3 x* E 8lIxll such that (y,x*) > 0.

By Proposition 5.5.2 a mapping f : D(f) C E --+ E is dissipative if and only if

llx - yll C llx - y - a(f(x) - .f(y))lI, d x, y E D(f), d a > 0. If, for instance, E = H is a Hilbert space with inner product mapping f : D(f) C E --p E is dissipative if and only if

a

(f(x) - f(y),x - y) < 0, d x, y E D(f). A mapping f is called strongly dissipative if there exists w > 0 such that f + wI is dissipative. A dissipative mapping f is called m-dissipative if the range of AI - f is the whole space E for some A > 0 (and then for any A > 0). Finally a mapping f is called almost m-dissipative if f - aI is mdissipative for some a E R. It is well known, see R. Martin [115], that any continuous dissipative mapping is m-dissipative. The Yosida approximations f, a > 0, of an m-dissipative mapping f are defined by

.fa(x) = f(JJ(x)) = aI (J,(x) - x), x E E,

(5.5.1)

where

J,,(x) = (I - a f)-'(x), x E E. We list some useful properties of Ja and fa.

(5.5.2)

Chapter 5

74

Proposition 5.5.3 Let f : D(f) --+ E be an m-dissipative mapping in E, and let Ja and fa be defined by (5.5.1) and (5.5.2) respectively. (i) For any a > 0 we have II Jax - Jayll <- Ilx - yII, V x, y E E.

(5.5.3)

(ii) For any a > 0 fa is dissipative and Lipschitz continuous:

IIMX) - fa(y)ll < a Ilx - yll,V x, y E E,

(5.5.4)

IIfa(x)II < IIf(x)II,dx E D(f).

(5.5.5)

and

(iii) We have

oJJ(x) = x,V x E D(f).

(5.5.6)

a

Proof - (i) follows from the dissipativity off and then (5.5.4) is clear by (5.5.1). We now prove dissipativity of fa. Let x, y E E and a > 0, we have

lix - y - Q(fa(x) - fa(y))ll (5.5.7)

(1 + a)(x - y) - a(Jc(x) - Ja(y))II

llx - yll,

which implies that fa is dissipative. Finally, since

fa(x) _ «(Ja(x) - Ja(x - af(x))), (5.5.5) follows from (5.5.3). Finally we have

II JJ(x) - xll = all fa(x)Il -< all f(x)ll ,d x E D(f), which implies (5.5.6). The following result will be useful in the sequel.

Proposition 5.5.4 Let E be a Hilbert space, f : D(f) -f E a mdissipative mapping in E, and let a, 0 > 0. Then we have (fa(x) - fa(y), x - y) < (a + 0) (I fa(x)I + I fa(y)I )2 , `d x, y E E.

Stochastic Differential Equations

75

Proof - Since x = Ja(x) + afa(x), y = J,3(y) +,3fla(y), we have

(f. (x) - fp(y), x - y) = (f. (x) - fp(y), Ja(x) - JO (y)) +

(.fa(x) - f,Q(y), afa(x) - /3f,3(y))

But (.fa(x) - .fla(y), JJ(x) - JQ(y))

=

(f(Ja)(x) - f(J,3)(y),Ja(x) - Ja(y))

<

0,

so that

(fa(x) - fp(y), x - y) < (fa(x) - f,3(y), afa(x) - Qfa(y)) (a + a) (I fa(x)I + I

5.5.2

)3(x)I )2

Existence of solutions for deterministic equations

Let us consider now the problem

y'(t) = Ay(t) + G(t, y(t)), t > 0, (5.5.8)

y(0) = xo E H,

under the following hypothesis.

Hypothesis 5.3 (i) A : D(A) C E -a E generates a semigroup S(t), t > 0 on E that is strongly continuous in ]0, +oo[. (ii) There exists w E R such that IIS(t)II < ew', t > 0.

(iii) G : [0, T] x E -# E is continuous. (iv) There exists rt E ]R such that A + G(t, ) - rl is dissipative for any t E [0, T].

Chapter 5

76

We say that u E C([0,T]; E) is a mild solution of (5.5.8) if t

u(t) = S(t)xo + f S(t - s)G(s, u(s))ds.

To solve problem (5.5.8) we need a lemma about the construction of approximate solutions, that is a straightforward extension to the nonautonomous case of a result proved in G. F. Webb [159], see also G. Da Prato [34]. We give the proof here for the reader convenience.

Lemma 5.5.5 Assume that Hypothesis 5.3 holds and let xo E E. Then there exists To E ]0, T] such that for arbitrary E > 0 there exist uE E C([O,To]; E) and O piecewise continuous on [0, To), such that uE(t) = S(t)xo + f tS(t - s)[G(s,us(s)) + 9 (s)]ds,

(5.5.9)

0

and

119E(t)II < e, d t E [0, To].

(5.5.10)

Proof - In the proof we set w = rl = 0 for simplicity. Let us first define for arbitrary e> 0, t E [0, T], x E E

pE(t,x) = sup {S > 0 : t1,t2 E [0,T], It, - tI < S, It2 - ti < b, x1, x2 E B(x, 6) = II G(tl, xi) - G(t2, x2)II < E}. As is easily checked,

PE(t, x) -

y)I < It - si + IIx - ylL, d t, s E [0, T], V x, y E E.

So p, : [0,T] x E --+ E is Lipschitz continuous. Moreover p,(t,x) > 0, V t E [0,T], x E E, since F is continuous. Since F is continuous there exist r > 0 and M > 1 such that II F(t, x)II < M, V t E [0,T], V x E B(xo, r).

(5.5.11)

We now define a function ue fulfilling (5.5.9) and (5.5.10). Set to = 0 and let t1 < t2 < ... be positive numbers. Define by recurrence

u,(t) = S(t - tn,-1)xn,-1 + f

S(t - s)F(s, x,_1)ds, t E [tn.-1,tn],

to-1

(5.5.12)

Stochastic Differential Equations

77

where

xn-1 = ue(tn-1).

Then setting 5.5.13)

ee(t) = G(t, ue(t)) - G(t, xn-1), t E [tn-l, tn], we have

ue(t) = S(t - tn-1)xn-1 + f

S(t - s)[G(s, ue(s)) + ee(S)]dS, o _1

(5.5.14)

for t E [tn_l,tn]. We want now to show that it is possible to choose {tk} such that 110e(t)II C E, t E [tn-l,tn]

Suppose we have chosen tn_1. To choose to note that by (5.5.12) we have Ilue(t)-xn-111 <-

IIS(t-tn-1)xn-l-xn_111+M(t-to-1), t E

[tn-l,tn].

Now let to = sn A An where (5.5.15)

and An is such that II S(t - tn_1)xn-1 - xn-111 =

sup tE[tn-1,A ]

Then if II ue(t) - toll

I Pe(tn-1, xn_1).

(5.5.16)

r on [tn_1i tn] we have

Ilee(t)ll = II G(t, ue(t)) - G(t, xn-l )II C E, t E [tn_l, tn].

It remains to show that the times {tn} can be chosen sufficiently large and such that Ilue(t) - x011 < r on [0, tn]. Let us distinguish two cases.

First case - There exists 7 > 0 such that Ilue(t)

- xoll

r, V t E [0,t], and Ilue(t) - xoll = r.

Chapter 5

78

By (5.5.12) we have n-1

ue(t) = S(t)xo + +

f

tk

S(t - s)G(s, xk_1)ds J k-1 k=1

(5.5.17)

t

to-I

S(t - s)G(s, xn_1)ds.

It follows that r = II ue(1) - xoll < II S(t)xo - xoll + M.

Thus there exists To > 0, depending only on xo, r, M, and not on e such that t > To > 0.

Second case - to T t* If t* = +oo the conclusion is obvious. Let assume that t* is finite. Then by (5.5.17) we have

xn - xn-1 = S(tn)xo - S(tn-1)XO n-2

tk

J -, [S(tn - s) - S(tn-1 - s)]G(s, xk-1)ds

+

k

ftn

+ Jto-1 S(tn - s)G(s,xn-1)ds. It follows that Ilxn - xn-111

<

II S(tn)xo - S(tn-1)x0II n-2

+

I

tk

k -1

II[S(tn - s) - S(tn-1 - s)]F(s,xk-1)Ilds

+ M(tn - to-1). Thus there exists x* E E such that xn -* x*, and, recalling that pE is continuous, we have Pe(tn, xn) - pE(t*, x*).

Stochastic Differential Equations

79

We notice finally that by (5.5.15)-(5.5.16) it follows that pE(t*, x*) _ 0, a contradiction.

Proposition 5.5.6 Assume that Hypothesis 5.3 holds. Then for any xo E E, problem ( 5.5.8) has a unique mild solution x) in [0, +oo[ (4).

Proof - For any e > 0 let uE be the continuous function in [0, To] defined in Lemma 5.5.5. Then uE is the mild solution to the problem

u' (t) = AuE(t) + F(t, ue(t)) + 9E(t), t E [0, To] uE(0) = xo.

For arbitrary E1,62 > 0 we have, by Hypothesis 5.3 dt IIu" (t) - uE2 (t)II <_ el + E2 -

Thus there exists u E C([O, To]; E) such that uE -> u in C([0, To]; E). Passing to the limit for e -* 0 in (5.5.9) it follows that u is a solution to (5.5.8) in [0, To]. By a standard extension argument one can show that there is a solution in [0, T]. Finally, the uniqueness follows from the dissipativity assumptions.

A stochastic version of Proposition 5.5.6 is the object of the next subsection. 4In fact one can show that the solution is strong, that is for any T > 0 there exists a sequence {

C Cl ([O, T]; E) n C([O, T]; D(A)),

such that

Y. -+

y - Ay. - F(yn) -+ 0, in

C([O, T]; E).

Chapter 5

80

5.5.3

Existence of solutions for stochastic equations in Hilbert spaces

In this section we present a method which implies existence and uniqueness of the solutions to the problem

dX = (AX + F(X))dt + BdW(t), (5.5.18)

where A, F satisfy some dissipativity assumptions on appropriate spaces and B is a bounded operator. As a byproduct we will obtain a proof of Proposition 5.5.6 with a slightly weaker concept of solution.

Let H be a Hilbert space and let K be a reflexive Banach space included in H. We assume that K is a dense Borel subset of H and such that the embedding of K in H is continuous. On the mappings A and F we impose the following condition. Hypothesis 5.4 (i) There exists q E R such that the operators A- 77 and F - rt are m-dissipative on H. (ii) The parts on K of A -,q and F -,q are m-dissipative on K. (iii) D(F) D K and F maps bounded sets in K into bounded sets of H.

We shall denote by AK and FK the parts of A and F respectively, that is D(AK) = {x E D(A) n K : AKX E K}, AKX = Ax, X E D(AK),

D(FK) = {x E D(F) n K : FKX E K}, FK(x) = F(x), X E D(FK). Let Z(t) = WA(t), t > 0, be the solution to the linear equation dZ = AZdt + BdW(t), (5.5.19)

X(0) = 0, given by

WA(t) = J0 S(t - s)BdW(s), t > 0, 0

where S(t), t > 0, is the semigroup generated by A in H. We will assume the following.

Stochastic Differential Equations

81

Hypothesis 5.5 The process WA(t), t > 0, is continuous in H, takes values in the domain D(FK) of the part of F in K, and for any T > 0 we have sup (II WA(t)II K + II F(WA(t))II K) < +oo, ? a.s.

tE[O,T]

Remark 5.5.7 Notice that Hypotheses 5.4 and 5.5 are satisfied if

(i) K = H. (ii) A is a linear operator almost m-dissipative on H. (iii) F is either Lipschitz continuous, or continuous and monotone with linear growth. (iv) WA() is an H-continuous process.

An H-continuous, adapted process X(t), t > 0, is said to be a mild solution to (5.5.18) if it satisfies lP'-a.s. the integral equation

X(t) = S(t)x +

J

t

S(t - s)F(X(s))ds + WA(t), t > 0.

(5.5.20)

If, for an H-valued process X, there exists a sequence {X,,,} of mild solutions of (5.5.18) such that ]P-a.s., X,,(.) -f X(.) uniformly on any interval [0, T], then X is said to be a generalized solution to (5.5.18). Note that each mild solution is also a generalized solution.

Theorem 5.5.8 Assume that Hypotheses 5.4 and 5.5 are fulfilled. Then for arbitrary x E K there exists a unique mild solution of (5.5.18) and for arbitrary x E H there exists a unique generalized solution of (5.5.18).

Remark 5.5.9 It will follow from the proof that generalized solutions X(t, x), t E [0, T],x E H, of (5.5.18) are all Markov processes in H with a Feller transition semigroup Pt, t > 0, given by PtW(x) = E[p(X (t, x))], x E H, p E Bb(H).

Proof of Theorem - We will show that the initial value problem ye(t) = Ay(t) + F(y(t) + WA(t)), y(O) = x,

(5.5.21)

Chapter 5

82

has a mild solution. That is there exists equation:

satisfying the following

y(t) = S(t)x + f S(t - s)F(y(s) + WA(s))ds, t E [0,T]. (5.5.22) t

Then the solution X to (5.5.18) is given by X (t) = y(t) + WA(t), t E [0,T].

For arbitrary a > 0 we consider the approximating problem

ya(t) = Aya(t) + Fa(ya(t) + WA(t)), y.(0) = x,

(5.5.23)

where Fa are Yosida approximations of F. Since F,,, are Lipschitz continuous, equation (5.5.23) has a unique H-continuous solution. We will show that lima-o ya(t) = y(t) exists uniformly for t E [0, T] and is the required solution of (5.5.22). The proof will be done in several steps. For simplicity we assume 7 = 0.

Step 1- A priori estimate in H. We have, by dissipativity of A and Fa if necessary replacing A by Yosida approximations A,, of A), that dt l y«(t)I H

= (Aya(t), ya(t))H + (FF(ya(t) + WA(t)), ya(t))H

= (Aya(t), ya(t))H + (F.(WA(t)), ya(t))H

+ (F.(ya(t) + WA(t)) - F.(WA(t)), ya(t))H <

(Fa(WA(t)), ya(t))H

C

I FF(WA(t))I HI ya(t)I H

<

IF(WA(t))IHiya(t)IH

Now by the differential inequality just obtained, Hypothesis 5.4-(iii) and Hypothesis 5.5 there exists a constant C1 such that Iya(t)IH < C1i t E [0, T], a > 0.

(5.5.24)

Stochastic Differential Equations

83

Step 2 - A priori estimate in K In a similar way, but using the subdifferential of the norm in K

instead of the scalar product of H, we obtain that there exists a constant C2 > 0, depending on cp, such that II ya(t)II K <-

C'2II "II K, t E [0,T], a > 0.

(5.5.25)

It follows from (5.5.25) that for a constant C3 > 0 I F(WA(t) + ya(t))IH < C3, t E [0,T], a > 0.

(5.5.26)

Step 3 - Convergence in H Let a > 0,3 > 0, then we have y. (t) 2 dt

I

- ya(t)iH = (Aya(t) - Ayo(t), ya(t) - ye(t))H

+(F.(ya(t) + WA(t)) - FQ(yp(t) + WA(t)),Y.(t) - YQ(t))H

(F.(y«(t) + WA M) - F,(y0(t) + WA(t)), ya(t) - ya(t))H. By Proposition 5.5.4 it follows that 2 dt I ya(t) - y0(t)I H

-

(a+0) (I Fc(ya(t) + WA(t))I

+ FF(y#(t) + WA(t))) 2 < 2(a +,3)C32, in virtue of (5.5.26)

It is therefore clear that ya(t) -+ y(t) in H as a -> 0 uniformly on [0, T].

Step 4 - We finally prove that we can pass to the limit as a --p 0 in the mild version of (5.5.23) :

Y", (t) = S(t)x+J tS(t-s)FF(ya(s)+WA(s))ds, t E [0,T]. (5.5.27)

Note that by (5.5.25) and reflexivity of K, for arbitrary t E [0,T], there exists a sub-sequence {ya,,,(t)} converging weakly in K to an

Chapter 5

84

element in K. Since {ya,n(t)} is strongly convergent in H, y(t) E K for all t E [0, T] and IIy(t)IIK
Let h E H, then (y«(t), h)H = (S(t)x, h)H+ f t(F(JJ(ya(s)+WA(s))), S*(t-s)h)Hds. (5.5.28)

Moreover

JJ(ya(s) + WA(s)) --} y(s) + WA(s) strongly in H, as a - 0, F(Ja(ya(s) + WA(s))) --+ F(y(s) + WA(s)) weakly in H, as a -+ 0. So, letting a tend to 0 in (5.5.28), we arrive at

(y(t), h) = (S(t)x, h) + j(S(t - s)F(y(s) + WA(s)), h)ds. The conclusion follows from the arbitrariness of h.

By arguments as above one can show that there exists a constant

c> 0 such that for arbitrary x, y E K sup IX(t,x) - X(t,yAH

c.Ix - yIH.

tE[O,T]

This easily implies uniqueness of the mild solution of (5.5.18) for arbitrary x E K and existence and uniqueness of the generalized solution of (5.5.18) for arbitrary x E H.

Remark 5.5.10 If F : K ' H is continuous, the hypothesis that K is reflexive can be dropped. Some useful variations of Theorem 5.5.8 are possible. To formulate one of them it is convenient to introduce the following hypothesis.

Hypothesis 5.6 (i) A separable Banach space K is continuously and densely embedded into a separable Hilbert space H.

(ii) For some w E ]R the operators A - w and AK - w are mdissipative on H and on K respectively.

Stochastic Differential Equations

85

(iii) The mapping F : K K satisfies a Lipschitz condition on, bounded sets and there is a continuous function a : R+ -a R+ such that for arbitrary x, y E K, x* E BIIxII

(F(x + y),x*) < a(IIyII) (iv) The process WA(t), t > 0, has a K-continuous version.

Theorem 5.5.11 Under Hypothesis 5.6 for arbitrary x E K there exists a unique continuous solution of (5.5.18) in K which determines

a Feller transition semigroup on K. If in addition for some w E R the mapping F - w is dissipative in H then for arbitrary x E H there exists a unique H-continuous generalized solution X(t, x), t > 0, of (5.5.18) with Feller transition semigroup on H.

Proof - Write y(t) = X(t) - WA(t), t E [0,T] and let SK() be the semigroup generated by AK. Then

y(t) = SK(t)x+10 SK(t-s)F(y(s)+WA(s))ds, t E [0,T], (5.5.29) and for x E K equation (5.5.18) is equivalent to (5.5.29). It is therefore enough to show that for an arbitrary Ii -continuous function z(.) the equation

y(t) = SK(t)x + 1 SK(t - s)F(y(s) + z(s))ds, t E [0,T], (5.5.30) t

has a unique global solution y. Local existence follows by a contraction mapping principle. To obtain global existence it is sufficient to deduce an a priori estimate for Ily(.)II If a solution z(.) exists on a time interval [0, To] then there exists a sequence { y,,} C C1([O, To]; D(AK)) such that

yn.(t) - y(t), d dtt) - AKyn(t) - F(yn(t) + z(t)) = Sn(t) - 0 as n oo, uniformly on [0,To]. Now for some xt,n E OIIyn(t)II, and t E [0, T],

Chapter 5

86

dt II yn(t)II

<

(AKyn(t) + F(yn(t) + z(t)),

+

(bn(t), xt,n)

<

wllyn(t)II + a (llz(t)II) + Ilbn(t)ll.

Consequently t

IIy(t)II <- etlly(0)II +

J

eW

(t-S [a (IIz(s)II) + Ilbn(s)II] ds.

Letting n tend to infinity we have

ft Ily(t)II < ewuIIxII +

eW(t s)a (Ilz(s)II) ds,

the required a priori estimate. To prove the final part of the theorem it is enough to notice that if X (t, a; and X (t, b) are two solutions of (5.5.18) with a, b E K then, going if necessary to smooth approximations of the solution,

1

dt

I X (t, a) - X (t, b) I2

< (A(X(t, a) -. X (t, b)) + F(X(t, a)) - F(X(t, b)), X (t, a) - X (t, b) )

< 2wlX(t,a) - X(t,

b)12.

So

IX(t,a) - X(t, b)12 < e2W1 la- bl, t > 0,

and the result easily follows.

Remark 5.5.12 For another version of Theorem 5.5.8 we refer to G. Da Prato and J. Zabczyk [43] and [44, Theorem 7.13].

Stochastic Differential Equations

5.5.4

87

Existence of solutions for stochastic equations in Banach spaces

In this section we consider the problem

dX = (AX + F(X))dt + dW(t), (5.5.31)

X(0)=x EE, on a Banach space (norm II - II) E C H. We assume

Hypothesis 5.7 (i) A : D(A) C E -4 E generates a semigroup S(t), t _> 0, on E that is strongly continuous in ]0, +oo[. (ii) There exists w E R such that

IIS(t)II < e"t, t > 0.

(iii) F : E -> E is continuous. (iv) There exists 77 E R such that A + F -,q is dissipative. (v) W(.) is a cylindrical Wiener process on H such that the stochastic convolution WA(t), t _> 0, belongs to C([0, T]; E) for arbi-

trary T > 0. We say that X E C([O,T); E) is a mild solution of (5.5.31) if

X(t) = S(t)xo +

t

S(t - s)F(X(s))ds + WA(t).

(5.5.32)

J0 Theorem 5.5.13 Assume that Hypothesis 5.7 holds. Then for any x E E problem (5.5.31) has a unique mild solution.

Proof - Setting Y(t) = X(t) - WA(t), equation (5.5.32) reduces to the problem

Y'(t) = AX + F(Y(t) + WA(t)),

Y(0)=x EE. Now it suffices to set C(t,z) = F(Y(t)+WA(t)) and to apply Proposition 5.5.6.

Chapter 6

Existence of invariant measures In this chapter we establish existence of invariant measures for stochas-

tic evolution equations by exploiting either compactness or dissipativity properties of the drift parts of the equations. A complete characterization of those linear equations for which an invariant measure exists is given as well. For dissipative systems convergence of the transition semigroups to equilibrium is obtained.

6.1

Existence from boundedness

We are here concerned with the problem

dX(t) _ (AX + F(X))dt + B(X)dW(t) (6.1.1)

X(0)=xEH, under Hypothesis 5.1. We denote by X(.,x) the mild solution of (6.1.1) and by Pt(x, ) the corresponding transition probability:

Pt(x, r) = .C(X(t, x))(r), r E 5(H), t > 0, x E H. It follows from Corollary 3.1.2 that if for some x E E the family of measures T

1

T

T > 1,

J 89

(6.1.2)

Chapter 6

90

is tight then there exists an invariant measure for the transition semi-

group Pt, t > 0. That property implies in particular that 1

VXEH Ve>O 3R>O VT>1

ITT

T

P(IX(t, x)l > R) dt < e.

(6.1.3)

We will show in this section that under some additional assumptions the easier to check condition (6.1.3) implies tightness of (6.1.2). A stochastic process X (t), t > 0, is said to be bounded in probability if

V e>o 3 R>o V t>o POX(t)1 > R) < e.

Note that if the process X(t, x), t > 0, is bounded in probability then (6.1.3) holds.

Remark 6.1.1 It is clear that (6.1.3) implies the existence of an invariant measure in the case dim H < +oo. If dim H = +oo then the implication is not true even for deterministic equations. In fact I. Vrkoc [157] constructed a bounded and Lipschitz mapping F : H H on a separable Hilbert space H such that all solutions X(., x) of the problem

X'(t) = F(X(t)), X (O) = x, t > 0,

(6.1.4)

are bounded. Nevertheless there is no invariant probability measure for problem (6.1.4). However we have the following theorem.

Theorem 6.1.2 Assume that (i) Hypothesis 5.1 holds.

(ii) There exists a E ]0,1/2[ such that

I

1

t-2aK2(s)ds < +oo,

0

where K is the function from Hypothesis (iii) The operators S(t), t > 0, are compact. (iv) Condition (6.1.3) is fulfilled. Then there exists an invariant measure for (6.1.1).

Existence of invariant measures

91

Remark 6.1.3 Existence of an invariant measure for (6.1.1) with condition (6.1.3) replaced by a stronger one, that for some x E H, X(t, x), t > 0, is bounded in probability, was proved in G. Da Prato, D. G.tarek and J. Zabczyk [39]. A similar proof, however, goes through under (6.1.3).

Proof - For the proof we use the factorization formula, see Theorem 5.2.5. For any a E ]0, 1] we define operators G" : LP(0,1; H) H by the formula 1

Gaf =

I(1 - s)"-1S(1 - s) f (s)ds, f E LP(0, l; H),

and we set

Y(t, x) =

j(t - s)S(t

t

s)B(X(s, x))dW(s).

(6.1.5)

Then one easily checks the identity

x)) + sin

X(1, x) = S(1)x +

r

x).

(6.1.6)

We have the following technical result.

Lemma 6.1.4 Assume that S(t), t > 0, are compact operators and let p > 2 and a > Then Ga is compact. 1.

Proof - Fore E ]0,1[, f E LP(0,1; H) define Gaf =

/1-e

J0

Then

(1 - s)a-1S(1 - s)f(s)ds.

-E

Gaf = S(e) f

1

(1 - s)'-1S(1 - e - s) f (s)ds,

0

and from the compactness of S(e), it follows that GI is a compact operator. Set q = pp l and note that

(a-1)q+1= ap-1 p-1 >0.

Chapter 6

92

By the Holder inequality, it follows that 1 (1 Gaf - Gaf I = J1 J

s)°`-1 S(1

- s)f(s)dsl lip

1

(1 -

s)(a-1'gjIS(1- s)IIgds)

<

(J'

<M

((a - 1)q+ 1)

e(a-1)q+1

If(s)Ipds)

1/q

Ifip,

where I' f p is the norm off in Lp(0,1; H) and M = sup$E[o,1]

iiS(s)Ij.

Consequently Ga -+ Ga as a -; 0 in the operator norm so that G. is compact.

It follows from the lemma that the mapping ry

: H x Lp(0,1; H) x Lp(0,1; H) -+ X, (y, g, h) -i S(1)y+Gig+Gah, (6.1.7)

is compact. Consequently, for arbitrary r > 0 the set

K(r) _ {x E H : x = S(1)y + Gig + Gah, IyI < r, Iglp < r, Ihlp < r} is relatively compact in H. The key to the proof of the theorem is the following lemma.

Lemma 6.1.5 Assume that p > 2, a E ] [ , and that the Hypothea a constant c > 0 such ses of Theorem 6.1.2 hold. Then there exists that for arbitrary r > 0 and all x E H such that I xI < r, p,

1P(X(l, X) E K(r)) > 1 - cr-p(1 + I xI'), r > 0.

Proof- Let Y(.,x) be defined by (6.1.5). Then, by Theorem 5.2.4,

Existence of invariant measures

93

there exists a constant k > 0 such that E

I'

kE

JY(s, x)jpds = IE

f

1

(J'(s -

< cPk2P/2E10 1

(10

I

1 IJ s(s - u)--S(s - u)B(X (u, x))dW(u)

u)-2aIIS(s

$(8

-

- u)II2 SII B(X(u,

u)-2aK2(s

x)II2

P

Sdu)P/2

- u)(1 +

ds

IX(u,x)I2)du)P/2

ds.

By the Young inequality, IE

fl IY(s,x)IPds

<

kcP2P/2

xE

f

1

(j't_2K2(t)dt)2

(1 + IX(u,x)I2)p/2 du

0

kl(1 + I xI P),

for some constant k1 > 0. Moreover, by (5.2.15) there exists k2 > 0 such that E

f1

IF(X(s,x))IPds < k2(1 + IxIP), X E H.

Now let Ixi < r. Since the mapping y defined by (6.1.7) is one-toone, one has X(1,x) K(r) if and only if either Y(., x) I P < sin «_ . It follows that I

?(X(1,x)

K(r)) <- ?(IF(X(-,x))Ip > r) +?

(iY()I> sin rair

and, by the Chebyshev inequality, P(X (1, x)

K(r)) < r-PE(I F(X (., x))I P)

+ r -P sine air >E(IY(., x)IP) < r-P (7r-Pkl + k2) (1 + IxIP),

ds

Chapter 6

94

and the lemma follows.

Proof of Theorem 6.1.2 - For any t > 1, we have by the Markov property, noting X (t, x) = X (t),

P(X(t) E K(r)) = EE(Pl(X(t - 1), K(r)))

> E {Pi(x(t - 1), K(r))xIX(t-1)I r1 > 0

?(X(t) E K(r)) > (1 - cr-P(1 + ri)) ](IX(t - 1)1 < r1). Consequently

P(X(t) E K(r))dt > (1 - c.(r-P(1 +ri))) x

jP(IX(t)I

< rl)dt.

Taking first r1 and then r > r1 sufficiently large we see that the family

Pt(x , )dt , T > 1 , 1

is tight. This finishes the proof. Property (6.1.3) is implied by various kinds of conditions implying boundedness of moments . If, for instance, for some x E H, To > 0

andp>0 sup EIX(t,x)1P < +x t>To

(6.1.8)

then, by the Chebyshev inequality, the condition (6.1.3) is satisfied. The final proposition gives sufficient conditions for boundedness of the second moment of the solutions of nonlinear equations. In its proof we use 11 . 112 as Liapunov function.

Proposition 6.1.6 Assume that A, F and B satisfy conditions of Theorem 6.2.1. Assume in addition that

Existence of invariant measures

95

(i) there exist constants a > 0, b, c such that

(Ax + F(x + y), x) < -alxl2 + bI yI2 + c, x E D(A), y E H,

(ii) we have f

+00

sup IIS(t)B(x)IIHS dt = k < +oo.

xEH

0

Then for arbitrary x E H sup ]EIX(t, x)I2 < +oo. t>0

Proof - Write Z(t) =

t 10

St - sBXsdWs,

Y(t) = X(t) - Z(t), t > 0, where X (t) = X (t, x), t > 0. It follows from (ii) that t

Supt>0 EIZ(t)i2 = sup )E t>0

f II S(t - s)B(X (s))IIHS ds 0

(6.1.9)

< k < +oo. Obviously

Y(t)- Stx+ f t St - sFXsds,t>0. Let YA(t) = A(A - A)-'Y(t) and F,\(x) = A (A - A)-1F(x), where A > 0 is sufficiently large. Since Y,(t) E D(A) for any t > 0 and

Y\(t) = A(A - A)'1S(t)x + j S(t - s)F(X(s))ds then Y,, satisfy the following equations:

tYY(t) = AYY(t) + FA(X(t)) = AYA(t) + F(YY(t) + Z(t)) + Sa(t),

where 6A (t) = FA(X(t)) - F(YA(t) + Z(t)) -+ 0 as A - +oo. We have 2 dt IYY(t)I2

=

(AYY(t) + F(YA(t) + Z(t)) + ba(t),YA(t))

<

-alYA(t)I2 + bIZ(t)I2 + C + I YA(t)II bA(t)I

< - 2 IY,(t)I2 + b(Z(t)12 + C + a216A(t)12.

Chapter 6

96

By a well known comparison theorem lYA(t)I2 <

2e-at/2IxI2

+2

J

t e-a(t-s)/2(bIZ(s)I2 + c+ a I 6a(s)I2)ds.

Letting A tend to infinity we obtain IY(t)I2 < 2e-at/2IxI2 +2 ft

e-a(t-s)12

(bIZ(s)12 + c)ds.

Finally, by (6.1.9) we find t

E (IY(t)I2)

<

2e-at/2Ix12 +2 fo

<

2+a(bk+c).

-a(t-s)/2(bk + c)ds

We will not go further into Liapunov type sufficient conditions implying (6.1.3), but we restrict ourselves to two important special cases which will be referred to later: to linear systems and dissipative systems.

6.2

Linear systems

We are here concerned with the linear equation, introduced in j5.2,

dX(t) = AX(t)dt + BdW(t), t > 0, (6.2.1)

X(o)_e, under the following assumptions.

Hypothesis 6.1 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H. (ii) B is a linear continuous mapping from U into H. (iii) For any t > 0 the linear operator Qt Q tx

=

j

S(s)QS*(s)xdt,, x

where Q = BB*, is of trace class.

H,

(6.2.2)

Existence of invariant measures

97

Obviously Hypothesis 6.1 implies Hypothesis 5.1 so that there exists a unique mild solution of (6.2.1),

S(t)d + f S(t - s)BdW(s), t > 0. 0t

A description of invariant measures

6.2.1

As is easily checked for any t > 0 and for any x E H, X (t, x) is a Gaussian random variable N(S(t)x,Qt), that is with mean S(t)x and covariance Qt. The transition semigroup corresponding to (6.2.1) is given by

Rtcp(x) = JH(y) (S(t)x,Qt)(dy), for all cp E Cb(H).

(6.2.3)

Clearly a probability measure on (H, 13(H)) is invariant if JH

Rtcp(x)1t(dx) = JH

V(x)tt(dx), for all cp E Cb(H), and t > 0.

Given h E H we set cph(x) = ei(h,x), x E H. Then Rtcph coincides with the characteristic functional of the measure N(S(t)x, Qt) and we have Rtcoh(x) = e=(h,s(t)x)-2(Q,h,h) x E H.

It follows that it is an invariant measure for (6.2.1) if and only if its characteristic functional µ is given by A(A) = µ(S*(t)A)e

2 (@ta,a)

, t > 0, A E H.

(6.2.4)

The following theorem is from J. Zabczyk [164]. The finite dimensional version is due to J. Snyders and M. Zakai. [143]

Theorem 6.2.1 Assume Hypothesis 6.1. Then the following conditions are equivalent. (i) There exists an invariant measure for problem (6.2.1).

(ii) sup Tr [Qt] < +oo. t>o

(iii) There exists a trace-class operator P in the cone E+(H), of all symmetric nonnegative operators on H, satisfying the equation 2(PA*x, x) + (Qx, x) = 0, for all x E D(A*).

(6.2.5)

Chapter 6

98

If any of the conditions (i), (ii), (iii) holds then any invariant measure for (6.2.1) is of the form v * N(O, Qom),

where v is an invariant measure for the deterministic system z' = Az and P = Q,, is the minimal nonnegative solution to (6.2.5) given by the formula +00

+

S(t)QS*(t)x dt, x E H.

JO

Proof - (i)ce (ii) Let us assume that y is an invariant measure for (6.2.1), and let µ be its characteristic functional given by (6.2.4). It follows that

(QtA, A) < 2 log (_Rel(A)) , for all A E H. Moreover, by the Bochner theorem, see e.g. G. Da Prato and J. Zabczyk [44, page 48], there exists a trace-class operator So E E+(H) such that A E H, (S0A, A) < 1 Re µ(A) > 2. Thus the following implication holds. A E H, (So A, A) < 1 = (QtA, A) < 2log 2, which yields

0 < Qt <2 log 2 So. So

Tr[Qt] < 2 log 2 Tr [So].

Let P E E+(H) be a solution of (6.2.5) and let x E D(A*). Then we have dt

(PS*(t)x, S*(t)x) = -(QS*(t)x, S*(t)x).

By integrating this identity between 0 and t we get ( Px,

x) =

j

(PS*(s)x, S*(s)x)ds + (Qtx, x), x E H,

(6.2.6)

Existence of invariant measures

99

which implies Tr Qt < Tr P.

(ii)=(iii) If (ii) holds then there exists a trace-class operator Q E E+(H) such that +00

Q."x =

+

t

S(s)QS*(s)xds = lim

tt+ooJo

o

S(s)QS*(s)xds, x E H.

On the other hand, if x E D(A*) we have

2(QtA*x,x) = j ds (QS*(s)x, S*(s)x)ds (6.2.7)

= (QS*(t)x,S*(t)x) - (Qx, x). Since fo (QS*(s)x, S*(s)x)ds < +oo, there exists a sequence t, I +oo such that 1 (QS*(tn)x, S*(tn,)) = 0. Thus, setting in (6.2.7) t = to and letting n tend to infinity we find 2(Q,,)A*x, x) = - (Qx, x), and (iii) is proved.

(ii)=(i) If (ii) holds then the linear operator +00

+ 10

S(t)QS*(t)xdt, X E H,

is of trace class. Let us prove that µ = N(O,Qoo) is an invariant measure. For this it suffices to show that (6.2.4) holds. We have in fact µ(A) = e-2 (Q00a,A) which implies

A(S*(t)A) = e-2

(6.2.8)

so (i) is proved.

We want to show now the last part of the theorem. Let µ be an invariant measure for (6.2.1) and let Q,, x = fo S(s)QS*(s)xds, x E H. Then, letting t tend to +oo in (6.2.8), we have µ(A) = e-2(Q-a,a). (a), A E H,

(6.2.9)

Chapter 6

100

where

O(A) =t lmo ji(S*(t)A), A E H. It remains only to prove that (A) is the characteristic function of a probability measure v, which is invariant for S(.) since v(S(s)A) =

tT mo

µ(S*(t + s)A) = P(A), A E H.

In fact by Bochner's theorem, given E > 0 there exists a positive operator S of trace class, such that Re 2(A)(x) > 1 -

if (Sx, x.) < 1.

Thus, if x E H is such that (Sx, x) < 1, it follows that Re

(A)(x) =

Re µ(A)(x)e1 (Q.a,a) > 1

-

So, using Bochner's theorem once again, there exists a probability measure v in H,L3(H) such that

as required. In conclusion

by (6.2.9) we have it = JV(O, Qom) * v.

Remark 6.2.2 Assume that there exists an invariant measure for fo ' S(t)QS*(t)xdt, x E H, is the minimal (6.2.1). Then solution of (6.2.5). In fact let P E E+(H) be a solution of (6.2.5); then, by (6.2.7),

(Px, x) > (Qtx, x), for all t > 0. Now, letting t tend to infinity, we get P > Qom. Equation (6.2.5) is sometimes called the Liapunov equation for equation (6.2.1). The following characterization shows that for linear equations boundedness in probability of a solution implies existence of an invariant measure, however, compare Remark 6.1.1.

Theorem 6.2.3 Assume Hypothesis 6.1. The following conditions are equivalent.

(i) There exists an invariant measure for problem (6.2.1).

(ii) There exists a solution X(t), t > 0, of (6.2.1), bounded in probability.

(iii) The solution X(t,0), t > 0, of (6.2.1) is bounded in probability.

Existence of invariant measures

101

Proof - (i)=(ii) If µ is an invariant measure for (6.2.1), and X is a solution such that C(X(O)) = µ, then ,C(_X(t)) =y for t > 0 and the boundedness in probability of X follows. Let

X (t) = S(t)d + X (t, 0), t > 0,

be a solution bounded in probability. Denote by µt,µ°, Vt the distributions of X (t), X (t, 0) and S(t)e respectively. Then

µt=vt* µ°, t>0. We claim that for arbitrary R > 0

?(IX(t)l > R) > 211(jX(t, 0)I > R), t > 0.

(6.2.10)

For B = B(0, R) one has

µt(B) = JHt(B - x)vt(dx), and therefore, for some i E H, µ°(B - x) > µt(B). It is clear that

B D (B - x) fl (-B + x), and therefore

µ°(B)

>-

µ° ((B - x) fl (-B + i))

> 1-µ°((B-x)`U(-B+i)c) >

1 - µt

µt

> -l+µ°(B-x)+µ°(-B+a). Since the measure µ° is symmetric and

µ° (B - x) > pt(B), we finally have

1 - P(IX(t, 0)l > R) > -1 + 2µt(B) > -1+ 2 (1 - P(IX(t, 0)I > R,)),

Chapter 6

102

and thus inequality (6.2.10) follows. Consequently To show that(iii)=(i) it is enough to prove, compare Theorem 6.2.1, that boundedness in probability of X(t, 0), t > 0, implies sup E (IX(t, 0)12) = sup Tr Qt < +oo. t>o

t>o

It follows from Fernique's theorem, see e.g. G. Da Prato and J. Zabczyk [44], that there exist constants R > 0, y E ]0, 1[, A > 0, and C > 0 such that if yo (B(0, R)) > ry then JH

eAIZ12 i4(dz) < C,

and in particular E (I X (t, 0) (2) <

This proves the required implication.

6.2.2

Invariant measures and recurrence

The concept of a recurrent set for a Markov process was introduced in §3.4. Here we will comment on recurrence for solutions to (6.2.1).

If dim H < +oo and the semigroup Rt , t > 0, is irreducible at some t > 0, then all solutions to (6.2.1) are recurrent with respect to all open nonempty sets if and only if all eigenvalues of A, with the exception of at most two, have negative real parts and the remaining eigenvalues either are pure imaginary of multiplicity 1 or are equal to 0, see H. Dym [58], R. Erickson [62] and J. Zabczyk [165]. There is also a simple connection between existence of invariant measures and recurrence. If there exists an invariant measure for an irreducible semigroup Rt, t > 0, corresponding to (6.2.1) then any solution to (6.2.1) is recurrent with respect to all open sets of H. In the infinite dimensional case such a characterization is not known. However, some results in this direction are possible, see J. Zabczyk [164]. In particular we have the following

Proposition 6.2.4 (i) Assume that the transition semigroup Rt, t > 0, corresponding to (6.2.1) is irreducible at some t > 0 and has an

Existence of invariant measures

103

invariant measure. If limt._..,+, S(t)x = 0 then any solution to (6.2.1) is recurrent with respect to all open sets of H.

(ii) There exists an equation of the form (6.2.1) such that the corresponding semigroup Rt, t > 0, is irreducible for all t > 0, and has an invariant measure, but a solution X (t, x), t > 0, to (6.2.1) is not recurrent with respect to a ball. Proof - (i) If µ is an invariant measure then p is unique, Gaussian with its support equal to H. Moreover for arbitrary x E H Pt(x, ) weakly as t -* +oo. Consequently, for any open, nonempty set

rcH lim inf Pt(x, r) = p(r) > 0,

t-+oo and it is enough to apply Proposition 3.4.5.

(ii) Let us consider the transition semigroup Rt, t > 0, corresponding to the linear equation

dZ(t) = AZ(t)dt + BdW(t), Z(0) = x. Fix R > 0. It is not difficult to find i E H such that II S(t)xII > t + R, t > 0. Then

Pt(x, B(0, R)) < Pt (0, B°(0, t)), t > 0. By Chebyshev's inequality Pt(0, B`(0, t)) < t2

(IX(o, t) 12)

,

t > 0.

It easily follows now that +00

J

Pt(i, B(0, R))dt < C f

+- dt

< +oc.

Assume that with probability 1 the set

It > 0 : IX(t,x)I < 2 } is unbounded and define TR = inf{t > 0 : IX(t,x)I < R}.

(6.2.11)

Chapter 6

104

Then inf

]E(rr) = a > 0.

I1sII
Therefore the expected time the process X(,) spends in B(0, R) after each visit to B(0,R/2) and before reaching OB(0, R) is at least a. Since by (6.2.11) the total time spent by X(., x) in B(0, R) is finite we easily obtain the contradiction.

6.3

Dissipative systems

Dissipative systems were introduced in f35.4. Here we study their long-time behaviour. We start from general comments on asymptotic properties of solutions to the deterministic problem

y'(t) = Ay(t) + F(y(t)), t > 0, (6.3.1)

y(0)=x E E, under the hypothesis of Proposition 5.5.4, with A = 0, which implies that f = A+F is dissipative. We denote by x) the strong solution to (6.3.1). Dissipativity of f implies an important contraction property for

the solutions of (6.3.1). In fact let y, z E E. Then, by Proposition 5.5.4, it follows that

at Il y(t, x) - y(t, z)II = (f (y(t, x)) - f (y(t, z)), xt,X,z ), for some xt

z

(6.3.2)

E all y(t, x) - y(t, z)II. Consequently

Il y(t, x) - y(t, z)II < Ilx - zll, for all t > 0, and x, z E E.

(6.3.3)

Assume now in addition that f is strongly dissipative and let w > 0 be such that f + wI is dissipative. Arguing as before, we get the estimate Il y(t, x) - y(t, z)II < e-"tll x - zll, for all t > 0, and x, z E E. (6.3.4)

Existence of invariant measures

105

Consequently, if f is m-dissipative and strongly dissipative, then there exists a unique x E D(A) such that f (x) = 0, that is x is a unique equilibrium point for f. Moreover, from (6.3.4) it follows that t

limo y(t, x) = x.

Consider now the corresponding transition semigroup Pt
lim Pt bx = bF, weakly for all x E E,

t-++oo

and therefore lim Pt *v = bx, weakly for all x E E,

t-.+oo

for arbitrary probability measure v. This proves the following result.

Proposition 6.3.1 If f is strongly m-dissipative, then there exists a unique invariant measure for the system 6.3.1 which is moreover strongly mixing.

6.3.1

General noise

A stochastic version of Proposition 6.3.1 is given by the following theorem.

Theorem 6.3.2 Assume that A, B and F satisfy Hypothesis 5.2. Assume in addition that there exists w > 0 such that

2(A,n(x - y) + F(x) - F(y), x - y) + JIB(x) - B(y)1 1z (6.3.5)

<-wlx-yj2, for all x,yEH, nEN, where A,, = AA(n - A)-i are the Yosida approximations of A. Then there exists exactly one invariant measure µ for (6.3.1), it is strongly mixing and for arbitrary v E Mj(H), Pt *v -p y weakly as t - +oo. Moreover there exists C > 0 such that, for any bounded Lipschitz continuous function cp, all t > 0 and all x E H, IPtc'(x) - (4o,i')I C C(1 +

IxI)e-"th/2IIsOIILip.

Chapter 6

106

Proof - It is useful to introduce another cylindrical Wiener process V(t), t > 0, independent of W(t), t > 0, and to define W(t) =

W(t) if t > 0, V(-t) if t-< 0,

(6.3.6)

Tt = a(W(s), s < t), t E R. Next, for any s E

H

and x E H, we consider the regularized equation

dX,,. = (A,X,, + F(X,,,))dt + B(X,,,)dW (t), t > s, (6.3.7)

X,(s) = X.

which has a strong solution X,(t) = X,(t, s, x), t > s. It is easy to see that as n --> oo, Xn(t, s, x) converges, in mean square, to the solution X (t) = X (t, s, x), t > s of the equation

dX = (AX + F(X))dt + B(X )dW (t), t > s, J

(6.3.8)

X(s) = X. We now proceed in two steps.

Step 1- A priori estimate We apply Ito's lemma to the process I Xn(t)12, t > s. Then dIX..(t)I2

= {2(AX.(t) + F(Xn(t)),Xn(t)) + II B(Xn(t))112}dt

+ 2(Xn(t),B(Xn(t))dW(t)). Consequently, integrating in [s, t], and taking expectations, we have E (IXn(t)12) = Ix12

+IEjt{2(AXn(a) + F(Xn(O1)),Xn(Q)) + II B(Xn(o,))112}do,.

Existence of invariant measures

107

It follows that ate (Ixn(t)12)

= ]E(2(AX.,(t) + F(X.(t)),X.(t)) + -w ]E (I X..(t)12) + 2IF(0)I E(I X..(t)I )

+ JIB(0)112 + 21IB(0)112](IB(X.(t))I2)

< - 2E (Ix.(t)I2) + C1, for a suitable constant C1 > 0. So there exists a constant C2 > 0 such that EI X,.(t)12 < e-w(t-s)l2(I x12 + C2), t > s.

As n tends to infinity we find ]EIX(t)12 <

e-W(t-s)12(I

xI2 + Cl) < C3(1 + Ix12), t > s,

(6.3.9)

for some constant C3 > 0.

Step 2 - For b > y > 0 define Z(t) = X(t, -y, x) - X(t, -b, X), t > -y. Using the Ito lemma and proceeding as before we find the estimate ElZ(t)12 <

t > -y

from which, recalling (6.3.9),

]EIX(0, -y, x) - X(0, -b, x)12 < C3(1 + IxI2)e-,-,, b > -y. (6.3.10) It follows that the sequence of random variables {X(0, -y, x)}.y>O, +oo, and theresatisfies the Cauchy condition in L2(St; H), as y fore it is convergent to a random variable i E L2(Sl, H), which is independent of x. We claim that the law of q is the invariant mea, sure with the required properties. To see this it is enough to remark

that

L(X (t, 0, x)) = £(X (0, -t, x)) ' £(t7) = µ, weakly as t - +oo,

Chapter 6

108

which is equivalent to

Pt b., -* p weakly as t - +oo. Finally, let cp E Lip (H), then by (6.3.10) Ptw(x) - Ps
=

(0, -s, x)))12

I E(c (X (0, -t, x))) -

IEIX(O, -t, x) - X(0, -s, x)12

<

< C3(1 +

Ix12)II(pIIL;p

e-wt

As s es+oo we find IPtV(x)

- (V,p)I2 < C3(1 + IxI2)IIkIILip e-wt

This finishes the proof.

6.3.2

Additive noise

Now we consider the setting of §5.5.2. We want to prove the existence

of an invariant measure for the problem

dX = (AX + F(X))dt + BdW(t), (6.3.11)

X (O) = x,

in the space H or in the Banach space K continuously and densely embedded in H. As before we set

WA(t) = J S(t - s)BdW(s). 0t

We shall assume, besides Hypotheses 5.4 and 5.5, the following.

Hypothesis 6.2 (i) There exists w1iw2 E R such that w = w1+w2 > 0 and the operators A + w1I, F + w2I are dissipative on H. (ii) We have sup E (I WA(t)I H + I F(WA(t))I H) < +oo. t>o

Existence of invariant measures

109

We now prove

Theorem 6.3.3 Assume that Hypotheses 5.4, 5.5 and 6.2 hold. Then there exists exactly one invariant measure p for (6.3.11, it is strongly mixing and for arbitrary v E M1(H), Pt *v -> it weakly as t --> +oo. Moreover there exists C > 0 such that, for any bounded Lipschitz continuous function 0 and all x E H I Pt4p(x)

Proof-

- (
Proceeding as in the previous theorem we consider prob-

lem (6.3.11) fort>s, sER. dX = (AX + F(X))dt + BdW(t), t > s

}

X(s)=x,

(6.3.12)

By Theorem 5.5.8 we know that there exists a unique generalized solution of (6.3.12), which we shall denote by X (t, s, x), t > s. We show now that there exists cl > 0 such that E(IX(t, -A, x)I) < C+I xl, for all A > 0, for all x E H, for all t > -A. (6.3.13)

For any A > 0 define WA,A(t) =

where

J

t S(t - s)BdW(s), a

is defined in (6.3.6). We first remark that

ZA(t) = X(t, -A, x) - WA,A(t), t > -A, is the mild solution of the problem dtZ=AZ+F(Z+WA,A),

Z(-A) = x, t > -A.

Chapter 6

110

It follows, denoting by xa t an element from the subdifferential of IZA(t)i, that dt IZa(t)I = (AZA(t) + F(Z,,(t) + WA,a(t))

-

F(WA,A(t)), xa,t) + (F(ZA(t)), xa,t)

<

-wIZ,,(t)I + IF(WA,A(t))I.

(6.3.14)

Consequently IZa(t)I <-

e--(t+A)IxI + ft e-"(t-s)IF(WA,A(s))Ids, t > -A,

-

a

and $ sup t] EIF(WA,A(s))I, t > -A. EIZA(t)I s Ixi + 1 W

-A,

Taking into account the definition of the process ZA and Hypothesis 6.2-(ii) one arrives at (6.3.13). In a similar way one shows that for all -A E t[

EIX(t, -A, x) - X(t, -7, x)I < e_W(t+-\)]EI X (-A, -,y, x) - xl <

e--(t+a)(2Ixl + C). (6.3.15)

Therefore there exists a random variable C, the same for all x E H, such that lim IEIX(0, -A, x) - (I = 0. (6.3.16) A-+-00

We claim that the law It = G(() is the unique invariant measure for Pt, t > 0. To see this it is enough to remark that, by (6.3.16), for arbitrary x E H,

A(x, ) = G(X(t, x))

= £(X(0, -t, x)) -> it, weakly as t -+ +oo.

Existence of invariant measures

111

Finally, let cp be a bounded Lipschitz function on H, then, by (6.3.15),

fors>t>0

P'
=

I E(V(X (t, 0, x)) - P(X (s, 0, x)))I

= IE(c'(X(0,-t,x)) - O(X(O,-3,x)))1 <

II'PIILip EI

<-

II'IILip

X(O, -t, x)) - X(0, -s, x)I

e--'(21x1 + C),

and the estimate follows.

Remark 6.3.4 Notice that assumptions of Theorem 6.3.3 are satisfied in particular if (i) A + wl is dissipative, (ii) F is Lipschitz continuous with constant W2 such that W1 +W2 = w > 0,

(iii) there exists an invariant measure for the linear system (6.3.11)

with B = 0. See also Remark 5.5.7. The following theorem is a variation of Theorem 6.3.3. It implies, however, only the existence of an invariant measure.

Theorem 6.3.5 Assume that the operators SK(t), t > 0, generated by AK are compact and that for an w > 0 IISK(t)II < e-wt, t > 0.

If in addition to Hypothesis 5.6 sup iE [IIWA(t)II + a(IIWA(t)II)] < +oo t>o

then there exists an invariant measure for the solution of (5.5.18) on K.

Chapter 6

112

Proof - Arguing as in the proof of Theorem 6.3.3 one obtains that II

Xt

x + JO - WAOtII <_ e-willII

t ew(t-s)a

(IIN'A(S)II) ds

and therefore sup

E [IIWA(t)iI + wa (II WA(t)II )]

EIIX(t)II <

< +c.

eup

Similarly as in §6.1 we will show that for arbitrary x E K the laws G(X(t, x)), t > 1, form a tight family of measures on K. This is certainly enough for the existence of an invariant measure for (5.5.18). For e E [0, 1] define an operator GE from C([0,1]; K) into K by the formula f 1 SK(u)c(u)du, (P E C([O, 1]; K). E

Since for E E ]0, 1]

SK(e) f SK(u - -)so(u)du, (P E C([0,1]; K), 1 E

and SK(e) is a compact operator, it is clear that GE is a compact operator for e E ]0, 1]. However, IIG°'-GECpII <_

sup IIc(u)II 10" IISx (v)II dv

uE ]0,1]

and consequently o the operator G° is also compact. Let us remark that for t > 1

X(t + 1,x) = SK(1)X(t,x) II

+

f1

SK(u)F(X(t + 1 - u, x))du + it

t

t+1

SK(t + (1 - u))dW(u)

= SK(1)X(t, x) + f SK(u)F(X(t 1 + (1 - u), x)) du + WA,t(1) 0

(6.3.17)

Existence of invariant measures where

t+u

WA,t(u) =

J it

113

S(t + u - r)BdW(r), u > 0.

-

However, compare the beginning of the proof,

IIX(t+u,x)-WA,t(u)II 5 e- JIX(t,x)II

+Ju e

"(u 9)a(IIWA,t(s)II)ds, u > 0,

an d therefore sup IIX(t+u,x)II <- IIX(t,x)II

uE[0,11

+ sup IIWA,t(u)ll+ sup a(IIWA,t(u)II). uE[0,1]

uE[0,11

But supt>o]EIIX(t,x)II < -boo and consequently for arbitrary e > 0 there exists C > 0 such that JF(IIX(t,x)lI > C) < E, for all t > 0.

Since the process has K-continuous trajectories and all the processes have the same distributions as the constant C > 0 can be chosen to give also

P

(

sup IIWA,t(u)II > C < E,

/

uE[0,11

]P

sup a(IIWA,t(u)II) >- C < E. uE[0,11

But then

]P( sup IIX(t+n,x)II >3C) uE[0,1]

+lP

P(IIX(t,x)II >C)

(uE[0,11 sup IIWA,t(u)II ? C +P ( sup a (II WA,t(u)II) > C uE[0,11

3r,

Chapter 6

114

for all t > 0. Taking into account this estimate, the identity (6.3.17), andthe compactness of the operators SK(1) and G°, we easily obtain that the family ,C(X(t + 1, x)), t > 0, is tight, as required. By Theorem 5.5.11 and Theorem 6.3.5 we get the following result.

Theorem 6.3.6 If in addition to the assumptions of Theorem 6.3.5 there exists w E R such that the mapping F - w is dissipative in H then there exists an invariant measure for equation (6.3.11) which in addition is concentrated on K.

6.4

Genuinely dissipative systems

We assume here that Hypotheses 5.4, 5.5 and 6.2 hold and moreover the following.

Hypothesis 6.3 (i) S(.) is of negative type. (ii) There exist r > 1 and C,. > 0 such that (F(x) - F(y), x - y) < -Cr I x - yI i+r. Hypothesis 6.3 implies not only that our system is strongly dissipative, but in addition that the nonlinear part is superlinear. In finite dimensions such systems have been considered by S. Cerrai [18]. Under Hypothesis 6.3 we will be able to prove that Ptcp(x) -+ fH W(x) t(dx) for all W E Lip(H) uniformly on x. Moreover, we show that the convergence of PtW(x) holds for all W E Bb(H), as t --). +oo, under the additional hypothesis following.

Hypothesis 6.4 For any cp E Bb(H) and t > 0 there exists Kt > 0 such that Ptcp is Lipschitz continuous and I Pt9 (x)

-

S Ktlx - yI

for all x, y E H.

Theorem 6.4.1 Assume that Hypotheses 5.4, 5.5, 6.2 and 6.3 hold and let p be the invariant measure for (6.3.11) given by Theorem

Existence of invariant measures

115

6.3.3. Then there exists Cl > 0 such that, for any bounded Lipschitz continuous function co, all t > 0 and all x E H IAW(x) - (sO,A) 1 <- C1e-"t/2IIwIILip.

(6.4.1)

If in addition Hypothesis 6.4 holds, then there exists C2 > 0 such that C2e-"t/2IIWIIo,

I

for all cp E Bb(H).

Proof-

Proceeding as in the proof of Theorem 6.3.3 we consider the unique generalized solution of (6.3.12), which we shall denote by X(t, s, x). We set

Z,\,.y(t) = X (t, -A, x) - X (t, -y, x), 0 < A < y. Then ZA,,y is the mild solution of the problem

d

Z,\,,y = AZa,.y + F(X (t, -A, x)) - F(X (t, -y, x)), ZA,y(-A) = x - X(-A,-y,x), t > -A

J

It follows, denoting by xa y t an element from the subdifferential of IZ,,.y(t)I, that dt IZA,y(t)I

<

(AZA,7(t) + F(X(t, -A, x)) - F(X(t, --r, x)), xa,y,t)

<

-'IZA,y(t)I - CrI Za,y(t)I'. (6.4.2)

By a well known comparison theorem it follows that ZA,y(t)I 5 77(t, X),

where rl is the solution to the initial value problem

r'(t) _ -wi(t)

- CTI 77(t)I', t > -A,

n(o) = Ix - X(-A,-'Y,x)l.

Chapter 6

116

By proceeding as in S. Cerrai [18], one can show that there exists a constant C2(t), independent of x, such that ]GI X (t, -A, x) - X (t, -7, x)I

< e-w(t+a)EIX (-.A, --y, x) - xI

< C3e-w(t+A), t > -A. (6.4.3)

Let now S be the random variable such that (6.3.16) holds, and let cp be a bounded Lipschitz function on H, then, by (6.4.3) we easily find (6.4.1).

Let us prove the last statement of the theorem. Let E B6(H) and let e > 0. Then, setting cp = P' in (6.4.1), we have, by Hypothesis 6.4, and by the invariance of p, IPt+eY'(x)

- (PeY',Ft)I

<

Cle-wt/2IIPeWIILip

<

C1Kee-wt/2II4'IIO.

It follows that IPtO(x)

-

xeCle-w(t-e)l2IIbIIo,

for all L' E C&(H) and for allt>r. Since IPt'i'(x)

- (',µ)I

<- 2II'bIIo,

the proof is complete.

We conclude this section by providing a sufficient condition for to Hypothesis 6.4 hold. We follow here G. Da Prato, D. Elworthy and J. Zabczyk [37].

Theorem 6.4.2 Assume that Hypotheses 5.4, 5.5 hold, and that B = I. Then Hypothesis 6.4 is fulfilled.

Proof - By Theorem 5.5.8 the equation dX = (AX + F(X))dt + dW(t), (6.4.4)

X(0) = x,

Existence of invariant measures

117

has a unique generalized solution X(-, x). Moreover X (t, x) is the L2 limit of the solution XX(t, x) of the problem

dXa = (AX,,, + Fa(Xc))dt + dW(t), (6.4.5)

X (0) = x, where Fa are the Yosida approximations of F. Since Fa are Lipschitz continuous we can apply Theorem 7.1.1 in next chapter, and conclude that for any T > 0 there exists a constant CT, independent of a, such

that IPt (p(x)

- Pt

Y1, x, y E H,

(6.4.6)

for any cp E Bb (H ), where

Pt,p(x) = Ecp(Xa(t, x)), t > 0, a > 0,

Remark 6.4.3 Differential stochastic equations in RI with polynomial drift have been studied by S. Kusuoka and D. W. Stroock [98], when the drift term is the gradient of a potential plus a Lipschitz continuous function.

6.5

Dissipative systems in Banach spaces

We are concerned with existence and uniqueness of invariant measures for the problem

dX = (AX + F(X))dt + dW(t), (6.5.1)

X (O) = x,

in a Banach space K (norm in H.

) continuously and densely embedded

Chapter 6

118

Theorem 6.5.1 Assume that Hypothesis 5.7 holds with w + rt = -b < 0 and that sup]E(IIWA(t)II) < +00. t>o

Then there exists exactly one invariant measure µ for (6.5.1), it is strongly mixing and for arbitrary v E M1(K), Pt *v -> µ weakly as t -> +oo. Moreover there exists C > 0 such that, for any bounded Lipschitz continuous function 'p, all t > 0 and all x E E, I Ptc (x) - (
(C + 2IIxII)e-sthl2II,IILjp.

Proof - As in the proof of Theorem 6.3.3 we consider problem (6.5.1) for t > -A, A > 0, dX = (AX + F(X))dt + BdW(t), t > -A, (6.5.2)

X(-A)=x, By Theorem 5.5.13 we know that there exists a unique generalized solution of (6.5.2), which we shall denote by X (t, -A, x), t > -A. We show now that there exists cl > 0 such that

]E(IIX(t,-A,x)II) 5 C+11x4, for all A > 0, for all x E H,for all t > -A. (6.5.3)

We first remark that ZA(t) = X(t, -A, x) - WA,A(t), t > -A, is the mild solution of the problem dt Z = AZ + F(Z + WA,A),

Z(-A)=x, t> -A

l J

It follows, recalling that F is dissipative, that there exists xa t

E

BIIZA(t)II such that dt IIZ),(t)II

= (AZA(t) + F(ZA(t) + WA,a(t)), xa,t) (F(WA,A(t)), xa t) + (F(ZA(t)), xa,t)

<

-wIIZA(t)II + II F(WA, (t))II

(6.5.4)

Existence of invariant measures

119

Consequently IIZA(t)II <- e-`"fit+A)IIxII +

J to

e ` (t-s)II F(WA,A(s))I ids, t > -A,

and

EIIZA(t)II <- Ilxll +

sup EIIF(WA,A(s))II, t > -A.

Taking into account the hypothesis imposed one arrives at (6.3.13). Now the proof is completely similar to that of Theorem 6.3.3.

Chapter 7

Uniqueness of invariant measures We are still concerned with invariant measures of the transition semi-

group Pt, t > 0, associated with problem (5.3.1). As we know from Theorem 4.2.1, uniqueness of invariant measure is a consequence of regularity. Moreover regularity of Pt, t > 0, follows from the strong Feller property and irreducibility for some t > 0, see Proposition 4.1.1. We will present here methods, applicable in several important cases, which imply the strong Feller property and irreducibility for Pt, t _> 0. We do not present the most general results because this would complicate the formulation, but restrict to consider typical situations. In several applications in Part III, some modifications of the arguments used here will be needed. We start from the strong Feller property.

7.1

Strong Feller property for non-degenerate diffusions

We are concerned with the problem

dX(t) = (AX + F(X))dt + B(X)dW(t),

X(0)=xEH. 121

Chapter 7

122

In all this section we will assume that U = H. We will need the following hypothesis, stronger than Hypothesis 5.1.

Hypothesis 7.1 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H. (ii) F is a mapping from H into H and there exists a constant co > 0 such that I F(x)I <- co(1 + Ixl), x E H,

IF(x) - F(y)I C colx - yl, x,y E H. (iii) B is a strongly continuous mapping from H into L(H) and there exists a constant K > 0 such that JIB(x)II <- (1 + Ixi), x E H,

JIB(x) - B(y)II < KI x - yI, t > 0, x, y E H. Moreover, for all z E H, B(z) is invertible and

IIB-1(z)II 0, and Jot II S(t) II HS dt < +oo.

We shall denote, as usual, by X(., x) the mild solution to (7.1.1) and by Pt, t > 0, the corresponding transition semigroup . The following theorem is taken from S. Peszat and J. Zabczyk [124].

Theorem 7.1.1 Assume that Hypothesis 7.1 holds. Then for any T > 0 there exists a constant CT > 0 such that for all 0 E Bb(H) and t E [0,T] I PtI(x) - PtO(y)I 5

II'IIo Ix - yI, x, y E H.

(7.1.2)

VIF

In particular Pt, t > 0, is a strong Feller semigroup for all t > 0.

Uniqueness of invariant measures

123

Proof - The proof will consist of two parts. In the first part we prove the desired estimate under the additional hypothesis that F, B and 0 are regular. In the second part we show how to dispense with that assumption.

Part 1

Here we assume that '0 E C6 (H), F E Cb (H; H) and

.

B E Cb (H; L(H)). We have divided this part into a sequence of lemmas.

Lemma 7.1.2 Assume that F , B and

are twice differentiable functions with bounded and continuous derivatives up to the second order. Then

1G(X (t, x)) = Pti(x) (7.1.3)

t

+ f (D.PP_8 b(X(s,x)),B(X(s,x))dW(s)), P-a.s.. 0

Proof - Let {en} be a complete orthonormal system in H. For each n let X,, (., x) be the solution of the problem

dXn(t) = (AnXn + F(Xn))dt + B(X,,,)QndW(t),

Xn(0) = x, where An = nA(n - A)-1 is the Yosida approximation of A and Qn is the orthogonal projection of H onto lin{e1, ..., en}.

It follows from Theorem 5.4.2 that the function

vn(t,x) = E(b(Xn(t,x))), (t,x) E ]0,+00 [ xH, is a strict solution to the Kolmogorov equation

d vn(t, x) =

2 Tr [B*(x)QnB(x)vyy(t, x)]

+ (Anx + F(x), vx(t, x)), vn(0, x)

= fi(x), x E H, t > 0.

Chapter 7

124

Applying the Ito formula, see e.g. G. Da Prato and J. Zabczyk [44], to the process vn(t - s, Xn(s, x)), s E [0, t], we obtain that ?/I(Xn(t, x)) = vn(t, x)

+ j(v(t - S, Xn(S, X)), B(Xn(S, x))QndW S

P-a.s..

Letting n -i +oo gives the desired result. Now we extend the finite-dimensional Elworthy formula, see D. Elworthy [61].

Lemma 7.1.3 Assume that F, B and b are twice differentiable functions with bounded and continuous derivatives up to the second order. Then the directional derivatives (D,,Ptb(x), h) are given by (D.Ptb(x), h) tE{(X(t,x)) j (B1(X(s, x))X(s, x)h, dW(s))}, P- -a. S.

=

(7.1.4)

Proof -

Fix h E H and set v(t,x) = Ptb(x), t _> 0, x E H.

Multiplying both sides of (7.1.3) by

j(B_1(X(s, x))Xx(s, x)h, dW(s)), and taking the expectation we get E

((x(t,x))J t(B-1(X(s,x))XX(s,x)h,dW(s))) o

_Ef

(B*(X (s, x))v,(t - s, X(s, x)), B(X(s, x))Xx(s, x)h)ds =

=

Ito (DxE(D,,Pt-s,O(X (s, x))), h) ds It

(V, (t - s, X(s, x)), h) ds =

= t(DxPtb(x), h),

J t (DxPtO(x), h) ds

Uniqueness of invariant measures

125

which yields (7.1.4).

Lemma 7.1.4 Assume that F, B and 0 are twice differentiable functions with bounded and continuous derivatives up to the second order. Then the estimate (7.1.2) holds true.

Proof - Fix T > 0, then by (7.1.4) we have I

(D.,PtO(x), h)j2

1

< t II0II2 E {

JOIB-1(X(s, x))Xx(s, x)h12ds}

and the conclusion follows from (5.4.2).

Part 2 We first show that if (7.1.2) holds for 0 E Cb (H) then it holds for all 0 E Bb(H). This follows from the lemma

Lemma 7.1.5 Let Pt, t > 0, be a Markov semigroup on Bb(H) and let c > 0 and t > 0 be fixed. Then the following conditions are equivalent.

(i) for all cp E Cb (H), for all x, y E H, I Ptcp(x) - Ptcp(y) < cII(pIIoIX - Y1.

(ii) for all cp E Bb (H), for all x, y E H, Ptyo(x) - Pt
I.

(iii) for all x, y E H, Var (Pt(x, ) - Pt(y, )) <

yI.

Proof - Let K1= {

11,

and II(pIIo<1}.

Since each bounded continuous function on H may be approximated pointwise by functions of Cb (H) we have sup VEK1

I

sup I Pw(x) PEr2

- Pts0(y)I

Chapter 7

126

for all x, y E H. As a simple consequence of the Hahn decomposition theorem, we have

sup I PtW(x) - Ptco(y)I = Var (Pt(x, ) - Pt(y, )) WE/CI

Therefore (i) implies (iii). However, if (iii) holds then for all cp E Bb(H) I Pty (x) -

Pt*y)I I

JH

p(z)(Pt(x, dz) - Pt (y, dz))

S

II'PIIo Var (Pt (x, ) - Pt (y, ))

<-

IIPIIoIx - yI,

and we see that (ii) is true. Obviously (ii) implies (i) and the proof of the lemma is complete. Therefore, to prove the theorem in the general case, we can assume that So E Cb (H). We will now construct approximations Fn of F and B,,, of B with the following properties. (i) Fn, Bn, n E N, are twice Frechet differentiable with bounded and continuous derivatives.

(ii) Fn, Bn, n E N, satisfy the Lipschitz condition with the constant c. (iii) for all K > K, 3 no E N such that the operators Bn(z), z E H, are invertible and sup II Bn 1(z)II < K, n > no. zEH

IIB(z) - Bn(z)II = 0. Before giving the construction we will show how to finish the proof of the theorem. Let x) be the solution of the equation (iv)

IF(z) - Fn(z)I = 0 and limn_.+

dXn(t) = (AnXn + F(Xn))dt + B(Xn)QndW(t), Xn(0) = x,

Uniqueness of invariant measures

127

and let Pn,t, t > 0, be the corresponding transition semigroup. Fix t > 0, E Cb (H) and a number Ft greater than ct. Lemma 7.1.5 applied to Pn,t, t > 0, shows that Ft4IbIIolx - yI,x,y E H.

IPn,tY&(x) - Pn,ty'(y)I

However, for all t > 0 and x E H there exists a sub-sequence IX,,,} such that Xn, (t, x) - X (t, x) a.s. as j -,. +oo. Since 0 is a bounded continuous function we have Pn,t4'(x) = ]E (4'(X.; (t, x))) - IE(O(X (t, x))),

as j -* +oo, and we have I Pt4(x) - Pt4(y)i s ctli4'Ilolx - yI, x, y E H,

which proves the theorem.

We return now to the construction of the sequences {Fn} and {B, }. We take a sequence of non-negative, twice differentiable functions {p,,} such that

supp IM C j

E

``

11

IIn < nJ

and

fin

1.

Identifying 1Rn with lin{el,..., en} we define

Fn(x) =

f p(

n

- Qnx)F E Siei

an

-

Bn(x)

i=1

d,

n

f n Pn( - Qnx)B

icl

biei

<.

Observe that Fn and Bn are twice Frechet differentiable functions with bounded and continuous derivatives. Moreover, for all x, y E H

Chapter 7

128

and n E N IFn(x) - FF(y)I

n

f Pn)

n

+ Qnx i=1

i=1

< cI Qn(x - y)I fin Pn(S) < CI X - yl.

In the same manner we can see that IBn(x) - Bn(y)I < Clx - yl. We next prove that, for sufficiently large n, the operators Bn, n E N, are invertible and that

limsup sup IIBn1(z)IIJ < K. n- +oo

zEH

To this end observe that the operators B(Qnz), z E H, n E N, are invertible, sup sup II B-1QnzII < sup 1IB-1(z)II nENzEH

zEH


and

sup II

B(Qnz) - Bn(z)ll

zEH

= sup zEH

f Ilf

n

[B J:Siei + Qnx - B

n

< C fin P. (

n

i=1

)

d<

Siei + Qny d ¢=1

C n.

Uniqueness of invariant measures

129

Consequently for sufficiently large n the operators B, n E N, are invertible and

lim sup sup l1Bn'(z)II

n-+oo zEH

= lim sup [zEH II [B(Qnz) - (B(Qnz) - B(z))]-' II, n-r+oo

< limsup sup II[B-'(Qnz)II n-r+oo

zEH

< K lim sup c* 1:

JIB-'(Qnz)(B(Qnz) - B(z))II j=O

(Kcn-1)j

n--r+oo j=0

= K.

It is also easy to check that (iv) holds.

Remark 7.1.6 The strong Feller property for equation (7.1.1) with f only bounded and continuous has been obtained by D. G4tarek and B. Goldys [77] by using the concept of martingale solutions, see G. Da Prato and J. Zabczyk [44].

7.2

Strong Feller property for degenerate diffusion

The invertibility Hypothesis 7.1(iii) may not be satified but, nevertheless the semigroup Pt, t > 0, corresponding to (5.2.1) can be a strong Feller one. We will examine this possibility for equations with constant diffusion coefficient:

dX(t) _ (AX + F(X))dt + BdW(t), (7.2.1)

X(0) = X.

We again assume a stronger hypothesis than Hypothesis 5.1.

Chapter 7

130

Hypothesis 7.2 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H. (ii) F is a mapping from H into H and there exists a constant co > 0 such that IF(x)l < co(1 + jxj), x E H,

IF(x) - F(y)I S coax - yl, x,y E H. (iii) B is a linear continuous mapping from U into H. (iv) For any t > 0 the linear operator Qt, Q tx

=

j

S(s)QS*(s)xds, x E H,

(7.2.2)

where Q = BB*, is of trace class. In some cases we shall assume further

Hypothesis 7.3 Image S(t) C Image Qt12 Hypothesis 7.3 is equivalent to the fact that the deterministic system

'(t) = Ae(t) + Bu(t) is null controllable, see e.g. G. Da Prato and J. Zabczyk [44, §B.3]. If Hypothesis 7.3 holds we denote by r(t) the linear bounded operator

['(t) = Qt 112S(t), t > 0. We start with the linear case,

dZ(t) = AZdt + BdW(t), (7.2.3)

Z(0) = X.

We remark that, for all t > 0, X (t) is a Gaussian random variable, H(S(t)x, Qt), that is with mean S(t)x and covariance Qt. We shall denote by Pt, t > 0, and Rt, t > 0, the transition semigroups corresponding to problems (7.2.1) and (7.2.3) respectively: A(P(x) = Ecp(X (t, x)), t > 0, cp E Cb(H), RtSo(x) = Ew(Z(t, x)), t > 0, yP E C6(H). We have the following basic result.

Uniqueness of invariant measures

131

Theorem 7.2.1 Let Rt, t > 0, be the semigroup corresponding to (7.2.1), and let t > 0. Then the following conditions are equivalent.

(i) Rt, t > 0, is t-regular. (ii) Rt, t > 0, is a strong Feller semigroup at time t. (iii) Hypothesis 7.3 holds.

Proof - Since transition probabilities are Gaussian Rt(x, ) = N(S(t)x, Qt), the equivalence (i)b(iii) is an immediate consequence of the CameronMartin formula.

To show that (ii) b (iii) assume that (ii) holds. If (iii) is not true then for some 7 E H, S(t)x # Im Qt/2. From the linearity S(t) (n x)

Image Qt/2, n E N. Since all measures Rt

(n,

)

are singular with respect to Rt(0, ) there exists a set F C 8(H) such that

,r=0, nEN, Rt(nT) and Rt(0, r) = 1. But this implies that the function RtXr is not continuous at 0. So (ii) =(iii). To show that (iii) i(ii), we first recall that, by the CameronMartin formula: dA((S(t)x, Qt) (z) = d(t, x, z), dN(O, Qt)

where

d(t,x,z) = exp (_tQh/'2s(t)xf2 + (Qt 1/2z,Qt 1/2 x)) eXp

(_Ir(t)xI2 + (r(t)x, Qt

1/2z))

132

Setting v(t, x) = Rtcp(x) it follows that

v(t, x) = IH)(5(t)x,Qt)) = JH cp(z)d(t, x, z).N(O, Qt)(dz). Set

N

exp

fN(x, z)

I

2

k=1

I(S(t)x,ek(t))I2 Ak(t)

N

X

(z, ek(t)) E (S(t)x, ek(t)) exp (k=1 Ak(t)

where {ek(t)} and {Ak(t)} are eigensequences relative to Qt. We recall that li

fN(, z)

m

d(t, , z),

for any t > O, z E H, in Lv(H, B(H), A((O, Qt)) and almost surely. If h E H, we have,

(d dx fN(x, z), h) = -d(t, x, z)

N (S(t)x, ek(t)) (S(t)h, ek(t)) k.1

Ak(t)

+d(t, x, z) N (S(t)h, ek(t)) (z, ek(t)) k=1

and it is easy to check that Nu oo(dx

in LP(H,8(H),N(O,Qj)).

fN(x,

z), h) _

Ak(t)

Uniqueness of invariant measures

133

By the dominate convergence theorem, it follows that (vx(t, x), h)

= fH W(z)d(t, x, z)(Qt 1/2z - r(t)x, r(t)h)N(0, Qt)(dz) =

cp(z)(Qt 1/2z

JH

- r(t)x, r(t)h)H(S(t)x, Qt)(dz).

Consequently the directional derivative of v(t, x) at x in direction does exist and is given by the formula (vs(t, x), ) _ J (r(t)e, Qt 112z)4P(S(t)x + z)w(O, Qt)(dz). (7.2.4) H

It follows from (7.2.4) that for arbitrary x, y E H and some s E [0, 1] I v(t, x) - v(t, y)I = I (vX(t, x + s(y - X))' X - y)I

H I (F(t)(x - y),Qt 1/2z)I I So(S(t)(x + s(y - x)) + z)IN(O,Qt)(dz) Ii

iio 11H I (r(t)(x - y), Qt1/2) I V/(O, Qt)(dz) 1/2

kP110 (IH I (r(t)(x - y),Qt1/2x)12 N(O,Qt)(dz)) < II0I0I!r(t)II 1(x - y)1

This proves (ii).

Remark 7.2.2 For general Markovian semigroups regularity is a consequence of the strong Feller property and irreducibility, but for the Gaussian case irreducibility is irrelevant. Take for instance H = L2(0,1) and S(t), t > 0, the shift semigroup:

S(t)x(e) =

x( + t) if E [0,1 - t], 0

ifE]1-t,l].

Chapter 7

134

Then Image S(t) = {0} for t > 1 and condition (iii) is satisfied for an arbitrary operator Q of trace class. If Q = 0 then certainly the transition semigroup Pt, t > 0, is not irreducible, but it is t-regular for t > 1. It is, however, an easy consequence of (iii) that t-regularity for some t > 0 implies regularity for arbitrary t > 0.

Before going to the nonlinear case let us remark that, although tregularity is a sufficient condition for the uniqueness of invariant measure it is not a necessary one. This is clearly shown by the following special case of Theorem 6.2.1.

Proposition 7.2.3 Assume that there exists an invariant measure for (7.2.3). Then any invariant measure for (7.2.3) is of the form

v*N(O,F), where v is invariant for S(t), t > 0, and

Fx = f

S(t)QS*(t)xdt, x E H.

0

We continue now our study of the strong Feller property for equation (7.2.1).

Theorem 7.2.4 Assume that Hypotheses 7.2 and 7.3 hold, and that IIr(-)II belongs to LioC(0,+oo), where r(t) = Qt1"2S(t). Then for arbitrary cp E Bb(H) and x, y E H

Pt a(y)I s e(t)Ix - yl, t > 0, where

(7.2.5)

is the solution to the integral equation,

t () t= - II I ()II

II

F Ilo

t

sds. s IItt(-)III()

In particular Pt, t > 0, is a strong Feller semigroup.

Proof - Denote by D(Ao) the set of all So such that (i) cp E Cb (H),

(ii) Wx(x) E D(A*) for all x E H and the mapping H - R, x --> (x, A*(px),

(..s) (7.2

Uniqueness of invariant measures

135

belongs to Cb(H). (iii)

W. E Cb(H,Li(H))

Now define a linear operator .Ao on Cb(H) by setting Aoco

= 2 Tr [Q(pxx] + (x, A*(p.),

D(Ao) _

{lp E Cb(H) : lpxx E Li(H), S11prEH IIVxxIILI(H) < +00,

and there exists 0 E CC(H) such that cp(x) = O(A-1x)

and the mapping x --+ (x, 0,,(A-lx)) is in Cb(H)}. It has been shown in S. Cerrai and F. Gozzi [20] that the set D(Ao) is dense in Cb(H) in the following sense: For arbitrary cp E C6 (H) there exists a sequence {cpn} C D(Ao) such that (1) IIPn1Io < 2II
(ii) cpn, -+ cp uniformly on any compact subset of H.

Moreover Rt(D(Ao)) C D(Ao) and AORt = RtAo on D(Ao), for arbitrary t > 0. Assume first that F is twice differentiable with bounded and continuous derivatives up to the second order and that cp E D(Ao). Then the function v(t,x) = Ptcp(x), t > 0, x E H,

is the strict solution of the Kolmogorov equation:

vt(t, x) = 2 Tr [Qvxx(t, x)] + (x, A*vx(t, x)) (7.2.7)

+ (F(x), vx(t, x)), t > 0, which can be written equivalently as dv

d = Aov(t)

+.Fv(t),

v(0)=yc, t>0.

7.2.8)

J

In (7.2.8) Ao is the operator defined above and

Fb(x) = (F(x), ix), 0 E Cb (H), x E H.

Chapter 7

136

For fixed t > 0 consider the function Rt_3(v(s)), s E [0,T]. Then, for s E [0, T], TS

[Rt-s(v(s))] = -Ao [Rt-3(v(s))] + Rt-s

\ dt (s)/

= -Ao [Rt-s(v(s))] + Rt_3 [Aov(s) +.Fv(s)]. Since the operators Ao and R3 commute we obtain that ds

[Rt-s(v(s))] = Rt_s F, s E [0, t].

(7.2.9)

Integrating (7.2.9) over the interval [0, t] one obtains that t

v(t) = Rtcp + f Rt_3(Fv(s))ds, t > 0.

(7.2.10)

o

The equation (7.2.10) is called a mild Kolmogorov equation. It fol-

lows that, for arbitrary 0 E C6(H) and t > 0, Rt'/ is a function bounded and continuous together with its first derivative and Ilr(t)ll Iloilo.

I

(7.2.11)

Note that Jot

IlFv(s)Ilods

<

f t IIr(t - s)II IIFv(s)llods t

IJF(t - s)II

Under our conditions on cp and F the function IID,v(s)Ilo, s > 0, is locally bounded. Taking into account (7.2.11) and taking derivatives of both sides of (7.2.10), we arrive at the inequality IID.v(t)Ilo S Ilr(t)II II 0,

valid for all t > 0. Since

is locally bounded by a generalization of Gron-

wall's lemma, II Dxv(t)Il o < fi(t), t > 0,

(7.2.12)

and (7.2.5) follows for all cp E Cb(H). The generalization to arbitrary Borel funtions
Uniqueness of invariant measures

7.3

137

Irreducibility for non-degenerate diffusions

Irreducibility, as a rule, requires some kind of non-degeneracy of the diffusion term. We divide the material into two sections. In the present section we consider the case when the mappings B have bounded inverses; next section will be devoted to the case when B is constant, but not necessarily invertible. The main result of this section is the following.

Theorem 7.3.1 Assume besides Hypothesis 5.1 that F is bounded and the mappings B(x), X E H are invertible, and sup (JIB(x)ll xEH

+ 1IB-1(x)ll) < +oo.

Then the transition semigroup Pt, t > 0, corresponding to problem (7.1.1) is irreducible.

Proof - We first prove, in the following lemma, that irreducibility of Pt, t > 0, is equivalent to irreducibility of the system

dZ = (AZ + cp(t))dt + B(Z(t))dW(t), t > 0, (7.3.1)

Z(0) = x, where cp is any bounded and adapted process. Then, by choosing cp in a suitable way, we prove that system (7.3.1) is irreducible.

Lemma 7.3.2 Assume, besides the hypotheses on A, B of Theorem 7.3.1, that the process cp is bounded. Then the laws in (H,a(H)) of X(t, x) and Z(t, x) are equivalent.

Proof- Let a(t) = B-1(Z(t))[cp(t)-F(Z(t))]. Then a is bounded and adapted, so that the equation

dM(t) = -(a(t), dW(t))M(t), t > 0, M(0) = 1,

has a unique solution M() and E(M(t)) = 1, t > 0. Now fix T > 0 and set dP*(w) = M(T)dP(w).

Chapter 7

138

Then P* is a probability on (0,Y) and therefore, by the Girsanov theorem (see e.g. G. Da Prato and J. Zabczyk [44]), the process

W*t- Wt

t

is a cylindrical Wiener process in H defined on the probability space (9,.F, I-). Note that P and P* are equivalent. Now ft

Z(t) = S(t)x +

ft

o

0 ft

= S(t)x +

S(t - s)B(Z(s))dW(s)

S(t - s)yp(s)ds + ft

S(t - s)B(Z(s))dW*(s).

S(t - s)F(Z(s))ds +

Consequently Z is the solution of (5.1.1) on the probability space (1, )'*, P*). Since the law of the solution does not depend on the particular choice of the probability space, for every Borel set r C H we have P(X (t, x) E r) = P*(Z(t, x) E r). Since P and P* are equivalent the laws of X (t, x) and Z(t) are also equivalent which is our claim.

We can now proceed to the proof of Theorem 7.3.1. We fix to > 0, r > 0, and x, a E H. We will choose a bounded process cp such that, if Z is the corresponding solution of (7.3.1), we have P(IZ(to) - al < r) > 0,

(7.3.2)

and we will apply Lemma 7.3.2. Let us consider an auxiliary system

dY = AYdt + B(Y)dW(t), Y(0) = x, which clearly has a unique solution Y. Let ti E [0, to[ be a moment to be chosen later. Now we choose V by setting 0 if t E [0, ti [, (P(t) =

f(t,Y(tl))

if t E [tl,to],

Uniqueness of invariant measures

139

where f : [ti, to] x H --r H is bounded, f (t, ) is Lipschitz, and moreover 0

if IyI > 2R, s E [ti, to],

f(s) y) = to i-t,

w. - t i )(a - y)

- A;;

- ,i,o ,

if y I < R s E [t t I

]

where R > 0 and a E D(A) will also be chosen later. We remark that, as is easily checked, the following implication holds: y E H, IyI <_ R

S(to - ti)y+

to

itt l

S(to - s).f(s,y)ds = a. (7.3.3)

We set moreover

Z(to) = Il + 12, where to

Ii = S(to - ti)Y(ti) + ft S(to - s).f (s, Y(ti ))ds, , and

to

I2 = Jt, S(to - s)B(Z(s))dW(s). Now we choose a E D(A) such that T la - aI < 3,

and claim that ]P(IZ(to) - al < r) _ P(I (Ii - a) + (I2 + a - a)I < r)

> ]P(Ii=a&II2I<-s). In fact if Ii = a and 1121 < s we have

I(Ii -a)+(12+a-a)I < 1121+Iii -al < 3r. Thus the claim is proved and we have

P(IZ(to)-aI P(Ii =a)-?(1121

?

3).

(7.3.4)

Chapter 7

140

We now choose tl E [0, to[ such that

l P

3

<

(7.3.5)

4'

and R > 0 such that

?(IY(ti)l < R) > 4.

(7.3.6)

R we have Il = a. Consequently,

By (7.3.3) it follows that if jY(tl)J b y (7.3.6)

?(Ii = a) ? P(IY(tl)I

R) >

-.

(7.3.7)

By (7.3.4), (7.3.5), and (7.3.7) it follows finally that

P(Iz(to) - al < r) > P(I1 = a) - ?(I2 > 3) >

3

1

-4 4

_

1

2

The proof is complete

Remark 7.3.3 Irreducibility for equation (7.1.1) for a much larger class of f including polynomial nonlinearities, has been obtained by D. G4tarek and B. Goldys [77] by using the concept of martingale solutions, see G. Da Prato and J. Zabczyk [44].

7.4

Irreducibility for equations with additive noise

Here we are concerned with nonlinear systems of the form

dX = (AX + F(X))dt + Q1/2dW(t), (7.4.1)

We assume that U = H, that B = Q1/2 and that the deterministic system

y'(t) = Ay(t) + F(y(t)) + Q1/2u(t), (7.4.2) y(0) = x,

Uniqueness of invariant measures

141

is approximately controllable. We denote by y(-, x; u) the solution of (7.4.2). We recall that the system (7.4.2) is approximately control-

lable in time T > 0 if, for arbitrary x, z E H and e > 0, there exists u E L'(0, T; H) such that Iy(T, x; u) - xj < e. Note that Hypothesis 5.1(iii) implies continuity of the solution Z(.) of the linear equation dZ = AZdt + Q112dW(t), (7.4.3)

Z(0) = 0. Theorem 7.4.1 Assume that the operatorA generates aCo -semigroup S(t), t > 0, F is a locally Lipschitz mapping, and that the deterministic system (7.4.2) is approximately controllable in time T > 0. Then the transition semigroup Pt, t > 0, corresponding to problem (7-4. 1) is irreducible in time T.

Proof - We first remark that the law £(Z(.)) of the process Z is a Gaussian measure on L2(0,T;H) with a covariance operator Q given by T

Qcp(t) = f g(t,s)cp(s)ds, P E L2(0,T:H); where

g(t, $)x =

J0

S(t - r)QS*(s - r)sdr. x E H.

JO

Moreover

t

Image Q112 = {v E L2(0,T; H) : v(t) = f S(t - s)Q1/2u(s)ds, 0

t E [0,T], u E L2(0,T;H)}. Since £(Z(.)) is concentrated on Co = {u E C([0, T]; H) : u(0) = 0}, the closure of Image Q1/2 in the Co-norm is the support of £(Z(.)), see G. Da Prato and J. Zabczyk [44, page 41]. Now fix x and z in H, and e > 0. We want to prove that

?(IX(T,x) - zj < e) > 0.

(7.4.4)

Chapter 7

142

We first remark that, by the approximate controllability assumption, there exists u E L2(0, T; H) such that l y(T, x; u) - zI < e.

Take R such that I y(s, x; u) I <

R 2

for all s E [0, T],

,

and let LR be the Lipschitz constant of F restricted to the ball {z E H : IzI _< R}. Fix e E ]0, 1[. Since the support of £(Z(.)) is the closure of ImageQh/2, we have, with positive probability, sup sE[O,T]

Z(s) -

J0

S(s - a)Q1 j2u(v)dv < e 2 e-LRMT

(7.4.5)

where M = suptE[o,]q IIS(t)II. We claim that if (7.4.5) holds, then we have sup IX(s,x)I < R (7.4.6) $E[0,T]

and

sup I X(s, x) - y(s, x; u)I < e 2 .

(7.4.7)

sE[O,T]

This will imply the result. Assume by contradiction that (7.4.5) holds but sup IX(s,x)l > R.

(7.4.8)

sE[0,T]

Let

T1 = inf{t > 0 : IX(t,x)I > R}. Then T1 < T and for all t E [0, T1]

IX(t,x)I < R, Iy(t,x; u) I <

R

Note that for t E [0, T]

X(t, x) - y(t, x; u) =

J0

t S(t

- s)(F(X(s, x) - F(y(s, x; u))))ds

/t + Z(t) - J S(t - s)Q112u(s)ds. 0

Uniqueness of invariant measures

143

Consequently

< LRM

IX (t, x) - y(t, x; u)l

-

Jo t JX (s, x) - y(s, x; u)lds

+

sup Z(s) - fo S (s - o)Q1j2u(o)do)

8

I

.

sE[O,t]

/' s

JX (t, x)

- y(t, x; u)I < eLRMT sup Z(s) - J S(s - o)Q1j2u(o)do, sE[0,t]

0

In particular, for all t < Ti I X (t, x)I

2

+ eMLRT

Re-MLRTe

< R.

On the other hand, IX(T1ix)l > R, a contradiction. Thus T1 = T. By similar calculations, for all t < T X (t, x)

- y(t, x; u)I < e 2

This finishes the proof.

Theorem 7.4.2 Assume that the operator A generates a Co-semigroup S(t), t > 0, F is a locally Lipschitz mapping, and that KerQ = {0}. Then the transition semigroup Pt, t > 0, corresponding to problem (7..4.1) is irreducible in time T.

Proof - It is enough to show that the deterministic system (7.4.2) is approximately controllable in time T > 0. Let cp E Q0, T]; H) be a function such that V(O) = x and cp(T) = z and let (t) = ca(t) - S(t)x -

j

S(t - s)F(cp(s))ds,t E [0,T].

Since Image Q1/2 is dense in H there exists a sequence {un} C Q[0, T]; H) such that the sequence t

bn.(t)

= 0(t) - f S(t - s)Q1 /2u,,(s)ds, t E [0, T], 0

Chapter 7

144

tends to 0 uniformly on [0, T]. Let yn = W(t) - yn(t) = bn(t) +

J

t

x; un ), then

S(t - s)(F(y(s)) - F(yn(s)))ds.

Therefore, for a constant L and all n E N T

1z - yn(T)I < I bn(T)I + LM f I y(s) - yn(s)I ds,

it follows that n-oo lim yn(T) = z. This proves approximate controllability.

Remark 7.4.3 Theorem 7.4.2, under Lipschitz conditions, was proved by B. Maslowski [116, Prop. 2.11]. We finish this section with a result, see J. Zabczyk [164], on the interplay between irreducibility and uniqueness of invariant measure, showing difference between finite dimensional and infinite dimensional systems. The proposition below shows also that irreducibility alone is not sufficient for uniqueness of invariant measures.

Proposition 7.4.4 Let Rt, t > 0, denote the transition semigroup corresponding to equation (7.2.3).

(i) If dim H < oo and Rt, t > 0, is irreducible at some t > 0 then there exists at most one invariant measure for Rt, t > 0. (ii) There exist an infinite dimensional Hilbert space H and an irreducible, at all t > 0, semigroup Rt, t > 0, for which there exist at least two different invariant measures.

Proof - (i) For sufficiently large t > 0 (in fact for all t > 0), Image Qt = H and therefore also Image Q... = H. Consequently by a control theoretic argument, see e. g. G. Da Prato and J. Zabczyk [44, Appendix D.1], existence of an invariant measure for the system implies limt,+c,, IIS(t)II = 0. By Proposition 7.2.3 uniqueness follows.

(ii) Define H = L2(0, +oo), U = R, and S(t)x(B) = etx(O + t), t > 0, 0 > 0,

Bu = bu, it

Uniqueness of invariant measures

145

where b(9) = e-02, 9 > 0. We first show that the semigroup Rt, t > 0, is irreducible. Assume in contradiction that there exists t > 0 such

that Image Q1t/2 = Image Qt # H.

Then there exists an element xo E H different of 0 such that (Qt/2z,xo) = 0 for all z E H, and consequently

Q1t/2x0

= 0. Since t

Q

J0

and

I (S(s)b, xo) I2 ds

+00

(S(s)b, xo) =

ese-(B+s)2xo(O)de

0

we have, for almost all s E]0, t[

0(s) =

j+°°e298 (e-02x0(9)) dO = 0.

From elementary properties of the Laplace transform we have xo = 0. This proves irreducibility of Rt, t > 0, for all t > 0.

Note that +00

Tr Qco

J

JS(t)b12dt ete-2tOe-tee-e2dt dO < +oo,

0

0

and therefore 1V(O,Q00) is an invariant measure for Rt, t > 0. How-

ever, if y(9) = e-B, 9 > 0, then S(t)y = y for all t > 0 and by Proposition 7.2.3 the measure .IV(S{y},Q00) is also invariant.

Chapter 8

Densities of invariant measures This chapter is devoted to the existence and regularity of the densities of an invariant measure with repect to a properly chosen Gaussian reference measure. Conditions are given under which transition functions for linear and nonlinear equations are absolutely continuous with respect to the reference measure. It is shown also that the density exists and satisfies an adjoint equation. Under additional conditions the density belongs to some Sobolev space. We examine also the so-called gradient systems for which an explicit formula for the density exists.

8.1

Introduction

We consider a stochastic equation on a separable Hilbert space H,

dX = (AX + F(X))dt + /dW, (8.1.1)

X(0)=xEH, having densities with respect to the invariant measure of the linear version

dZ = AZdt+V1Z7dW, (8.1.2)

Z(O) = x E H. 147

Chapter 8

148

We will assume

Hypothesis 8.1 (i) A is the infinitesimal generator of a strongly continuous semigroup S(t), t > 0, on H of negative type. (ii) F is a Lipschitz continuous and bounded mapping from H into H. (iii) Q is a bounded symmetric nonnegative operator on H. (iv) W is a cylindrical Wiener process on H. (v) We have

f

+oO

Tr [S(t)QS*(t)] dt < +oo.

0

Under Hypothesis 8.1 equations (8.1.1) and (8.1.2) have unique mild solutions X (t, x), t > 0, and Z(t, x), t > 0, respectively, in virtue of Theorem 5.3.1. We shall denote by Pt, t > 0, and by Rt, t > 0, the corresponding transition semigroups, Ptcp(x) = E[(p(X (t, x))], t > 0, x E H, (P E Bb (H),

Rtcp(x) = ]E[p(Z(t, x))], t > 0, x E H, cp E Bb(H).

By Theorem 6.2.1 there exists a unique invariant measure it for the semigroup Rt,t > 0, being a Gaussian measure, it = with mean vector 0 and covariance operator Qom: 00

Q.x = j S(s)QS*(s)xds,

X E H.

0

Since µ is an invariant measure for the semigroup Rt, t > 0, we can extend Rt, bya standard argument, to a contraction semigroup in L2(H, y), which we still denote by Rt, t > 0. We have in fact IRt o(x)12 = (E[ o(Z(t,x))])2 < >E

[,p2(Z(t,x))] =

so that IH

I Rty(x)I2i(dx) < fH Rt(4p2)(x)Ft(dx) = JH 1P 2(x)p(dx).

In Theorem 8.4.4 we give conditions under which there exists an invariant measure µF for the solution of (8.1.1) absolutely continuous

with respect to p with density dµ dYF

dµ

E

,

L2(H,l)

Densities of invariant measures

149

Under additional sets of conditions, see Theorems 8.7.5, 8.8.1 and 8.8.2, we show that

dfF dµ

or even

dt t

E W6' Q1'2( H µ) , E WQ'2 (H,

a),

where WQ'2(H,1) and WQ'2(H, p) are appropriately defined Sobolev spaces of functions on H. For generalizations to LP and W1,n of the results on regularity of the densities, see A. Chojnowska-Michalik and B. Goldys [26]. Regularity of invariant measures for equations of the form

dX = I

-2X + F(X))

dt + Q112dWt, X(O) = x E H,

(8.1.3)

on H = L2(0,1)' was an object of paper by I. Shigekawa [139] and R. V. Vintschger [153]. These papers treates the case of the operator Q defined by the formula 11( A r7)(p(rl)d17,

E H,

E [0, 1],

(8.1.4)

and the F nonlinearity having values in Image Q1/2. In particular in I. Shigekawa [139] the mapping was assumed to be bounded. We remark that the cases covered here are different and motivated by applications. In §8.6 we investigate the so-called gradient systems for which explicit formulae for the density of an invariant measure can be derived.

8.2

Sobolev spaces

We start by recalling definitions and basic properties of some function spaces. We assume Hypothesis 8.1 and set µ = We denote by {e,,,} a complete orthonormal basis of eigenvectors of Q... and by {q,,,} the sequence of its positive eigenvalues.

Chapter 8

150

We are also given a self-adjoint operator R with the same set of eigenvectors {en} and with positive eigenvalues {rn}. Let P be the set of all functions cp on H of the form cp(x) = p((x, el), . . ., (x, en)), x E H,

where p is a polynomial in n variables, n E N. The spaces WR'2(H, µ) and W2ri 2(H, µ) are defined as the closures of P with respect to the norms I R1/2D.'(x)I2µ(dx) and IICPIIwR2(H,µ)

=

w 2(H,µ) + III II R1 2 DrxP(x)R1 "2IIHS µ(dx),

II(PII2

respectively. Here II - II HS denotes the Hilbert-Schmidt norm of an operator.

If R = I we set WR'2(H,IL) = W1,2(H, p) and W2 2(H, M) W2'2(H 11).

Let

n!n

(-1)n Hn(e) =

e4 ;

d"

),n E N,

(e_

E R,

be Hermite polynomials forming, as is well known, a complete orthonormal basis on L2(R,N(0,1)). Denote by r the set of all mappings

ry:N--GNU{0},n- -yn, such that

00

171=Eyn<+00. n=1

For any y E r we define 00

H.y(x) =

H'Yn

n=1

I In qn)

,

xn = (x,en).

(8.2.1)

Densities of invariant measures

151

Then it is well known that {H.y : y E I'} is an orthonormal complete system on L2(H, p). For any cP E L2(H, p) we set P7 - (cP, H7)LZ(H,µ)

Proposition 8.2.1 (i) For any W E WR'2(H, µ)

j H I R1/2DxV I2i(dx) _ 'YEr (r,7) q (ii) For any cO E WR'2(H, ic)

J II Rl /2DxxW(x)R11211 Hs µ(dx) = 1

(8.2.2)

I c -I2.

Y) (2 ryEr LIS'

- (I

2.

(8.2.3)

In the above formulae we have used the notation 00

r

q

rn

n=1 qn

(7) r

1n,

q

2

00

_

=

2 rn 27'n

n=1 qn

Proof - Recalling that Hn(x) =/ Hn_1(x), n E N,x E R, we have: 00

DxcP(x) =

E

(qnn

h#n

H y, ( qh) en,

and so

j: 00

fxI

1:

rn'ln I.ryl2

n=1 'yEr,-yn>O

,

qn

and (8.2.2) follows. To prove (8.2.3) we remark that

E

yEr,'Yn>O,ryn-1>0

x

rr jl Htih-1

h¢n

h

P71 qn

Yn(7'n - 1)H-vn-2

n

qn

Chapter 8

152

and, if m 54 n,

1:

Dxn Drm

1

1E r, yn>O, Ym>0 T r

x

rr 11

M. 11.Yn-1 (_) H'ym-1 ( x

Xh

H-r,,_1

qh

.

h$n,m

It follows that

JH Tn IDn

T

2 10

p(dx) =

Y

I2

ryEr,ryn>O,'Yn-1 >0

qn2ryn(7n

and

JH

rnrm7n7m

rnrm I DxnDxmPI2 µ(dx) ^YEr,ryn>O,^Ym>O

gn gm

which yields (8.2.3).

The following proposition was proved in G. Da Prato, P. Malliavin and D. Nualart [40] and independently in S. Peszat [123]

Proposition 8.2.2 For an arbitrary Gaussian measure µ on a Hilbert space H the embedding WR'2(H, p) C L2(H, M)

is compact if and only if rn = +00. n-oo qn 1im

(8.2.4)

In particular W1,2 (H, u) is compactly embedded in L2(H, µ).

Proof - Note that (8.2.4) holds if and only if for any ordering y(k), k E N, of I', limk-.,,o(Qo , y(k)) = +oo. This easily implies the required result.

Densities of invariant measures

153

Remark 8.2.3 Note that the embedding W2,1 (H, M) C W1,2 (H,

µ),

is not compact. To see this define cpm(x) = (x, em), m E N, x E H.

Then

Dxpm(x) = e-m, D2,pm(x) = 0. Therefore II0mIIL2(H,,) = q,,,12

and II(PmIIW1,2(H,,U) = II(mIIW2.2(H,p.) = (1 + gm)1h/2

So the sequence {cpm} is bounded in W2'2(H,µ), converges to 0 in L2(H, p) and does not contain any subs-equence convergent to 0 in W1,2 (H, z).

8.3

Properties of the semigroup Rt, t > 0, on L2(H, µ)

For our study of the transition semigroups Rt, t > 0, and Pt, t > 0 we will need, besides Hypothesis 8.1, the following.

Hypothesis 8.2 (i) Image S(t) C Image Qt 12, t > 0.

(ii) Setting F(t) = Qt 112S(t), the function IIr(t)II, t > 0, is

Laplace transformable, i.e. for sufficiently large A, is finite.

f0+00 e-"tjIr(t)Ildt

Note that if Hypotheses 8.1 and 8.2 hold, then the assumptions of Theorem 7.2.4 are fulfilled, so that Rt, t > 0 is a strong Feller semigroup.

Remark 8.3.1 Note that if Q coincides with the identity operator I, then Hypothesis 8.2 is satisfied, see G. Da Prato and J. Zabczyk [44, Corollary 9.22]. On the other hand if H = Rd, d E N, then Hypotheses 8.1 and 8.2 hold if and only if Q is invertible, see T. Seidman [137].

Chapter 8

154

We introduce, following [26], the linear operator in L2(H, p) : Aow

e

D(Ao) _

1

2

Tr [QV.) + (x, A*wx),

{w(x) = f((x,a,),...,(x,a,a) : f E Cb2(R"),

a1i ..., a. E D(A*), n E N} . It is easy to see that D(Ao) is invariant with respect to the semigroup Rt, t > 0, and is dense in L2(H, p).

The following result is proved in G. Da Prato and J. Zabczyk [47].

Theorem 8.3.2 Assume Hypothesis 8.1. (i) Ao is closable in L2(H, p) and its closure A is the infinitesimal

generator of a Co-semigroup etA,t > 0, in L2(H, p) identical with

Rt, t > 0: etASO(x) = E [ca, (S(t)x + f t S(t - s)dW(s) )J

/

o

Rttp(x),

t>-O, cp E L2(H, p) (8.3.1)

(ii) If moreover Hypothesis 8.2(i) holds, then for all t > 0 and cp E L2(H, p) we have Rtcp E W1'2(H, p), and II DxRtwII L2(H,µ) : IIr(t)II IIPIIL2(H,,L),

(8.3.2)

and for any t > 0, Rt is a compact operator on L2 (H, p). (iii) If, in addition, Hypothesis 8.2(ii) holds, then D(A) C W1'2(H, p)

Proof - (i) If cp E D(Ao) then, for arbitrary x E H Rtw(x) = (x) +

j

Rs(Aow)(x)ds.

Therefore D(Ao) C D(A) where A is the infinitesimal generator of the semigroup Rt, t > 0. From Theorem 1.9 in E. B. Davies [49], it

Densities of invariant measures

155

follows that D(Ao) is dense in the graph norm of D(A) and therefore the closure of AO is A, and (i) follows. Now we prove (ii). Let co E Cb(H). Since IHW(y)N(S(t)x,Qt)(dy),

it follows by the Cameron-Martin formula that, for any h E H, (D=Rt,P(x),h) = J(r(t)h, Qt 112y)(P(y)N(S(t)x,Qt)(dy) Consequently (D.Rtw(x), h) l

JH I

(r(t)h,

Qt

112Y) l

1 v(S(t)x + y)I N(0, Qt)(dy)

and by the Schwarz inequality

I (D.Rtp(x),h)12 < JH I (r(t)h,Qt 1/2y)12H(O,Q,)(dy) X

<-

y)12H(O,Qt)(dy)

JH

IIr(t)112Ih12(Rtso2)(x).

Thus, integrating both sides with respect to µ, we arrive at (8.3.2). The compactness of Rt follows from Proposition 8.2.2. Finally as far as (iii) is concerned, it is a simple consequence of (8.3.2) and the formula

Dx(A - A)-'w _ j

+oo

e-atD.,etAcadt,

0

which yields

+00

L2(H,µ)

The proof is complete.

f0

e--'t IIF(t)II dt

II (PII LZ(H,µ)

Chapter 8

156

8.4

Existence and absolute continuity of the

invariant measure of Pt, t > 0, with respect to t Now we are going to show that the semigroup Pt can be extended to a Co-semigroup on L2(H, µ). We will need the following result which is a direct extension of Theorem 3.5 of E. B. Davies [49].

Proposition 8.4.1 Let A be the infinitesimal generator of a Cosemigroup on a Hilbert space 1-l, and let C be a closed operator on 1-l such that IICetAIldt < +oo.

D(C) D U etA(x), t>o

0

Then D(C) D D(A) and for an arbitrary bounded operator F on 7-l the operator

,C = A+ .PC, D(C) = D(A),

(8.4.1)

is the infinitesimal generator of a Co-semigroup etc, t > 0, on 7-l and we have

00

etr_ p

=

Tn(t)4o, t > 0, yO E f,

(8.4.2)

n-o

where etA 0, Tn(t)cp =

t e(t-s)AFCTn_l(s)
J0

n E N. (8.4.3)

Remark 8.4.2 Theorem 3.5 in E. B. Davies [49] covers the case of F being the identity operator, however, its proof is valid in the present more general situation. Now we can prove the result

Theorem 8.4.3 Assume that Hypotheses 8.1 and 8.2 hold. Define B,p =

E Wl'2(H,µ),

and

L = A + 13, D(L) = D(A),

Densities of invariant measures

157

where A is the infinitesimal generator of the semigroup Rt, t > 0, on L2(H, µ). Then the operator C generates a Co-semigroup et1C, t > 0, in L2(H, y) which is an extension of Pt to L2(H, µ). Moreover (i) D(G) C W1,2 (H, µ), (ii) Operators et1 are compact for all t > 0.

Proof - Note that the operator C, CV = D,,V, D(C) =

W1,2 (H,

µ),

is closed on L2(H,µ) and the operator .1, -PAP = (F(x),'P), D(F) = L2(H,µ), is bounded. Moreover, by (8.3.2) we have IICet-''P L2(H,µ) < I117(t)II II'PII L2(H,µ)-

Therefore the assumptions of Proposition 8.4.1 are satisfied and the operator,C generates a Co-semigroup etC, t > 0, in L2(H, µ). To show that et", t > 0, is the required extension, consider cP E

D(Ao) and define u(t,x) = Pt'(x), t > 0, x E H. Assume, for a moment, that F E Cb (H; H). Then the function u is the unique strict solution of the associated Kolmogorov equation

ut =

1

2

Tr[Qu.,x] + (Ax + F(x), ux), (8.4.4)

u(0, x) = c0(x).

In particular the L2(H, µ)-valued function u(t, ), t > 0, is a solution of the problem u'(t) = Gu(t), t > 0,

u(0) = V,

and therefore u(t, ) =

etf-' for t > 0. If F is only a Lipschitz continuous function one can find a uniformly bounded sequence of mappings

{F7L} C C6 (H) such that limn., F,,(x) = F(x), x E H. It easily follows from (8.4.2) and the definition of Pt that for the corresponding semigroups etc^, t > 0, Pt, t > 0, and cP E D(Ao), we will have et1C

= lira etc°cP = lira Pt cP = Pt(p. n-*oo

Chapter 8

158

So Pt = L2(H,p).

etf- cp, t > 0 for cp E D(Ao) and therefore for all cp E

Property (i) follows directly from the definition of C. To show (ii)

it is enough to remark that 00

E JjD.Tn(t)jj < +oo

n=0

for t > 0, where Tn is defined in (8.4.3), and to use (8.4.2).

We are now in a position to prove the existence of an invariant measure pF for (8.1.1) and that it is absolutely continuous with respect to A.

Theorem 8.4.4 Assume that Hypotheses 8.1 and 8.2 hold. Then there exists 'k E D(,C*),i/> > 0, such that G*V)

0, J O(x) p(dx) = 1.

Moreover 0 is the density of an invariant measure pF for (8.1.1). Finally, if ImageQ is dense in H, the invariant measure is unique.

Proof - Since the operators et'c, t > 0, are compact on L2(H, p), the same is true for their adjoints etc*,t > 0. Since 1 belongs to the spectrum a(etc) of etc, 1 is also an eigenvalue for etr*. Repeating the argument from the proof of Theorem 7.8 in E. B. Davies [49], one obtains that the set

G = {*EL2(H,p):etC*b= Vt>0}

= {'EL2(H,p):G*0=0}, is linear and of finite, positive dimension. It is easy to show that elements of G are densities of possibly signed invariant measures for ett', t > 0. That there are nonnegative and nonzero elements in follows from Lemma 3.4 in I. Shigekawa [139].

To prove uniqueness, it is enough to show that Pt, t > 0 on Bb(H) is a strong Feller and irreducible semigroup; however this is so by Theorems 7.1.1 and 7.2.4.

Densities of invariant measures

8.5

159

Locally Lipschitz nonlinearities

We assume here Hypothesis 8.1(i)(iii)(iv)(v) but, instead of 8.1(ii), we assume that F : H - H is locally Lipschitz continuous and that problem (8.1.1) has a unique mild solution X(t, ), t > 0. We want to extend Theorem 8.4.4 to this more general situation. For this we define the mappings Fn, n E N, Fn(x) =

F(x), Ixi < n, F ' Ix) >

(nn/

(8.5.1)

n,

which are Lipschitz continuous and bounded. Then we consider, besides equation (8.1.1), the following equation corresponding to Fn : dXn = (AXn + Fn(Xn))dt + v/¢dW,

(8.5.2)

Xn(0)=xEH. Let Pt(x, ), Pt (x, ), t > 0, n E N, x E H BE the transition probabilities corresponding to solutions of (8.1.1) and (8.5.2) respectively. We first prove a result on absolute continuity of transition probabilities.

Proposition 8.5.1 Assume that Hypothesis 8.1(i)(iii)(iv)(v) hold, that F is locally Lipschitz continuous and that problem (8.1.1) has a unique mild solution X (t, ), t > 0. If for some t > 0 and all n E N the measures Ptn(x, ) are absolutely continuous with respect to a measure v, then the measure Pt(x, ) is absolutely continuous with respect to v.

Proof - Let F C H be a Borel set such that v(r) = 0. Then Pt(x, r) = ]P(X(t, x) E r) = 1P(X(t, x) E r, r,,i > t)

+ P(X(t, x) E r, rn < t), where

rn = inf{s > 0 : IX(s,x)I > n}.

Chapter 8

160

Since for all s < rn, X(s,x) = X,,,(s,x), we have

Pt(x, r) <

?(X,,, (t, x)

E r, r,, > t) + P(rn < t)

< Pt (x, r) + P(rn < t).

But Pt (x, r) = 0 and P

lira r,, = +oo) = 1, and consequently

(n-r+00

Pt(x, r) = 0.

//

Now we prove the main result of this section under the following assumptions.

Hypothesis 8.3 (i) Hypotheses 8.1(i)(iii)(iv)(v) and 8.2 hold and Image Q is dense in H. (ii) F is locally Lipschitz continuous and problem (8.1.1) has a unique mild solution.

(iii) Problem (8.1.1) has a unique invariant measure µF.

Theorem 8.5.2 Assume that Hypothesis 8.3 holds. Then the invariant measure µF to problem (8.1.1) is absolutely continuous with respect to the invariant measure y to problem (8.1.2).

Proof - From Theorem 8.4.4 for any n E N problem (8.5.2) has a unique invariant measure µF << y. Then the conclusion follows from Proposition 8.5.1.

8.6

Gradient systems

Before investigating the problem of regularity of the density d we devote this section to an important class of stochastic systems, called gradient systems, for which an explicit expression for the density exists. Again let H (norm I) be a separable Hilbert space and E (norm 11) a separable Banach space densely and continuously embedded into H.

Densities of invariant measures

161

Gradient processes are solutions X on E of the equation

dX(t) = (AX(t) + U'(X(t)))dt + dW(t), t > 0, (8.6.1)

X(0)=x EE, under the following hypothesis.

Hypothesis 8.4 (i) A is a self-adjoint negative definite operator on H such that A-1 is of trace class.

(ii) For any t > 0, E is an invariant subspace of S(t) = eta Moreover for any x E E the mapping S(.)x is continuous on ]0, +oo[, with the topology of E. (iii) The Ornstein-Uhlenbeck process WA(t) = fo S(t - s)dW(s), W being the cylindrical Wiener process on H, has a version. concentrated on E.

(iv) U : E -+ R is continuous and there exists the directional derivative DU(x; h) of U at any point x E E and at any direction h E E. Moreover there exists a mapping F : E - H such that DU(x; h) = (F(x), h). (v) U is bounded from above, and F : E -* H is a locally Lipschitz

mapping: for arbitrary r > 0 there exists k,. > 0 such that IF(x) - F(y)IH s KrII X - YIIE-

(vi) For arbitrary x E E there exists an E-continuous solution of the integral equation t

X(t) = S(t) + f S(t - s)F(X(s))ds + WA(t)x, t > 0.

(8.6.2)

Example 8.6.1 Define E as the space C[0, 7r] of continuous functions on [0, ir] and H = L2(0, 7r). Let 2

Ax = d22, V x E H2(0,7r) n Ho(0,7r).

Let U be of the form U(x) =

f(x())de,

(8.6.3)

Chapter 8

162

where p is a polynomial of even order with negative leading coefficient. Then, recalling Theorem 5.5.11, it is not difficult to see that Hypothesis 8.4 holds.

Equations of the form (8.6.1) were studied in particular by R. Marcus [112], B. Maslowski [116], T. Funaki [75], P. L. Chow [27] and by J. Zabczyk [168]. Here we follow J. Zabczyk [168]. It turns out that, under Hypothesis 8.4, there exists an invariant

measure µF for (8.6.1) on E that it is absolutely continuous with respect to the measure µ which in the present situation is of the form

AI(O,Qoo)=N 0,-

1 2A-1).

Before proving this result let us introduce some notation. Let {en} and {an} be the sequences of all eigenvectors and all eigenvalues of A. We assume that

0>-al>-A2>

.

We denote by Hn the closed subspace of H spanned by {el,..., en} and we set

lIn(x) = E(x, ej)ej, x E H, n E N. j=1

We assume moreover that the cylindrical process W(t) is such that 00

(W(t),x) = E #j (t)(x, ej), V x E H, j=1

where {,Qj } is a sequence of mutually independent, standard, real Wiener processes. We have the following lemma.

Lemma 8.6.2 For arbitrary n E N, Hn C E and we have lim n-r+oo JJx - Hn(x)II E = 0,

for µ-almost all x E E.

Densities of invariant measures

163

Proof - Since the measure it is concentrated on E, E contains the reproducing kernel of p which is Image Q/2 =

D((-A)1/2).

Thus D((-A)1/2) C E and so {ek} C E. Moreover the random variables j E L2(µ, E) given by

.j(x) = 2Aj(x, ej), x E E, are Gaussian, normalized and independent and therefore the sequence

n

e j(x)V,xEE,nEN,

j=1

converges p-almost surely in E norm to a random variable C such that G(() = it, see G. Da Prato and J. Zabczyk [44, §2.2.3]. Since E is continuously embedded in H, the limit has to be ((x) = x. We can now prove the result

Theorem 8.6.3 Under Hypothesis 8.4 the measure (8.6.4)

pF(dx) = e2U(a)µ(dx), x E E,

is invariant for (8.6.1) and the corresponding transition semigroup Pt, t > 0, is µF-symmetric JE SO(x)PtO(x)AF(dx) =

JI'(x)Pt(x)PF(dx),

(8.6.5)

for all cp, 5 E Bb(E). For some comments on symmetric semigroups see the end of §2.2.

Proof - It is enough to show the identity (8.6.5). Define n

WW(t) = 1:Pj(t)ej, j=1 and t

Zn(t) = f S(t - s)dWn(s) _

t

J

df3j s e

Chapter 8

164

It is clear that the processes Zn are E-continuous and satisfy the equation dZn = AZndt + dW,,,, Zn(0) = 0.

It follows from Hypothesis 8.4(vi) that on an arbitrary finite interval [0, T]

lim ZZ() = Z(.),

n-++oo

uniformly in E-norm. Let Pt, t > 0, be the transition semigroup corresponding to the solution Xn(t, x) of the equation

Xn(t) = S(t) +

t

Jo

S(t - s)llnF(llxn(s))ds + Z,,,(t)x, t > 0.

-

It is well known that the theorem is true in finite dimensions. Consequently, if ttn are the projections of p on Hn = llnH and Un(x)

U(ln(x)), x E H, n E N, then setting i4 (dx) = kneunxp(dx), x E E,

one has JE

V(x)Pt ,O(x)eUlnnxl µn(dx) = JE O(x)Pt cp(x)eU(nnx)µn(dx). (8.6.6)

But JE P(x)Pt O(x)eU(nnx) Pn(dx) = JE 4p(Hnx)Pt O(Hnx)eU(nnx)/tn(dx,). (8.6.7) Moreover one can check, see J. Zabczyk [168], that for any sequence {xn} C E convergent to x in E and for any function E Cb(H) one has lim Pt O(xn) = Pt'(x). (8.6.8) n-.+oo

By (8.6.7) and (8.6.8) we can pass to the limit in (8.6.6) provided y0, 0 E Cb(H). Therefore

JE o(x)PtV)(x)eU(x)y(dx) =

JEO(x)PtP(x)eU(x)p(dx), d (o, 0 E Cb(H).

(8.6.9)

By a monotone class argument (8.6.9) follows for arbitrary yo, O E Bb(H).

Densities of invariant measures

8.7

165

Regularity of the density when C is variational

Theorem 8.4.4 implicitly states that the density d is in the domain D(L*). Therefore, one way of obtaining information about the regularity of d"F is to characterize elements of D(L*). In general this seems to be a difficult task. In the present section we restrict ourselves to systems such that the infinitesimal generator of the semigroup Pt, t > 0, is variational. We recall that L is said to be variational if there exists another Hilbert space V continuously and densely embedded in i _: L2(H, µ) and a continuous bilinear form

a: V x V - R, such that (LAP) O)L2(H,µ) = -a(cp, 0), V

E D(L), V 0 E V.

If the operator L is variational then obviously D(L) C V, and since L* is also variational we also have

D(L*) c V. We assume here that

Hypothesis 8.5 (i) ImageQ,, C D(A).

(ii)Q=I. Remark 8.7.1 If S(.) is analytic, and there exists w E JR such that IIS(t)II < ewt for all t > 0, then Hypothesis 8.5 is fulfilled, see G. Da Prato [35, Proposition 2.5].

Remark 8.7.2 More general hypotheses were considered in V. I. Bogachev, M. Rockner and B. Schmuland [12], see also M. Fuhrman [73].

Chapter 8

166

Following [12], we introduce the space E of all exponential functions: E _ {cp E Cb(H) : W(x) = e°(h'x), x E H, h E D(A*)}.

Clearly E is linearly dense in L2(µ, H) and in D(C). Moreover if cp(x) = e'(h°x), x E H, we have Gcp(x) = - 2 Ih12 + i(x, A*h).

(8.7.1)

We now set V = W12(µ, H) and introduce the bilinear form in

VxV, JH(DxcP(x),AQ.DxO(x))µ(dx), W, V) E W1'2(µ,H)

a(V,V) =

(8.7.2)

Under Hypothesis 8.5, a is clearly well defined and continuous. Let us check, following M. Fuhrman [73], that it is coercive.

Proposition 8.7.3 We have a(cP, y') =

-2 I1cPII2y1,2(,,H)

I

V y' E Wl'2(µ,H).

(8.7.3)

Proof - We first remark that for any x E D(A*) the Liapunov equation holds:

-Ixl2

= (Q,, x, A* x) + (QUA*x, x)

By Hypothesis 8.5(i) this implies

AQ,, +

-I.

(8.7.4)

We can now prove the statement. We have a((P, c)

=

2

2

f(D(x), AQ.Dxcp(x))µ(dx)

I

(AQooDxcP(x), DxcP(x))µ(dx)

JH(DxcP(x), (AQ. + (AQ.)*)Dxw(x))µ(dx)

2 1

2IIcPI,W1.2(µ,H),

y' E u'l'2(µ, H).

Densities of invariant measures

167

We finally check that a is the Dirichlet form associated with the infinitesimal generator C of Pt, t > 0.

Proposition 8.7.4 We have D(C) _ {cp E W1'2(µ, H) : a(., c) is continuous in L2(µ, H)}, (8.7.5) and

a(4', ,O) = (ASP, v')

,

d *p E D(C), V 0 E W"' (y, H).

(8.7.6)

Proof - Since £ is linearly dense in D(L) it is enough to prove the identity in (8.7.6) for all c0, 0 E C. For this purpose we notice that (k, x) a(dx) = i(Q 12h, Q 00/2k) e- 21e I2h12, h, k E H. (8.7.7) This follows by (setting ea(h,x)

JH

F(E) = 1 e'(h'x)+e(k'x) y(dx) = e_ , H

E > 0,

and by computing F'(0). Let now W(x) = e'(alx), O(x) = e'(a'x), a, 0 E D(A*). By (8.7.1) and (8.7.7), we have (L
)L2 (µ,H)

ei(,+)3,x) p(dx) 2 JaI2 IM

+i f J(

A*a)e'(a+a,x) p (dx)

x'

_ - 2 IaI2 + (QUA*a + -2(a,3)

We can now prove the result

a+ a(w,V))

Chapter 8

168

Theorem 8.7.5 Assume that Hypotheses 8.1, 8.2 and 8.5 hold. Then D(,C*) C W16'2(H,µ), and there exists an invariant measure µF such that dµF E W16'2(H µ) dµ '

Proof -

It is enough to notice that D(A) = D(G) and that

the adjoint C* is the variational operator in V corresponding to the bilinear form a*(V, 0) = a(i, p), V, 0 E V. This implies D(G*) C W1'2(H,µ).

8.8

Further regularity results in the diagonal case

We consider here the case when the operators A and Q are symmetric and diagonal with respect to a given orthonormal basis. We assume here

Hypothesis 8.6 (i) The operator A is negative definite with eigenvalues and eigenvectors equal respectively to -an, en, n E N. (ii) There exists a sequence {qn} of positive numbers such that Qen = green, n E N.

Let us remark that regularity of the transition densities was studied, in the present situation, by B. Gaveau and J. M. Moulinier [78] and 1. Simao [141]. Their results are not, at least directly, applicable to invariant measures. Their methods are based on Girsanov's theorem, ours are more analytical.

Theorem 8.8.1 Assume that Hypotheses 8.1, 8.2 and 8.6 hold and moreover that ImageF(x) C ImageQ1/2 for all x E H, and Q-1/2F is bounded. Then D(G*) C WQ'2(H,µ), and there exists an invariant measure µF such that dµF E W16'2(H µ). dµ '

Densities of invariant measures

Proof -

169

Set V = WQ'2(H,p) and let a be the bilinear form on

V x V, a(cp,

Q1/2Dx. )

&) = H

(Q-1/2F(x),Q1/2Dxp),O] u(dx).

+

Let, in addition, A be the infinitesimal generator of the semigroup

Rt, t > 0, on L2(H; u). Recalling the identity H,"(x) - xH,(x) _ -nHn(x), n E N, x E R, we have, for y E IF, see also §8.1, 00

qn 82H.y 2 8xn

AHY n=1

-an x 2a

an H "

n1 X

11 H7n

nOH,

8xn

nxn

qn

[2-a h

- an

tan xnH'yn

qn

xn I

J

xh)

qh

00

E anynH'y(x)n=1

Thus Hy is an eigenvector for A and 00

anynHy = -(a,y)Hy.

AH.y = -

(8.8.2)

n=1

Consequently Ayo =

- 1: -YEr

D(A)

+oo},

E L2(H,,u) : ryEr

and

D((-A)1/2) _ {Cp E L2(H, µ) : 1: I (a, y) I k I2 < +()O l. -yEr

Chapter 8

170

It easily follows from Proposition 8.2.1 that WQ'2(H, µ) = D((-A)1"2), and

D(A) C WQ'2(H,µ). Moreover a(W, &) = (ASP + (F(x),

0L2(µ), V (p,

E

D(A).

The form a is obviously continuous on V x V. It is also coercive since, W61,2 (H, µ), we have

for all cp E

a(W, o) ?

IQ1I2DXplL2(µ)

IIQ-1/2FIIo f I

&(x)I µ(dx)

> _11Q-1/2FIIoIQ1/2DxpIi2(A)

-

It follows that D(A) = D(L). Since .C* is the variational operator (with respect to V = WQ'2(H,it)) corresponding to the bilinear form (p, V) E V,

we have D(,C*) C W1,2(H, p). The proof is complete.

We consider finally a situation where even better regularity of the density can be obtained. For this we give a complete description of D(G*) under the following hypothesis.

Hypothesis 8.7 There exists a constant C > 0 such that 00

I(x,AF(x))I < C, E

OFk

k=1 axk

< C, for u-a.e. x E H.

(8.8.3)

Densities of invariant measures

171

Theorem 8.8.2 Assume Hypotheses 8.1, 8.2 and 8.7. Then D(,C*) = D(C) and there exists an invariant measure µF for equation (8.1.1) such that d1uF

E W$'2(H µ)

dpµ

It is sufficient to show that D(,C*) = D(G) and the proof of this identity is a consequence of the following two lemmas.

Lemma 8.8.3 Assume that an infinitesimal generator L on a Hilbert space H is of the form

,C=A+B, where A is a negative definite operator,D(A) = D(G) and B is a closed operator with domain D(B) D D(-(A)1/2). If D(B*) D D((-A)1/2) then ,C* = A + B*,

D(G*) = D(A).

Proof - It follows from the closed graph theorem that B, B* E L(D((-A)1 12); x) Consequently the operator BR(A, A) is well defined for all A > 0 and

for a constant C > 0

s CII -A 112R(A,A)II

IIBR(A,A)II

Moreover

R(A, A+ B) = R(A, A)(I - BR(A,A))-1. So if A > C2 then 00

R(A, A + B) =

R(A, A)(B(R(A, A + B)))n. n-o

Chapter 8

172

In a similar way, considering A + 8* with the domain D(A),

R(A,A+ 8*) =

00

R(A,A)(8*(R(A,A)))n.

n=0

But

R(A,(A+B)*) = R*(A,A+8) 00

E [(,UR(A, A))*]nR(A, A). n=0

Therefore, to show that A+13* = (A+1)* it is enough to prove that for arbitrary n E ICY U {0}

R(A, A)(B*(R(A, A)))n = [(BR(\, A))*]nR(A, A).

(8.8.4)

We check (8.8.4) for n = 1, the general case follows by induction. Let x, y E R. Then (R(A,A)B*R(A,.A)x,y) = (B*R(A,A)x,R(A,.A)y) = (R(A, A), BR(A, A))

because D(B**) = D(8) D D(A). Since ((BR(A, A))*R(A, A)x, y) = (R(A, A)x, 8R(A, A)y)

Identity (8.8.4) for n = 1 follows.

Lemma 8.8.4 If B is the operator given by (F, Dxv), (P E D(8) = W1 " (H; and Hypothesis 8.7 holds then

D(6*) D W1'2(H;p) and

5*0 =

Dx0) + [2 (Ax,

Div F(.)]1,

for all 0 E W1,2 (H; u), where

k(x), xEH. DivF(x)=E 00

(8.8.5)

Densities of invariant measures

173

Proof - Take cp smooth and depending only on X1, ..., XN, then N 13c p=

k

k=1

If &EW1"2(H,µ) we have, N

LW

BW(x)b(x)µ(dx) = ?lJH Fk axk Oµ1(dxl) ... itN(dxN) JH xz

-

H N k=1

(x)aak Fk(x)e 2

a [9Fk JH4O(x) 8xk + Fk xk

xk qk

µl(dxl) ..dxk ..µN(dxN)

Fk µ(dx), J

and the conclusion follows.

Proof of Theorem 8.8.2. - By Lemma 8.8.3 the conditions of Lemma 8.8.4 are satisfied and therefore D(,C)=D(G*).

Part III

Invariant measures for specific models

175

Chapter 9

Ornstein-Uhlenbeck processes This chapter is devoted to the existence and uniqueness of invariant measures for linear equations under various specific conditions on the coefficients and on the Wiener process, motivated by applications. In particular the stochastic wave equation, a linear equation of mathematical finance, and Ornstein-Uhlenbeck processes in random environments are considered.

9.1

Introduction

The Ornstein-Uhlenbeck processes introduced in §5.2 and §6.2 are of great importance in applications. Let H be a separable Hilbert space and (St, a probability space with a filtration Ft, t > 0. Ornstein-Uhlenbeck (0-U) processes are solutions X(., x) = X to the equation

dX = AX + BdW, X(0)=xEH,

(9.1.1)

where W is a cylindrical Wiener process, usually on another separable Hilbert space U, and B E L(U; H). We denote by Pt, t > 0, the transition semigroup Ptcp(x) = E [ (X (t, x))], t > 0, .x E H, 4o E Bb(H). 177

Chapter 9

178

We first summarize results on invariant measures for O-U processes

obtained so far in Theorem 6.2.1, Theorem 7.2.1 and Proposition 7.2.3.

Theorem 9.1.1 Assume Hypothesis 6.1. (i) There exists an invariant measure for (9.1.1) if and only if the process X is bounded in probability and if and only if X is bounded in second moment. (ii) The invariant measure for (9.1.1) is unique if and only if the only invariant measure for the deterministic equation

z'=Az is 8{o}.

(iii) Assume, in addition, that for some s > 0 Ps is a strong Feller transition semigroup. Then an invariant measure for (9.1.1) exists if and only if there exists M > 0, w > 0 such that IIS(t)II < Me-wt, t > 0.

(9.1.2)

Moreover if p is an invariant measure then it is unique and lim Ptcp(x)

t- +00

(9.1.3)

for every xEH,cpEBb(H). Remark 9.1.2 By Theorem 7.2.1, P8 is a strong Feller semigroup if and only if Hypothesis 7.3 with t = s holds.

9.2

Ornstein-Uhlenbeck processes of wave type

9.2.1

General properties

We start from a general result on stochastic linear systems

dX(t) = AX(t) + BdW(t), (9.2.1)

X(0)=xEH,

Ornstein-Uhlenbeck processes

179

with the operator A generating a Co- group of operators S(t), t E R. Linear stochastic wave equations provide a typical application of this study. As in the preceding section, W(.) stands for the cylindrical white noise on a Hilbert space U, and B E L(U; H).

Theorem 9.2.1 Assume that the operator A generates a Co- group of linear operators S(t), t E R. (i) Equation (9.2.1) has an H-valued solution if and only if B is a Hilbert-Schmidt operator. (ii) If B is a Hilbert-Schmidt operator and dimH = +oo, the transition semigroup corresponding to (9.2.1) is never a strong Feller semigroup.

Proof - (i) If there exists a solution to (9.2.1) then, for any T > 0, the stochastic integral T

J0

S(T - s)BdW(s), t > 0,

should be a well defined Gaussian random variable. In particular its second moment T

2

E f T S(T - s)BdW(s) 0

_

JIs(r)B11Hs ds,

should be finite. From the group assumption it follows that there exist constants cl > 0 and c2 > 0 such that c2IxI < JS(r)xj < cljxj, r E [0,T], X E H.

Let {en} be an orthonormal and complete basis in U. Then II S(r)BII Hs

00 = E IS(r)Ben12,

n=1

and consequently 00

00

C2 > BenI2 < II S(r)BII Hs < ci > Ben12, r E [0,T]. n=1

n=1

Since f o IIS(r)B112 Sds < +oo, part (i) follows.

Chapter 9

180

(ii) It is enough to show that the control system

y' = Ay + Bu, y(O) = x,

(9.2.2)

with B a Hilbert-Schmidt operator is not null controllable in any time T > 0. For this purpose we recall a well known result, that we prove for the reader's convenience.

Proposition 9.2.2 Assume that U and H are separable Hilbert spaces and B is a linear compact operator from U into H. Then the operator G : L2(0,T; U)

rT

--- H, Cu = J S(r)Bu(r)dr, 0

is also compact.

Proof - Since the operator B is a limit, in the operator norm of linear operators with finite dimensional range, one can assume that

U=Band Bu = bu, u E R, where b E H. Then the adjoint operator C* to L has the form

L*x(r) = (S(r)b,x), x E H, r E [0,T]. It is enough to show that C* is compact. Let {xn,} be any bounded sequence in H and let {xnk } be a sub-sequence weakly convergent

to x. Then for arbitrary r E [0,T], (S(r)b, xnk) -+ (S(r)b,x), and therefore, by the Lebesgue dominated convergence theorem, ,C*xnk --+ C*x in L2(0,T; R).

This finishes the proof of the proposition. To complete the proof of (ii) notice that the solution of (9.2.2) at moment T > 0 is zero if and only if

-S(T)x = Lu. Since the range of S(T) is the whole of H, the operator L would be surjective. This is impossible if dim H = +oo since L is compact.

Ornstein-Uhlenbeck processes

181

Assume now that B is a Hilbert-Schmidt operator. Then a necessary condition for existence of an invariant measure for (9.2.1) is

that

f+oo

IIS(r)BII Hs ds < +oo.

0

In particular liminf II S(r)BIIHS = 0. r-.+oo

This means that if B # 0, then the semigroup S(r), r > 0, should have a dissipation property, at least for some x $ 0: liminf IS(r)xI = 0. r-+oo

(9.2.3)

Such systems are considered in the next sub-section.

9.2.2

Second order dissipative systems

Let H be a separable Hilbert space, A a positive definite symmetric

linear operator on H, and C : U -+ H a linear bounded operator. We consider the stochastic equation

dX(t) = Y(t)dt,

dY(t) = (-AX(t) - aY(t))dt + CdW(t),

(9.2.4)

X(0) = x, Y(0) = Y. The system (9.2.4) is of the form (9.2.1) where the Hilbert space H should be replaced by 1-l = D(A1/2) x H, with the norm 2

= IA1/2xI2 + Iy12.

Moreover the infinitesimal generator A can be defined by

D(A) = D(A) x D(A1/2), Al y

l

-(

0 a) (y A

y

) ED(A),

Chapter 9

182

and B is given by

B( y)-(Cx) If a = 0 one can write an explicit formula for the corresponding group of transformations

S°(t)

x

cos t

_

Y

t1(x

A-1/2 sin

_A1/2 sin vrA- t

cosy

t

/f

y

, t > 0.

For the general a write

Cy(t))=S(t)(y ), x

x y

Then by direct calculations we get

ld 2 dt

2

_ -aly(t)12, t > 0.

C y(t) x(t) )

(9.2.5)

So, if a < 0, then

rc

for arbitrary (

ai

x) E 1-l, and (9.2.3) cannot hold. Therefore if C 54 0 Y

and there exists an invariant measure for (9.2.4), then a > 0. But then, see e.g. A. Pritchard and J. Zabczyk [126, Proposition 3.5], IIS(t)117y < Me-"t, t > 0,

for some positive M and w. Taking into account Theorem 9.1.1 we get the following result.

Theorem 9.2.3 Assume that C : U --> H is a non-trivial HilbertSchmidt operator. Then there exists an invariant measure for (9.2.4)

if and only if a > 0. If a > 0 then the invariant measure is unique and Gaussian.

Ornstein-Uhlenbeck processes

183

Example 9.2.4 (Damped wave equation) Formally this can be written in the form a2

ate

X(t, E) = OX(t,x)-c

X(0' )

= x(e),

X (t, )

=

0,

at X (0, e) =

E 0, t >

E 0, t > 0,

where 0 is a bounded closed subset in Rd, and WQ is the Wiener process with covariance Q = BB*. To reduce this equation to the form (9.2.4), we set H = U = L2(0) and define

AY = -Ay, y E D(A) = H2(0) n Ho(0). Then D(A1/2) = H0(O) and the norm 11 JJj is merely the energy norm of the classical theory. The operator Q is usually given in the integral form Qx(e) = 1n q(f, q)x(rl)dq, f E H,

E 0,

with Q being the spatial covariance of the noise process.

9.2.3

Comments on nonlinear equations

Of great interest are nonlinear equations

dX(t) _ (AX(t) + F(X(t)))dt + BdW(t), (9.2.6)

X(0) = 0,

with A generating a Co-group S(t), t E R, on a Hilbert space H. If F is Lipschitz continuous there exists a unique solution X(., x) of (9.2.6) by Theorem 5.3.1. Since the operators S(t) are not compact one cannot use Theorem 6.1.2 for establishing existence of invariant measures. By Theorem 9.2.1 we cannot expect that the solution to (9.2.6) will be a strong Feller one and uniqueness, if it occurs, should be investigated with

Chapter 9

184

different methods from those developed in Chapter 4. In some situations however dissipativity arguments from §6.3 can be adapted and used. Note for instance that by formula (9.2.5) the semigroup S(t), t > 0, is dissipative. However, it can happen that an estimate of some moment of X(.,x) in a norm 11 Ily with Y compactly embedded in H is available. In this case the existence of an invariant measure follows from a tightness argument. See e.g. H. Crauel, A Debussche and F. Flandoli [30] where the existence of an invariant measure, and also of an attractor, for the wave equation is proved.

9.3

Ornstein-Uhlenbeck processes in finance

A slight modification of Theorem 9.1.1 is needed to discuss ergodic properties of a process describing the evolution of a term structure of interest rates as proposed by M. Musiela [121], see also A. Brace and M. Musiela [14]. Let t > 0, _> 0, be the rate at which one could enter a

contract at date t > 0 to borrow or to lend for a period of time at date t + e. The risk related to borrowing and lending is represented by a real, nonnegative volatility function satisfies, see [121], the following equation:

dX(t, e) _

r(e)

X(0,e) = x(e),

Jo

0. The process X

r(,q)dq I dt +

> 0, t > 0, (9.3.1)

where W (t), t > 0, is a one dimensional Wiener process. To cover a large class of functions, we choose as the state space H a weighted

Hilbert space L? = L2(0,+00), 'Y > 0, with weight e-'K,

0.

Thus x E L7 (0, +oc) if and only if +00

J

Ix12

< +00.

(9.3.2)

Let A denote the infinitesimal generator of the (left-shift) semigroup

S(t)x(e) = x(t + ), t > 0,

> 0, x E L,.

Ornstein- Uhlen beck processes

185

Then IIS(t)IIG(L2) < e21, t > 0.

Let moreover B be the one dimensional operator from U = R in L., given by

Bu(d) = ur(e), u E R, e > 0. Then equation (9.3.1) can be written in an abstract way:

dX = (AX + a)dt + BdW(t), (9.3.3)

X(0) = x E H, where

One of the main concerns of the paper [121] was existence and uniqueness of an invariant measure for (9.3.1). The following result is a direct extension of Theorem 6.2.1.

Theorem 9.3.1 (i) There exists an invariant measure for (9.3.3) if and only if

(il) supt>o TrQt < +oo. (i2) There exists an invariant measure for the deterministic system

z'=Az+a.

(9.3.4)

(ii) If conditions (ii), (i2) are satisfied then any invariant measure for (9.3.3) is of the form

y = v * N(0, Q.) S(r)QS*(r)ds, and v is invariant for (9.3.4). (iii) If (il) holds and a E Image A then any invariant measure for (9.3.3) is of the form where

µ= v* 6{b} * A1 (O, Q,,)

where v is invariant for z' = Az and b E D(A) is a vector such that a = Ab.

Chapter 9

186

Proof - The first two parts can be proved in a similar way as Theorem 6.2.1 and therefore the proof will be omitted. To show (iii)

notice that a function z is a solution to z' = Az + a if and only if y(t) = z(t) + b, t > 0, is a solution to y' = Ay and therefore the formula for the invariant measure for (9.3.4) follows. To apply the theorem to equation (9.3.1) notice that the condition (i) is equivalent to +00

(S(t)rl2,dt < +oo.

J0

(9.3.5)

Moreover the condition that a E Image A means that a E L22 and b(E) = fo a(rl)drl, > 0, belongs to L' as well. Starting from these observations T. Vargiolu [152], proved the following result.

Proposition 9.3.2 (i) Assume that -y = 0 and T is a bounded function. Then a necessary and sufficient condition for the existence of an invariant measure for (9.3.3) is

f

+00

ulT(u)ldu < +oo.

0

If this condition is satisfied then there exists exactly one invariant measure for (9.3.3). (ii) If y > 0 then a necessary and sufficient condition for the existence of an invariant measure for (9.3.3) is +00

72(u)du < +oo. JO

If this condition is satisfied then there exists an uncountable number of invariant measures for (9.3.3).

9.4

Ornstein-Uhlenbeck processes in chaotic environment

We assume here that H D U = L2(]Rd ) and that A is the operator

A=A - mI,

Ornstein-Uhlenbeck processes

187

where A is the Laplacian, and m _> 0. In particular, if H = U the corresponding heat semigroup is e-mt

S(t)x(O =

J pt(S (9.4.1)

= e-9"tPt *

x E U, 6 E Rd

where

pt(y) =

(4irt)-d/2e-t, t > 0,

E Rd.

We will consider the equation

dZ = (A - m)Zdt + JdW, Z(O)

(9.4.2)

in two important cases: when Q = I and when Q is a convolution operator,

Qu=q*u, uEU, see Example 5.2.1. Moreover J is the embedding operator of U into H.

9.4.1

Cylindrical noise

If Q = I then equation (9.4.2) has no solution in H = L2(Rd) because T

Tr QT = J e-2mt Tr S(2t)dt = +oo, 0

for arbitrary T > 0. It is therefore of interest to look for solutions of (9.4.2) in larger spaces. As a natural candidate we consider weighted spaces H = L2(Rd), where the weight p is a continuous, positive and integrable function. However, other choices are possible and have been considered by several authors. For instance the space S(Rd) of tempered distributions is taken as the state space for a generalized Ornstein-Uhlenbeck process by R. A. Holley and D. Stroock [88]. This space is natural in some applications to branching diffusions and to statistical mechanics [89], see also D. A. Dawson [50], [51], D. A. Dawson and H. Salehi [53], D. A. Dawson and G. C. Papanicolau [52].

Chapter 9

188

Proposition 9.4.1 Assume that p is a positive, continuous and integrable function on Rd such that, for some w > 0,

pt*p<ewtp,Vt>0.

(9.4.3)

Then, for arbitrary p > 1 and m E R, the formula

S(t)x = e-"(Pt * x), t >_ 0, defines a Co-semigroup on LP(Rd) spaces, and II S(t)II L(LP(j d)) < ep-m)t, t > 0.

Proof - We can assume m = 0. Consider first

(9.4.4)

p > 1. Setting

q = ppl and taking x E LP(Rd) we have

j

p

Rd

(

pt(e - 9)x(77)d?7

f-

(Jf

n)dn )

l (JPt(

ii)I x(ii)I pdri)

p

p /q

- n) d?7)

Consequently

fd I'd pt( - n)I x(n)I pp(e) drl d

I S(t)xI LP(d)

fed I

<

p

Pt(l -

ewtI'dlx(?l)Ipp(rl)d?l ewtIxIpLP(d).

Also we see that II S(t)II

ewtI', t > 0,

d) dn

Ornstein-Uhlenbeck processes

189

and therefore the estimate (9.4.4) holds. Continuity of S(t)x, t > 0, is straightforward for functions x which are continuous and with have compact support. The case of general x follows by a standard density argument. Finally, the case p = 1 can be handled in the same way.

Definition 9.4.2 A positive, continuous, integrable weight p is said to be admissible for the operator A - mI if and only if the operators S(t), t > 0, form a Co-semigroup in LP(Rd). Remark 9.4.3 Nonnegative, continuous functions p satisfying (9.4.3) are called w-excessive in probabilistic potential theory, R. M. Blumenthal and R. K. Getoor [10]. They are solutions of the inequality (9.4.5)

AP :5 wP,

understood in the sense of distributions, see L. Schwartz [135]. We will need the following lemma.

Lemma 9.4.4 Assume that p(x) = i(Ixl), x E Rd, where 0 is a C1 function on [0,+oo[, with 0" continuous and integrable on any bounded closed subinterval of ]0, +oo[. Then the estimate (9.4.3) holds if and only if

0"(y) +

d

r

1

'(y) < WOW, y > 0,

and in addition if d = 1 then 0'(0) < 0.

Proof - If d = 1 then, see L. Schwartz [135], Ap(x) = 2L'(0)b{o} + "(Ixl), x E R

and if d > 1 then Op(x) = 0"(I xl) + and the result follows.

dlxl1 b'(Ixl),

x

Chapter 9

190

We will consider two specific families of weights

pr(x) = e-"lVl, x E Rd, where ic > 0 and 1

1 +roll , x E R

p,c,r(X) =

d

where x > 0 and r > d. When the number r is fixed we will write for

short p, We write H" = L'. (Rd) and Hr = Lp,t r(E8d). The latter notation is justified since for fixed r > d the weights pK,r, for all c > 0, give rise to isomorphic spaces. If it is not stated otherwise Hr is equipped with the norm of L2K r(Rd). For the weights introduced we have the following result.

Proposition 9.4.5 For the heat semigroup S(t), t > 0, given. by (9.4.1) we have the following estimates (i) If p = p" and p > 1, then

2

- t > 0. it S(t)II LPP(Xd) < e ()t (ii) If p = p,c,r and p > 1, then e(KP-M)t

IIS(t)IILP Rd <

where

E=b(r,d)=

t > 0,

r-d-}-2 (r-1)2(1-*).

r Proof - The proof follows from the previous lemma. We give only basic steps of the calculations.

(i) Here l(y) = e-"y, y > 0. Consequently 1'(0) Moreover Oy)_0(y)1K2-dy1KJ

(y)+dy1 and

0(y)

(K2

if and only if r. 2 < w.

-dy

1

# ) < wO(y), V y > 0,

< 0.

Ornstein-Uhlen beck processes

191

(ii) In this case 1

O(y) = T+ nyr' y > 0, icryr-1

(y) N

(1

+,yr)2, y >

K2r2y2r-2

(y) - 2 (l

0,

/Gr(r - 1)yr-2 (1 + ,cyr)2

+ ,cyr)3

y>0.

Note that 0'(0)=0andforally>0 WOW-,O"(y)-dy101(y) = (1 + 1 r)3

y[w(1

Yvyr)yr-2,r(r

+(1 +

- 2n2r2y2r-2)

+ ryr)2

+ d - 2)].

Therefore

wO(y) - 0ii(y) -

d

1

V), (Y) > 0

Y

for all y > 0 if, for instance, w(1 + kyr)2 > 2n 2r2y2r-2 - ic2r(r + d - 2)y2r-2, y > 0,

or equivalently, if 2r-2 w > l£2r(r + d - 2) (1 +

y > 0.

(9.4.6)

The right hand side of (9.4.6) attains its maximum for y such that yr = r-1. So (9.4.6) holds for all y > 0 if and only if 2

W > K,

as required.

r-d+2 (r - 1) 2(1-T) r

Chapter 9

192

Proposition 9.4.6 (i) Assume that p is an admissible weight for 0 - mI. Then the solution to equation (9.4.2) exists in L2(Rd) if and only if d = 1. (ii) If d = 1, then there exists an invariant measure for equation (9.4.2) in H" if and only if m > 0. (iii) If d = 1, and m > 2 , then the invariant measure for equation (9.4.2) is unique in H (iv) If d = 1, and 0 < m < 4 , then there are many invariant measures for equation (9.4.2) in H'. (v) If d = 1 then there exists exactly one invariant measure for equation (9.4.2) in Hr.

Proof -

(i) Let {ej } be an orthonormal and complete basis in

L2(Rd) and {/3j} independent real-valued Wiener processes. Define, see §5.2, the cylindrical Wiener process W(t) by the formula 00

W(t) = E

(t).

j=1

Assume that WW(t) is L22(Rd)-valued. Then EIN'o(t)I2 = F

P( e)IW(t,e)I2de) (Ld

= J'd P( ) E

I

j

2

S(t - s)ej(e)d(s) j=1

t00

PO d

E I S(O')

d

=1

t 00

_

P(f) J0

j=1

I (ej,p'( - .))I2do,<,

d.

Ornstein-Uhlenbeck processes

193

where (ej, is the scalar product in L2(Rd) of e7 and Consequently, by Plancherel's identity, EI Wo(t)12 =

Ld

I(- )I2(Rd)dadIt

)f

t

[

JLd' (S)dS f I pcI22(ld)dQ.

But 1p

,

e

1

_ (4jrv)d JJ,d

L2

_d -

1

2d

1

(27rs)d

Consequently EIWo(t)12 < +00,

if and only if d = 1. (ii) Assume d = 1. In the same way one has

'

IWo-m(t)12

=

Jd

1

d 2

1

2zr 10

So if and only if m > 0 we have Sup EI Wt _m(t)12 < +00, t>O

and the result follows.

(iii) It is enough to notice that for the semigroup S(t), t > 0, corresponding to A - m one has HS(t)HL(HK) <

e-(m

2 )t, t > 0,

2z

with m - > 0. (iv) Let S(t), t > 0, be the semigroup corresponding to A - m with m > 0. Note that the function equation Ox = nix. Moreover

E R, satisfies the

Je_'I(e)I2de = Je_?e2i'de < +oo

Chapter 9

194

if and only if rc-2/ >0.If rc-2/>0then S(t)x=x for all t > 0 and b{7) 54 b{o) is an invariant measure for z' = (0 - m)z. By Theorem 9.1.1(ii) this finishes the proof of (iv). (v) Let us choose r > 0 such that "p 6 < m. Then by Proposition 9.4.5(ii) the semigroup S(t), t > 0, is exponentially stable in L2 and Theorem 9.1.1 implies the result.

9.4.2

Chaotic noise

We assume now that the covariance operator Q, of the Wiener process W, is of convolution type:

Qu=q*u, uE U=H=L2(R ),

(9.4.7)

where q is a continuous, integrable, positive definite function on l[8d see Example 5.2.1. In fact we will assume the following:

Hypothesis 9.1 The operator Q is of the form (9.4.7) where the function q is integrable and of the form

q(6) = Jd e1(f>n)g(q)di, 6 E Rd,

(9.4.8)

with g a nonnegative integrable function.

It follows from Hypothesis 9.1 that q and g are continuous and bounded. The representation (9.4.8) is natural in the light of Bochner's theorem.

Theorem 9.4.7 (i) Assume that Hypothesis 9.1 holds and that p is an admissible weight for 0 - m. Then equation (9.4.1) has a weak solution on Lp(Rd).

(ii) If in addition m > 0 then there exists an invariant measure for equation (9.4.1). Moreover if m = 0 then there exists an invariant measure for equation (9.4.1) if and only if f1d g(17)drl < +oo. In the latter case the invariant measure is never unique.

(iii) If p = p" and m > 2 then invariant measures are unique and if m < 2 then they are not unique. (iv) If r > d and m > 0 then there exists exactly one invariant measure for equation (9.4.2) in Hr.

Ornstein-Uhlenbeck processes

195

Proof - Using the same notation as in the proof of Proposition 9.4.6 we have that W (t'

t > 0,

Q112ej

E Rd.

j=1

By Hypothesis 9.1, Q112u = ql * u, u E L2(Rd), where

e2rl)drl,

fLd

E Rd.

So we have, compare the proof of Proposition 9.4.6(i), that E I Wo-m.(t)I2 = E 00

IWo-m.(t, S)I2 = E

it

00

((ps j=1

f (P(e) I Wo-m(t, E)I2 d ,d

F

e-m(t-s)pt-s

110 t

* ql * ej(S)dOj(S)

* qlej(.))i2ds 2

ft

J e-2mslps * g1I22ds,

E

0

Consequently ]E

t

L2

IWo-m(t, )I 2 =

f p(71)dr7 f e-2mslps * gIIL2ds d

0

By Plancherel's theorem ps * q1 IL2 = (27r)d lid e-2sIt

Therefore >E IWO-.(t, t

x J.d

Cfo e

22 = (2.)d fed p(i)dq -2(.+IC12)s

ds g(ri)dri.

2

Chapter 9

196

Since g is an integrable function, lE IWA_m(t,

e)12

p < -boo for arbitrary t > 0.

This shows (i). To prove (ii) it is enough to remark that sup E IWo-m(t, )ILP = t>0

(27r )d fit p(rl)dq d 1

2

g(rl) Rd nz+I7I a dry < +oc,

provided that m > 0, and for m = 0 one has to require that

Jd

(E)

e < +oo.

The proofs of (iii) and (iv) are completely analogous to those of (iii),(iv) and (v) of Proposition 9.4.6 and will be omitted. We will now construct functions q and g such that (9.4.5) holds.

This way we will show that even if m = 0 invariant measures for (9.4.1) may really exist.

Example 9.4.8 For arbitrary y > 1 the even function (1 - Ixx'1) if

11

0 if ICI > 1,

is concave on [0, +oo[ and by a result of Polya, see W. Feller [64, Vol. II], and Plancherel's theorem, it is a characteristic function of an even, integrable, nonnegative continuous function V& : e'1n,0(ii)d?1,

f E R.

We define

g(f) = 2 (h( - 1) + h( + 1)) ,

Ornstein-Uhlenbeck processes

197

Then

f

etEn

q(rl) = J

2

_

1)d +

2ei77

11

(h( - 1) + h(f + 1)) d 2e-in fig ei(£+1)nh( +

2ir cos r1o(i ), rl E R.

It is therefore clear that Hypothesis 9.1 is satisfied. If, in addition, y > 1 then

Jfd< f 12

2-ry

d+ f

5II

so the condition (9.4.5) holds as well.

1

1>1

P

Chapter 10

Stochastic delay systems In this chapter we study invariant measures for stochastic delay equations by the dissipativity method.

10.1

Introduction

Let d, m E N, r > 0, a() be a finite d x d matrix valued measure on [-r, 0], b a d x m matrix, f a mapping from Rd into Rd and W(.) a standard m dimensional Wiener process. In this section we are concerned with a stochastic delay equation of the form

dy(t) = Cf a(d9)y(t + 9) + f(y(t))) dt + bdW(t), 0r

(10.1.1)

Y(O) = xo E Rd,

Y(O) = x1(9), 9 E [-r,0].

For simplicity of the presentation we assume that f is Lipschitz continuous. It is known, see A. Chojnowska-Michalik [24], that if H = Rd x L2(-r,0,Rd) then the H-valued process X,

t>0,

X(t) where

yt(9) = y(t + 9), t > 0, 9 E [-r, 0], 199

Chapter 10

200

is a mild solution of the equation

dX(t) = (AX(t) + F(X(t)))dt + BdW(t)

(10.1.2)

where the operators A, F and B are defined as follows A

P

J

for

=(

1

J

dW

de

(10.1.3)

D(A) _

F(xo X1

c0(0)

E Wl'2(-r,0,Rd) } ,

'(0o) Bu

b0u

X1 ) EH, X1

I, uERm.

(10.1.4) (10.1.5)

The process X will be called an extended or a complete solution to (10.1.1) and denoted by X(t, X) = C y(t; xo, xi)

yt(.;xo,xi)

Stochastic delay equations can be studied and have been studied as equations on Rd, without any reference to the theory of stochastic evolution equations. Basic work in this direction is due to K. Ito and M. Nisio [92]. From more recent publications let us mention M. Scheutzov [131], [132], [133] and S. A. Mohammed and M. Scheutzov [119].

In the present chapter we apply general results developed in the first two parts of the book. We are concerned with linear systems and with applicability of dissipativity method to nonlinear ones.

10.2

Linear case

We start from linear delay equations for which the abstract version is

dX(t) = AX(t)dt + BdW(t), (10.2.1)

X(0)=xEH.

Stochastic delay systems

201

Note that the system (10.2.1) is genuinly infinite dimensional while the noise process W is finite dimensional. To simplify the following presentation we restrict our considerations to the discrete delay case, N

dy(t) _ (aoY(t) +

icl

ay(t + 8)) dt + bdW(t), (10.2.2)

Y(O) = xo, y(8) = x1(8), 0 E [-r,0],

when -r = 01 < 02 <

< ON < 0 = 8N+1. Then the measure a(.)

is of the form N

aiboi (r), F E B([-r, 0]).

a(F) = a050(F) +

(10.2.3)

i=1

We have the following result on the generator A and the correspond-

ing semigroup S(t), t > 0, relevant to the asymptotic behaviour of (10.2.1).

Proposition 10.2.1 (i) The operator A given by (10.1.3) generates a Co-semigroup S(t), t > 0, such that the operators S(t), t > r, are compact.

(ii) The spectrum a(A) of A is of the form N

Q(A)= AEC:

det

AI-ao-Eeao,aj =0

,

(10.2.4)

j=1

and the semigroup S is exponentially stable if and only if sup{Re A : A E Q(A)} < 0.

(10.2.5)

For the proof we refer to J. Hale [82].

Example 10.2.2 Assume that d = 1, and N = 1, then the stability condition (10.2.5) holds (for the corresponding generator A) if and only if

ao < 1, ao < -a1 < y2 + ao, where 0 < y < ir, and y coth 7 = ao, see N. D. Hayes [84].

(10.2.6)

Chapter 10

202

Theorem 10.2.3 Let Pt, t > 0, be the transition semigroup on Bb(H) corresponding to (10.2.2).

(i) Pt, t > 0, is a strong Feller semigroup for all t > r if and only if

N

rank [AI -

e' B' ai, b]

= d,

(10.2.7)

:-1

for all A E C.

(ii) Pt, t > 0, is irreducible for all t > r if and only if N

rank [AI - ao -

i.l

e'e'ai, b] = d, rank [al, b] = n,

(10.2.8)

for all A E C.

Proof - It follows from A. W. Olbrot and L. Pandolfi [122] that (10.2.7) is equivalent to the null controllability of the control system

corresponding to (10.2.2). Then by Theorem 7.3.1 (i) is true. In a similar way the proof of (ii) follows from a characterization of the approximate controllability due to A. Manitius [110] (for an alternative

proof see It. F. Curtain and H. J. Zwart [32]), and the fact that the corresponding control system is approximately controllable in time t if and only if the image of Qt = fo S(t)BB*S*(t)dt is dense in H. Proposition 10.2.4 A sufficient condition for the strong Feller prop-

erty of Pt, t > 0, for all t > r is that rank [b, aob,...,

ao-1 b]

= d, and Image b D Image ai, i = 1, ..., N. (10.2.9)

Proof - Consider the control system N

z'(t) = aoz(t) +

aiz(t + Bi) + bu(t), i=1

(10.2.10)

z(0) = xo, z(9) = x1(9), 9 E [-r, 0],

and introduce new control functions uo and v such that u(t) = uo(t)+

v(t), t > 0, and E"_'1 aiz(t+9i)+bu(t) = bv(t),t > 0. Then (10.2.10)

Stochastic delay systems

203

becomes z'(t) = aoz(t)+bv(t), and, by the rank condition, the system can be steered to zero.

The main result of the present section is formulated in the following theorem.

Theorem 10.2.5 (i) Assume that (

sup

t

N

r

l

1

ReA:det[AI-Eeoai] =0} <0,

i-

(10.2.11)

JJJ

then there exists exactly one invariant measure for (10.2.2). (ii) If there exists an invariant measure for (10.2.2), and N

rank [AI_

ll

eAe* ai, b]

=d

for all A E C then (10.2.11) holds.

Proof - The result is a direct consequence of Theorems 9.1.1, 10.2.3 and Proposition 10.2.4.

Remark 10.2.6 (i) If d = 1 and b

0 then there exists an invariant measure for (10.2.2) if and only if (10.2.6) holds. (ii) If condition (10.2.9) holds then there exists an invariant measure for (10.2.2) if and only if (10.2.11) holds.

10.3

Nonlinear equations

We conjecture that if there exists a solution to (10.1.1) which is bounded in probability then there exists an invariant measure for (10.1.1). However, this result does not follow from Theorem 6.1.2 since the operators S(t) are compact only for sufficiently large t > 0. Some results in this direction can be found in K. Ito and M. Nisio [92]. It is also possible that if the transition semigroup for the linear equation is a strong Feller one, see §4.2, then the semigroup corresponding to the nonlinear equation (10.1.1) is likewise but the proof

Chapter 10

204

of this result is also laking. Consequently, for the proof of the uniqueness of the invariant measure, Doob's theorem cannot be applied at the moment. Results on this topic, based on Girsanov's theorem, are contained in M. Scheutzov [132]. In this section we discuss applicability of the dissipativity method for the equation N

dy(t) =

aoy(t) + E aj y(t + 9j) + f(y(t))

dt + bdW(t),

j=1

y(0) = xo E Rd,

y(9) = x1(9), 0 E [-r,0]. (10.3.1)

We start from the crucial question: under what conditions does exist

a positive number w such that the operator A + wI is dissipative, compare §6.3.2, Unfortunately, with respect to the original norm in H = Rd x L2(-r, 0; Rd), the operator A does not necessarily have this property. If, however, we replace the original norm by an equivalent one of the form 0 In(

20

)III2=Ixol2+ J TT(9)Ixl(9)I2 d9,

xo x1

) E H,

with the function T satisfying 71 < T(0) <- 'Y2, 0 E [-r, 0],

y1 and 72 being positive numbers, the situation can change dramatically. It has been shown in G. F. Webb [161] that if N -r(O)

= E (lal l +

... + l aj l) X]A,.,A,+,,' 0 E [-r, 0],

j=1

and

(aoz,z) < -aoIzl2, z E Rd, then

IIIS(t)III < e"t, t > 0,

(10.3.2)

Stochastic delay systems

205

for

m(o,ijI ax l-ao Th us in the estimate (10.3.2), the speed 7 is nonnegative, a result not satisfactory for our purposes. We will therefore introduce a different function T, N

r(9)

7e2,Qe

=

(10.3.3)

+ E TjX]ej Bj+il , 9 E [-r, 0], j=1

where 3 > a and

Tj=y(a-1) e2aej, j=1,2,...,N.

(10.3.4)

Theorem 10.3.1 Assume that the matrix ao+aoId (1) is dissipative on Rd with ao > 0 and that for positive numbers ,3 > a > 0,

ao > a+

/2

a

a

1/2

la112

a

t3 e2Qei + 0

X

(10.3.5)

N E e2Qej - e2Qej-1 =2 1aj12

a

Then there exists y > 0 such that in the corresponding norm

Ill

III

one

has

lHS(t)111 < e-"", t > 0

(10.3.6)

Proof - Denote by -< , >- the scalar product X1

I,\

0

Z1

X0

where

x0 X1

>- = (x0, zo) + f T(O)(xi(O), zi(0))dO, r

o

zo

, ( z1fI E H. Then the operator A+ald is dissipative

with respect to the scalar product -
X(O)

x(0)

,

>- if and only if

>-< -a (1x(0)12 + J T(8)1 x(9)12d9 (10.3.7)

11d represents the identity on Rd.

206

for arbitrary x E W1'2(-r, 0; Rd). Note that A

X(O)

(x(0))

),

N

_ (aox(0),x(0)) + E (aj x (0j), x (0)) j=1

+ Jeo T(0)

(de (0), x(0))

dO

= J1 + J2 + J3.

We have J3

jr(O)

=

y

(de (0),x(0)) dO

Joe''" ((0),x(0))d9 01

N + >j=1 r,

f

dx

Bi+1

(T(0),

6i

x(0))

d0.

But

j°e2/39((9),x(9))d9=

e2P6

2 Jeo

= and

(1x(0)12

dB

- e2,111, 1x(01)12) -,310e2119 IX(0)12 dO,

2

e3+1 / ddx (0),x(0)>

JJ

d IX(0)12 dO

de = 2 (Ix(ej+1)12 - I x(0j )12)

Moreover N

Jl < -aolx(0)I2, j2 E IajI Ix(0j)I 1x(0)1 j=1

N.

Stochastic delay systems

207

Therefore x(0) -< A

x

X

x(0)

x

N

< -aplx(0)I2 + E Iajl ix(ej)I Ix(0)I j=1

N +2 E Tj (Ix(ej+1)I2 - I x(Bj)I2) + /

j=1

--0 J

(Ix(0)I2 - e211,

IX(

01)12

2 0e2R°Ix(9)I2dO

< (-aO + TN2

)

Ix(0)I2

N

+2 1:(Tj_1 - Tj)Ix(O )I2 J=2 N

1

2

(71

+'Ye2)3e1)

Ix(81)I2

f

+E IajI I x(Oj)I Ix(0)I - yae2111Ix(0)I2d9. ° j=1

Consequently, for (10.3.7) to hold it is sufficient that

Cap-a-TN2 y )Ix(0)I2 N

+2 j=2

r _i)Ix(Bj)I2 + 2 (71 + ye2001) Ix(91)I2

(10.3.8)

N

E I aj I I x(ej)I Ix(O)I >- 0, - j=1

and ° (,y,3e20e 181

- ar(9)) IX(0)12 dO > 0,

(10.3.9)

Chapter 10

208

for all x E W1'2(-r, 0; Wi). The inequality (10.3.9) holds if and only if _ N e a 200 > 0 E ]01, 9N+1] a

j=1

Tjxloj.Bj+1l'

Taking into account (10.3.5) one easily sees that (10.3.9) is satisfied.

Moreover if Ix(0)I = 0 then the inequality (10.3.8) holds because Tj > Tj_1 for j = 2,...,N and Tl + 7e2Qe1 > 0. If Ix(0)I 0 then dividing both sides of (10.3.8) by Ix(0)I2 we get -Tj-F)-ye2,301)CIx(01)I12

12 EN(7-j N

(Ix(Oi)I) E - aj j=

Ix(o)I J

2

Ix(o)I /l1

j=1

7- a0.

I

(10.3.10)

By the definition (10.3.5), the minimal value of the left hand side of (10.3.10) is _1 4

N E (Tl+7e2P91 + j=2 Tj 2Ia112

1

a

a Ia1I2

27

2Iaj12

e2001 +

T7-1

N .7=2

- e2Qej-1 Iajl2

e2130j

Therefore (10.3.10) holds provided there exists 7 > 0 such that

ao-a> 2 (0-ae 200N + 1 1 1a lal12 +27

a

N

e2/301 + 3 - a =2 j

Iaj12

(10.3.11)

e210j-1

Minimizing the right hand side of (10.3.11) with respect to 7 > 0 one gets the desired result.

Stochastic delay systems

209

Remark 10.3.2 One gets even more explicit conditions implying (10.3.6) if one takes for instance /3 = 2a (or 0 = 3a, /3 = 4a,...). Different choices for the weight r are possible. For instance if N = 1 and T(0) = ye2a0, 0 E [-r, 0],

one finds that the inequality a0 > a + e' lall

implies (10.3.6).

The main result of this section is a direct consequence of Theorem 6.3.3, see Remark 6.3.4, and of Theorem 10.3.1.

Theorem 10.3.3 Assume that condition (10.3.5), holds and that the Lipschitz constant of f is smaller than a. Then, there exists exactly one invariant measure y for the extended solution, of (10.3.1).

Moreover there exists w > 0 such that for all x =

I

xo XI

E H,

cp E Bb(Rd) and t > 0

((y(t;xo,xi)) - f IP(z)µd(dz)/ `- (C + 2lxl)e-"tlIPIILip, d I

where µd is the projection of a on the Rd component of H.

Chapter 11

Reaction-Diffusion equations In this chapter we study invariant measures for stochastic reactiondiffusion equations. First equations on a finite space interval are studied and then equations on the whole of Rd and with the space homogeneous Wiener process are investigated.

11.1

Introduction

Deterministic reaction-diffusion equations are usually written as systems of equations

atk

akAyk(t, e) +

yi(t, c), ..., YK(t, c)), t > 0,

k=1,...,K, (11.1.1)

where fk, k = 1, ..., K, are given real functions from 0 x RK to R, Uk, k = 1, ..., K, are positive constants, and 0 is an open subset of Rd. If 0 # Rd one also adds boundary conditions of Dirichlet, Neumann or mixed types. 211

Chapter 11

212

The stochastic version of (11.1.1) is of the form

dXk(t, ) = (okzXk(t, E) +

X1(t, c), ..., XK(t, ))] dt

+ bk(e, X1(t, ), ..., XK(t, ))dWk(t, ), = xk(e), e E 0 C Rd, k = 1, ..., K, t > 0,

Xk(0,)

(11.1.2)

plus boundary conditions if 0 54 Rd. Moreover W1, , WK are independent Wiener processes and bk, k = 1, ..., K, are given real functions. Setting 0'10

X1(t, E)

X(t,C) =

A

W(t)

Wi(t)

,

XK(t,C)

WK(t)

CJ xd(S))

XI(S)

xl(b),...xd(S))

xd(S)

f1(b,xl(S),..

F(x)(C) _ fd(S, b1 (S, xl

.xd(S ))u1

ul (')

u()

(B(x)u)(C) = bd(C, xl (C), ...xd(C))ud

ud(')

one can write problem (11.1.2) in the standard form

dX(t) _ (AX + F(X))dt + B(X)dW(t), (11.1.3)

X(0) = X. To formalize this procedure one should give precise definitions of the operators A, F and B, of the state space F (or H in the Hilbert space case) and of the space U on which the cylindrical Wiener processes are defined. We will consider only one equation, K = 1, and distinguish two main cases:

Reaction-Diffusion equations

213

(i) d= 1,O=]a,b[, -oo 1, 0 _ R. In both cases the operator A will be the infinitesimal generator of a strongly continuous semigroup, S(t), t > 0. For each case various conditions on F and B can be considered. The diffusion mapping B can be independent of the state variable (then we have the socalled additive noise case) or can depend on the state variable in an essential manner. In the latter case the dependence might be of Lipschitz type or be less regular. A number of different conditions can be imposed on the mapping F. F can act as a Lipschitz mapping and therefore have linear growth, or it can have nonlinear growth. F can be locally Lipschitz or merely continuous or even less regular. All the above mentioned types of assumptions have an important impact on the existence and uniqueness of the solutions to (11.1.2). Additional requirements will be needed for existence and uniqueness of invariant measures. In some cases it will be possible to throw additional light on the limit behaviour of the transition semigroup. As our aim here is to illustrate applicability of the various techniques discussed in Part II, we will restrict our considerations to typical situations and examples. Possible extensions will be indicated in remarks and comments.

11.2

Finite interval. Lipschitz coefficients

In this section we assume that K = d = 1. We start from the analysis of the equation

a

2

X(t, 0) = X(t,1) = 0, X(0,E) = X(01 (11.2.1)

Chapter 11

214

for t > 0, E ]0, 1[. We will require that the Wiener process W is cylindrical on U = L2(0,1). This space will also be our state space H.

In this section it is natural to assume that 2

Ax = d-2 , x E D(A) = Ho'2(0,1) n H2'2(0,1), f(x(E)), Thus problem (11.2.1) is equivalent to problem (11.1.3).

11.2.1

Existence and uniqueness of solutions

We assume that f and b are Lipschitz continuous mappings. Consequently, F is likewise, but B is not from H into L(H). We will check, however, that conditions of Theorem 5.3.1 are satisfied. As the functions

=v sin

n E N,

E [0,1],

form an orthonormal sequence of eigenfunctions of the operator A, corresponding to the eigenvalues -ir2n2, n E N, we have II S(t)(B(x) - B(y))II

Hs

= E I00((B(x) - B(y))en, S(t)em) I2 n,m=1

e-2tl2m2I

(B(x) - B(y))eml2

m=1

But

I(B(x) - B(y))em I2

=2 <

I

1

2IIbIILipIx -

b(y(e)))2(sin m7r')2de y12.

Consequently I I S(t)(B(x) - B(y))II HS 5V IIbIILip I x - yI II S(t)II Hs,

so we see that Hypothesis 5.1(iii) holds.

Reaction-Diffusion equations

215

Moreover, for arbitrary T > 0 00

J

00

T

T II S(t)II Hsdt =

Za2 n=1

n=1

n < +00.

Thus Theorem 5.2.5 is applicable. In this way we have proved the following result, see J.B. Walsh [158], M. I. Freidlin [69], G. Da Prato and J. Zabczyk [44], R. Sowers [144].

Theorem 11.2.1 If f and b are Lipschitz continuous, then equation (11.2.1) has a unique mild solution and the corresponding transition semigroup is a strong Feller one.

11.2.2

Existence and uniqueness of invariant measures

We will prove the following result, see It. Sowers [144], as in G. Da Prato, G4tarek and J. Zabczyk [39].

Theorem 11.2.2 Assume that f and b are Lipschitz continuous functions such that (i) IIf IILip < 7r

2

(ii) b is bounded.

Then there exists an invariant measure for (11.1.3) on H.

Proof - The operators S(t), t > 0, are compact. Then to apply Theorem 6.1.2 it is enough to check condition (6.1.2). For this we use Proposition 6.1.6. Assumption (ii) of this proposition is automatically satisfied because 00

10

Iis(t)II HS dt = E n=1

J0

e-2t'r2n2dt =

1 n=1 E1 < +0 n2 00

2

and sup. IIB(x)II = supC +oo. To check condition (i) notice that

(Ax,x) < -ir2IxI2, x E D(A),

Chapter 11

216 and

(F(x + y), x) = (F(x + y) - F(y), x) + (F(y), x) <_ IIfIILipIxi2 + IF(y)IIxI < IIfIILipIXI2 + 1 e1x12 +

JF(y)I2J , X, y E H, e > 0.

It is therefore enough to use the fact that F has linear growth and choose e > 0 sufficiently small.

Remark 11.2.3 Using the factorization procedure one can prove that solutions of (11.1.3) take values, for t > 0, in E = C'([0, 1]) with a E ]0, 1/2[. Consequently if ti is an invariant measure for (11.1.3) then

µ(E) = JHPt(x,E)() = 1, t > 0, and the measure u is concentrated on E, see It. Sowers [144]. For uniqueness we have the following theorem, see R. Sowers [144], C. Mueller [120], S. Peszat and J. Zabczyk [124]. The present proof and the limit formula for Pt, t > 0, are taken from [124].

Theorem 11.2.4 Assume that f and b are Lipschitz continuous functions and for some constants rc, k > 0

Then there is at most one invariant measure for (11.1.3). If µ is an invariant measure for (11.1.3) then, for arbitrary x E H, I' E ,3(H),

t

iim Pt(x, r) = p(r). 00

Proof - By Theorem 4.2.1 and Proposition 4.1.1 it is enough to show that the transition semigroup corresponding to the solution of (11.1.3) is a strong Feller one and irreducible. The strong Feller property is not a direct consequence of Theorem 7.1.1 because B is not a Lipschitz transformation from H into

Reaction-Diffusion equations

217

L(H). However, as we saw in subsection 11.2.1, B satisfies Hypothesis 5.1(iii), and the proof of Theorem 7.1.1 workes with only one exception. That is we do not know if the mappings B,,,(x), x E H,

are invertible. But in the present particular case their invertibility is a simple consequence of the fact that either b(C) > m for all or -m for all . Irreducibility follows from Theorem 7.3.1.

11.3

Equations with non-Lipschitz coefficients

As in §11.2 we set K = d = 1; however, we no longer assume that the function f is Lipschitz continuous, but to simplify presentation we set the diffusion coefficient to be equal to 1. Thus we will consider now the problem dX (t, £) _

X (t, 0) + f (e, X(t, ))

12

dt + dW(t, e), (11.3.1)

X(t,0) = X(t,1) = 0, X(010 = X(01

for t > 0, E ]0, 1[. We will need the following conditions on

f.

Hypothesis 11.1 (i) The function f is locally Lipschitz continuous and (f ( + i) + l; + )) sgn < a( rl ), for all , q E R, where

a(r) < co + clr", for some co > 0, cl > 0 and p > 1. (ii) For some w E R, the function f - w is decreasing. By K we denote the space Co([0,1]) of all continuous functions on [0, 1] vanishing at 0 and 1 with the usual supremum norm. As in §11.2 we take H = L2(0,1). We define d2

Ax = dC2 - x, x E D(A) = H2(0,1) n Ho(0,1),

Chapter 11

218

and E [0, 1].

Theorem 11.3.1 Under Hypothesis 11.1(i) the solution of problem (11.3.1) on K has an invariant measure concentrated on K. If in addition Hypothesis 11.1(ii) holds then the solution of equation (11.3.1) on H has an invariant measure concentrated on K.

Proof - We will check the hypotheses of Theorem 6.3.5 and Theorem 6.3.6. The semigroup S(t), t > 0, generated by A is of the form

S(t)x = Y, 00 e-«"ten(en,x), x E H,

(11.3.2)

n=1

where an = -a2n2 - 1, n E N, and f sinnir , n E N. The same formula defines the restriction SK(t), t > 0, of S(t), t _> 0, to K. Both semigroups are compact repectively in H and K and I S(t)I

< e-2t, IISK(t)IIK < e-t, t > 0.

It is also clear that the transformation F has the desired properties. Therefore it is enough to check that the process

WA(t) = f S(t - s)dW(s), t > 0, ot

-

has an E-continuous version such that for arbitrary p > 1 supEIIWA(t)Ilfc < +oo. t>o

This follows however from Theorem 5.2.9.

Remark 11.3.2 In I. Gyongy and E. Pardoux [81], C. Donati Martin and E. Pardoux [55], and I. Gyongy [80], equation (11.3.1) is studied under weaker assumptions by using a comparison theorem for SPDEs. For a result about the existence of an invariant measure in this setting see G. Da Prato and E. Pardoux [41].

Reaction-Diffusion equations

11.4

219

Reaction-diffusion equations on d dimensional spaces

In this section we use the notation of §9.2. We will consider equations of the form

dX(t,E) = [(Ax(0, E) =

t > 0, E Rd, t > 0,

(11.4.1)

where A is the Laplace operator, m a nonnegative constant and the covariance operator Q of the Wiener process W is either I or a convolution operator

Qu(a) = q *

E Rd, U E U,

satisfying Hypothesis 9.1.

As far as the function f : R -p IR is concerned we will require that the following is satisfied.

Hypothesis 11.2 The function f is of the form

f=10+fi where fo(b)+ii , e E R, is continuous and decreasing for some rt E R, and for some s > 1 and co > 0 Ifo(E)I <- co (1 +

E R.

Moreover f1 is Lipschitz continuous.

For instance if f1 is Lipschitz continuous and 2n fo(tt S)

=

-e2n-F1 + r S

bktS2n-k, S` E R,

k=O

then Hypothesis 11.2 is satisfied with s = 2n + 1. We start from an existence result.

Theorem 11.4.1 Assume that f satisfies Hypothesis 11.2 and either

Chapter 11

220

(i) d = 1, and either Q = I or Hypothesis 9.1 holds or (ii) d > 1 and Hypothesis 9.1 holds. Then equation (11.4.1) has a unique generalized solution in H', s; > 0, and H,., r > 0. If in addition x E LP8(Rd) where p = p" or p = Pr, then the generalized solution is mild.

Proof - We will apply Theorem 5.5.8 with H = L2.(Rd), H = and K = LpK(][8d), Is = Lpr(l[8d). By Proposition 9.4.5 the heat semigroup can be extended to semigroups on all those spaces. If A,c,2 i Ar,2 i AK,2a i A,.,2s are the genera-

tors of the semigroups then, again by Proposition 9.4.5, the operators A.,2 + 1), Ar,2 + 17, A,c,2s + 77, Ar,2s +77 are m-dissipative for sufficiently small q7.

To see that the operator F + 77 where

F(x)(e) = f(x(e)), f E D(F)

_

J X E L 2(Rd) :

Rd,

f

E L2(][8d)},

sastisfies Hypothesis 5.4, note that D(F) J K and the domain D(FK) of the part FK of F in K contains LP3(Rd). It is also easy to see that F + 71 and FK + 11 are m-dissipative for small 77 and that F maps bounded sets in K into bounded sets in H. Then Hypothesis 5.4 holds. The following proposition shows that Hypothesis 5.5 is satisfied as well and consequently finishes the proof of Theorem 11.4.1.

Proposition 11.4.2 Let Ap,P, where p = p', K > 0, or p = pr, r > 0, and p > 1, be the generator of the heat semigroup in LP(Rd). Then the Ornstein-Uhlenbeck process WA,,, (t), t > 0, has continuous

trajectories in LP(Rd), provided that condition (i) or condition (ii) of Theorem 11.4.1 holds.

Proof- We will consider only the more difficult case (ii). Without any loss of generality, we can assume that p > 2 and that 00

C'`Q1/2ej(e)Nj(t), t > 0, W(t,S) = 3=11

E Rd.

Reaction-Diffusion equations

221

We will use the factorization formula, Theorem 5.2.5, and investigate first the process t

Ys(t) = f S(t - s)(t - s)-SdW(s), t > 0,

-

o

[. By Gaussianity of Y6(t, e), t E] 2 cl > 0, suitable constant where 5

p,

EIYs(t)I LP(] d) = IE t

lid P(E)E

f

f (t -

E W', for a

_> 0,

)IPdf

d

P

00

s)-s>(S(t

-

d

j=1

Pt -

s)-s E(S(t - s)Q1,2hj)(e)dij(s) j=1 P/2

P(S) t (10

< c1

t U-28 E 00

C

I S(S)Q1/2hj(S)I2ds

j=1

But

S(S)Q112hj(e) = e-asps * ql * hj(e), where

E Rd'

Q1/2u=g1*u, uEU.

Consequently IS(S)Q112hj(S)I2 = e-2as(ps * ql(S

and by Parseval's and Plancherel's identities

E

2 IS(S.)Q112hj(f)

=

-2asjjps * g1IIU

jElld e-2as f, d

AU

Chapter 11

222

Therefore, for a constant c2, P/2

EIYS(t)IPP(Ld) < c2 [g(n) (e-2s(m+InI2)s-2ads)

d(i)J

Since m > 0, for another constant c3,

]EIY6(t)ILP(jd) < [cs f

d (m + g(ij) IrlI2)2s+1

lp/2

+oo

for arbitrary t > 0.

Taking into account, see G. Da Prato and J. Zabczyk [44, p.128],

that WA, (t) =

sin'rbG5Y5(t), ir

t > 0,

where Gb is the operator given by

GsV(t) = f

t S(t-s)(t-s)b-1Cp(s)ds,

t E [0,T], p E LP(O,T;LP(R'1)),

JO

we see that II WAP.P(t)II LP(IlE.d)

` X

sin orb

(ft s4(&-1)IIs(s)IIL(LP(

1

4

d))ds)

7r

(Jot II Ys(s)II iP(nd)ds)1/P

where q = P The function IIS(s)I)L(LP(Ld)), s > 0, is locally bounded and q(b1) > -1. Therefore for arbitrary T > 0 there exists a constant c4 such

that >E

sup tE[0,T]

II WAP,P(t)II LP(]Ld))

< c4 JOT EIIY6(s)IILP(]&d)ds < +oo,

and the required continuity follows.

The next theorem is devoted to the existence and uniqueness of the invariant measures.

Reaction-Diffusion equations

223

Theorem 11.4.3 In addition to the conditions of Theorem 11.4.1 assume that the function fo is decreasing and

m-IIfiIIL,p>LO >0. Then there exists yo > 0 such that the transition semigroup Pt, t > 0, corresponding to the solution of (11.4.1 has a unique invariant measure in both H = L',, (Rd) and H = L2, (Rd), for any ic, r E ] 0, eco[. Moreover there exists c > 0 such that for any bounded Lipschitz function yp on H, all t > 0 and all x E H

IPto(x)

- JH W(x)p(dx) I < (c + 2IxI)e-WtII4oIILip.

Proof - We apply Theorem 6.3.3. Arguing as in the proof of Theorem 11.4.1 one easily shows that condition (i) of Hypothesis 6.2 holds. It remains to check that (ii) holds as well. We will prove that

for arbitrary p > 1 sup ]E

(IiwAP.P(t)IILP(]&d))

< +o0-t>0

Note, compare the proof of Theorem 11.4.1, that for all t > 0 t 00

E II YYAP, (t)II LP(Rd)

=J

d

p(S)

f0 E S(t j=1 t 00

f

< cl

- cl

J d p(e) (10 Ej=1I S(t - s)Q1/2hj(6)I2ds

I

) P!2 g(ij) dp() 1\ 2(m + I7]I2) d d

(

p/2

d6

Chapter 12

Spin systems Invariant measures for classical and quantum spin systems on d dimensional lattices Zd are considered. Using the dissipativity method, conditions for the existence of invariant measures are given. The exponential convergence of the transition probabilities to the invariant measure is established as well.

12.1

Introduction

Let Zd be the d dimensional lattice whose elements will be interpreted as atoms. A configuration is basically any real function x defined on

Zd. The value x(y) of the configuration x at the point -y can be viewed as the state of the atom -y and it is sometime called the spin

of y. Physical properties of a lattice system are determined by a matrix (a.yj),yjeZd ofglobal interactions and a function g : R -> ]R, of local interactions. For an arbitrary finite set I' C Zd the matrix (a.y,j),y jEzd is assumed to be negative definite. The Hamiltonian Hr of the restricted spin system is given by the formula

Hr(x)

a.y,jx.yxj 1',jerd

g(x.y), x E RZ",

(12.1.1)

ryEr

and the corresponding restricted Gibbs measure pr has a density, with respect to the Lebesgue measure on Rr, of the form Cre{-Hr(x)) = Cre'

r,iEZd a"r,ix7xi+E7Er

225

9(X7))'

x E Rr, (12.1.2)

Chapter 12

226

where Cr is a normalizing constant such that

µr(Rr) = 1.

(12.1.3)

For this to be true some conditions on the behaviour of g at infinity have to be imposed. Identifying the space R" with the set of those configurations x E RZd which vanish outside of F, every measure µr can be regarded as a probability measure on ]R . Assume now that {r,,} is any increasing sequence of finite subsets of Zd. All probability measures on the space of configurations that are weak limits of are called Gibbs measures. Of special interest are those Gibbs measures which are concentrated on configurations with polynomial growth. Denote by yr the Gaussian measure on Rr with mean vector zero and covariance matrix Sr =

2

((-a'Y,.7)-Y,.1Er)-1

Then for a possibly different constant Cr

yr(dx) = CreF7er9(x7)vr(dx).

(12.1.4)

In general the sequence of densities

{Crel:,Er s(x,) }f EN

does not converge to a density with respect to the limit of Gaussian measures µr,,. This makes the problem of finding a Gibbs measure rather difficult. Let f be a primitive function of Zg:

d (z) = 2 g(z), z E R.

(12.1.5)

Under general conditions on g the measure µr is the unique invariant measure for the stochastic differential equation

dXry(t) = E a.yjXj(t) + f(Xry(t)) dt + ;Er

7Er, t>0,

(12.1.6)

Spin systems

227

where W.y, y E t, are independent Wiener processes. It is therefore reasonable to expect that Gibbs measures corresponding to ((a1j), g) can be invariant measures for the infinite system of equations dX1(t) _

a-,j XX(t) + f (X1 (t))

dt + dWy(t), (12.1.7)

X,y(0)=x,y, yEZd,

t>0.

J

In the following section we will study system (12.1.7) regarded as a stochastic evolution equation

dX = (AX + F(X))dt + BdW(t), (12.1.8)

X(0)=xEH on a properly chosen Hilbert space H of functions on Zd. We not only prove existence of solutions to (12.1.7) but give conditions under which the corresponding transition semigroup has a unique invariant measure. Generalization to quantum spin systems will be the object of §12.3. We follow G. Da Prato and J. Zabczyk [48].

12.2

Classical spin systems

Let W(t) = {W.y(t)}lEzd, t > 0, be a Wiener process on (1,.F,?) with values in U = 12 (R') and with covariance operator Q E L(U). In particular if Q is the identity operator then the processes W.y, y E Zd, are independent standard real valued Wiener processes. However, Q

can be any nonnegative definite operator on U and for any t > 0 the family {W.y(t)}1Ezd can form a general Gaussian random field including stationary ones if Q is shift invariant. To study system (12.1.7) we apply results on dissipative systems from subsections 5.5.2 and 6.3.2 to (12.1.8). The operators A and F will be given by the formulae A(x.y) _ (>AEzd a9'yx9) , x = (x1) E H, (12.2.1)

F(x.y) = (f (x.y)) , x = (x.y) E H,

J

Chapter 12

228

and H = 1p(Zd) will be a weighted Hilbert space of sequences (x y) with positive summable weight p : Zd R+. B will denote the embedding operator J from U into H. Cb(H) and Bb(H) are respectively the space of all bounded continuous functions in H and the space of all bounded Borel functions in H. The following proposition, which goes back to Schur, see P. R. Halmos [83, Pag. 36], gives sufficient conditions under which the matrix (ayj) defines a bounded operator on 1P(Zd) for arbitrary p E [1, +oo].

Proposition 12.2.1 Assume that (I) sup T, Ia-YjH =a<+oo, 'YEZd jEZd

> 0 such that

(II) There exists

l a-,jl p(y) < 1p(j),

E Zd.

'yEZd

Then the formula

ayjxj

Apx =

, x E £P(Zd),

(12.2.2)

jEZd

defines a linear bounded operator on fp(Zd), for all p E [1, +oo], with norm not greater thAn a1/9131/p, 1 + 1 q p

= I.

Proof - Assume, for instance, that p E ] 1, +oo[ and that x = (x j ) has only a finite number of coordinates different from 0. By Holder's inequality and (i), for arbitrary -y E Zd p

a-rjxj jEzd

p

I a7j l I xjI

<

p

<

(I

/n a-Y.7

I1

l xj j) I aryj l ll e

jEZd

jEZd p/9

<

E Ia'yjI IxjIpI jEzd

Ia-vjI I

(

jE1d

<W11:Iar.iI jEVd

lx.iIp.

Spin systems

229

Consequently, by (ii) P

P(y)

IEZd

1: a.yjxj

ap/q E I a"Yj I P('Y) I xj I P

jEZd

'Y,j EZd

apl9Q r

P(.7) I xjI P,

jEZd

and the result follows. We will impose several conditions on the matrix (a,j).Y jEZd and on the positive weight p, compare S. Albeverio, Y. G. Kondratiev and T. V. Tsycalenko [4], and B. Zegarlinski [170]. We will assume that there exist constants R, M > 0 such that

a7,j=0 iffy - jI>R, I a.y, j I< M for all y, j E Zd

}

(12.2.3)

and P(y)

<M ifly - jI
P(j) (12.2.4)

Pet) < +00. 'YEZd

The next result follows directly from Proposition 12.2.1.

Proposition 12.2.2 If conditions (12.2.3) and (12.2.4) hold then the operators AP given by (12.2.1) are bounded. Condition (12.2.4) is satisfied for two specific families of weights,

Pk(y) = e-'111, y E Zd, K > 0,

(12.2.5)

and Zd , yE Pr(y) = P.,(-Y) = 1 + r IyjT

> 0, r > d.

(12.2.6)

If r > d is fixed then all spaces £r(Rd) , ic > 0, are isomorphic. Therefore the index K will often be dropped.

Chapter 12

230

Remark 12.2.3 It is useful to notice that for arbitrary is > 0 and

r>d

G

PK(Zd) D

.

D'PK(Zd))

where 7-l,,, is the space of configurations with at most polynomial growth. Moreover the two spaces QPK(Zd) and £ (Zd) are shift invariant: if x E £PK(Zd) (resp. tPK(Zd), then for arbitrary y E 7Gd, X( + y) E fpK(Zd), (resp. EPK(Zd), and therefore they are appropriate for physical considerations.)

As far as the function f : R -> R is concerned we will require that it satisfies Hypothesis 11.2. Our first theorem is the following existence result.

Theorem 12.2.4 Assume that conditions (12.2.3), (12.2.4), and Hypothesis 11.2 are satisfied. Let H = fl (Zd) and let the operators A and F be given by (12.2.1). (i) For arbitrary x E £Ps(Z' ), there exists a unique strong solution X(t, x), t > 0 of (12.1.8) and (12.1.7).

(ii) For arbitrary x E H = £P(Zd), there exists a unique generalized solution X (t, x), t > 0, of (12.1.8) and (12.1.7), and the transition semigroup

Pt 0,x E H, y0 E Cb(H), has the Feller property.

Proof We will apply Theorem 5.5.8 with

H = 12(Zd) and K = lps(7Ld). Note that, by Proposition 12.2.1, the linear operators A and A23 are bounded on H and K respectively and therefore, without any loss of generality, we can assume that A = 0. The operator F + rll, with the domain

D(F) =

x E QP :

P(7)1 f(xry)12 < +0 yEZd

Spin systems

231

is m-dissipative in H and its part in K is m-dissipative in K provided

rii is small enough. It is also clear that F maps bounded sets in K into bounded sets in H. So Hypothesis 5.4 is satisfied in our situation. We show now that Hypothesis 5.2 holds as well. Let Q'/2 = (qyj) be the nonnegative square root of Q. Then W-1(t) =

q-tjWry(t), 'Y E Zd, jEZd

where W.,, y E Zd, are independent real valued Wiener processes. By Gaussianity of W, for any p > 1, lE IWry(t)I '

= cp (i

I W.

(t)I2)pl2 p/2

= c,tPl2

q7j

,

t > 0.

7EZd

Consequently p/2

E p('Y) E q j YEzd

,

t > o.

jEZd

Since Q'/2 is a bounded operator on Q2(Zd), we have sup

q21Y3

< +00,

'YEZdjEZd

and W(t), t > 0, is an £P(Zd)-valued Wiener process. By Kolmogorov's theorem W has a continuous version in any £p(Zd), p > 1, and Hypothesis 5.5 is satisfied.

Remark 12.2.5 Similar results, but with different concepts of solutions, are contained e.g. in H. Doss and G. Royer [57] and in G. Royer [128]. Existence of strong solutions is treated in G. Da Prato and J. Zabczyk [48]. The conditions we impose on f are slightly weaker than those in B. Zegarlinski [170] as we do not require that fo and fi are differentiable. By (ii) the semigroup Pt, t > 0 has

Chapter 12

232

the Feller property. Note however, that by Theorem 9.2.1 it is not a strong Feller one even in the linear case, f = 0. Moreover if f = 0

the transition probabilities Pt(x, ), x E H, t > 0, are in general singular and therefore Doob's theorem, see §4.2, is not applicable in the present context.

The following theorem gives precise information on the rate of convergence to equilibrium.

Theorem 12.2.6 Assume, in addition to the conditions of Theorem 12.2.4, that for an rl > 0, the operator A + rl, restricted to £2(Zd), is dissipative, that fo is decreasing and q - 11A IILip > w > 0. Then there exists ico > 0 such that in the spaces f2(Zd) = H, with p = p'£ given by (12.2.5), or with p = pr given by (12.2.6), and K, r E ]0, tco[, the equation (12.1.8) has unique generalized solutions. The semigroup

Pt, t > 0, has a unique invariant measure p on H and there exists c > 0 such that, for any bounded and Lipschitz function cp on H, all

t > 0 and all x E H,

Ptv(x) - JH 4p(x)µ(dx)I < (c +

2IxIH)e-"tIIVIILip.

To prove the theorem we need the following two elementary lemmas.

Lemma 12.2.7 Assume that conditions (12.2.3) and (12.2.4) are satisfied and in addition

-1 <6, forty - iI < R.

If (Ax,x)e2(zd) < «IIXIIi2(zd), X E L2(7Ld),

then (Ax,x)ep(zd)

(a+

2M6

(2R

1)`t)Ilxlle2(zd), x E

ep(d).

Spin systems

Proof -

233

Take x E £P(Z'), then

(Ax,x) P(ad) = P(Y) p,,,jXj P('Y)a7,j

E

P(j)

h jl
P(Y)a

P(j

aI
Iry

+ E a j,j

PU)xj P(7)x

7,j - ay,j

) P(j)xj

x.v

r

P(Y)xr

P(7)xj P(-t)x-r

Iry-jI
2M

E (PWx + P('Y)x2)

+ all xll eP(Zd)

h-j I
= 2 (2R

1)dIxI

(Ld)+allXIIQP(Zd).

Lemma 12.2.8 For weights p" and p,,,, given by (12.2.5) and by (12.2.6) we have sup lim K-0 Iry-jl
(7)

VP P""U

)

lim c-0

sup [I-y-jl-
The proof follows by direct calculations.

Proof of Theorem 12.2.6 - We will consider only the weights p" as the case of p = pc,, can be treated in the same way. It is clear that (F(x) - F(y),x - y) < 1111IILiplix - y112, X, y E D(F),

and therefore F - IIf1IILip is m-dissipative. By Lemma 12.2.7 and Lemma 12.2.8 for arbitrary e > 0 there exists 'E > 0 such that for

Chapter 12

234

' E ]0, iE [

(A(x-y)+F(x)-F(y)+(rl-E)(x-y), x-y)t2

(zd)

< 0, x, y E j2"(Zd).

Consequently if tc E 10, r.,[ the operator A + q - e is m-dissipative. Since -IIf1IILip +q - E > w for sufficiently small E > 0 Hypothesis 6.2 is fulfilled.

We will show now that the process WA has bounded p-moments in £P(Zd) spaces for all p > 1: (12.2.7)

+oo.

sup 1EIWA(t)I t>o

Write WA(t) = (Zy(t))dyEz and S(t) = (ary(t)),q-YEZ. Then t

Jsj(u)dWj(u),

Z,y(t) _ jEZd

EIIWA(t)IIeP(zd)

u E(IZ',(t)Ip)-

yEfd

By Gaussianity and assuming, to simplify notation, that Q = I,

E I Z-r(t)I P= cr (E I Z-r(t)l2)pl2 <

P

(fs(u)du jEZd

However,

1: s7j(u)

IIS(u)IIi(t2(zd))

jEZd

and therefore p/2

t

EIIWA(t)Il

(Zd)

< CP E P(7) (10 jELd

II S(u) IIL(12(Zd))du

Since

II S(u)II L(t2(zd)) < e-'", u > 0,

1l

f

Spin systems

235

the estimate (12.2.7) follows. From (12.2.7) sup (EII WA(t)II + II F(WA(t))II) < +oo. t>o

This completes the proof.

Remark 12.2.9 A theorem similar to Theorem 12.2.6 was proved earlier by B. Zegarlinski [170, Theorem 4.2]. His method, based on logarithmic Sobolev inequalities, does not require that IIf1IILip is small. On the other hand we can cover less regular local interactions f like sgn E R, and our basic estimate holds for a more general class of functions W. Since for all r > 0 the spaces £p(Zd) are identical as sets, see Remark 12.2.3, the invariant measure t is supported by sequences (x1,) such that x'Y 2

1+I'YIr <+00 -YEZd

for arbitrary r > d. This set is smaller than 'H

Thus working with weighted Hilbert spaces not only is technically convenient but gives additional information about the spin system. More general nonlinearities could be treated with the help of appropriate Orlicz .

spaces (replacing CO(Zd)).

We remark finally, that the noise process W can be absent in our approach, but not in the one based on logarithmic Sobolev inequalities.

12.3

Quantum lattice systems

Quantum lattice systems are a mixture of systems studied in §12.1 and §12.2 They were introduced in S. Albeverio, Y. G. Kondratiev and T. V. Tsycalenko [4) and are described by systems of equations

Chapter 12

236

of the form dX. (t)

=

AX-y(t) + E ay,X;(t) + .F(X.y(t))) dt, l

jEZd

+ dWy(t) X.y(0)

= x y E f, y E Zd, t> 0. (12.3.1)

In the equation (12.3.1), A and F are respectively linear and nonlinear, in general unbounded, operators on a Hilbert space ?-l, (a.yj),y JEzd is a given matrix with real elements and (W.y),yEzd is a family of independent, cylindrical Wiener processes on 1I. Following [4] we will

require that the space f and the operators A and F are of special character although it will be clear that our general schema works in the more abstract setting. We will thus assume that 7.1 = L2(0,1), A =

dZT

(12.3.2)

- a,

D(A) = {x E H2(0,1) : x(0) = x(1), x'(0) = x'(1)},

f(x(e)),

E [0, 1], (12.3.4)

D(F) = {x E L2(0,1) : f (x) E L2(0,1)}, and on f and on the matrix (ayj)'y,.7E Gd we will impose Hypothesis 11.2 and conditions (12.2.3)-(12.2.4). To see that (12.3.1) is of the general form (12.1.8) define x(2d)

H = £ (L2(0,1)) _ {(x y)

P(Y)II xti II 2 < +°O},

E yEZd

K =

£22s(L2s(0, 1))

{(x.y) E (L2s(0,1))(7d)

E,yE2d

p(y)IIx_YIIL2s(o,i)

< +001,

where p = p" or p = pr, K,T > 0, see the previous section.

Spin systems

237

Let A = AO + Al where Al is a bounded linear operator on H given by

Al(x.y) =

a.yjxj

, x E D(A1) = H,

jEZd and

Ao(x.y) _ (Ax.y), x = (x.y) E D(Ao).

D(Ao) _ {(x.y) E H :

P(y)IIAx. II' < +oo}. 'YEZd

Boundedness of Al follows from an obvious generalization of Proposition 12.2.1. Let, in addition,

F(x..) = (Fx.y), x = (x.y) E D(F), D(F) = {(xry) E H :

+oo}.

It is easy to check that the operator AO + 77 on H and its restriction Aop +,q to K are m-dissipative for sufficiently small rt.

Let S(t), So(t), t > 0, be Co-semigroups on H generated by A and AO and write

WA(t) = J S(t - s)dW(s), t > 0, ot

-

WAo(t) = J So(t - s)dW(s), t > 0, ot

where W(t), t > 0, is the Wiener process H.

-

embedded into

Proposition 12.3.1 The processes WA0(t), t _> 0, WA(t), t > 0, have continuous versions with values in fp(LP(0,1)), p > I.

Proof -

Existence of a continuous version of WAo (t), t > 0, in £p(LP(0,1)) can be obtained by factorization, as in the proof of Proposition 11.4.2, and using the diagonal character of the semigroup So(.).

Chapter 12

238

Note that for Z(t) = l4'4(t), t > 0. Z(t) = WA,(t) + JOt So(t - s)A1Z(s)ds, t > 0.

-

(12.3.5)

Since the semigroup go(.) has a Co -continuous restriction to £ ( L ( 0 , 1)), the equation (12.3.1) has a unique solution in

C([O,T);t;(LP(0,1)), for arbitrary T > 0 by an elementary fixed point argument. This 0 proves the result. As a corollary from our discussion and Proposition 12.3.1 we set the following result.

Theorem 12.5.2 Assume that Hypothesis 11.2 and conditions (12.3.2)-(12.3.4) hold. Then for arbitrary x E H the equation (12.3.1)

has a unique generalized solution X(-,x). If x E K the solution is mild.

As in §11.4 and in §12.2 one can derive a result on exponential decay for the transition semigroup Pt, t > 0, corresponding to the solution X(., x) of (12.3.1).

Theorem 12.3.3 In addition to the conditions of Theorem 12.3.2, assume that fo is decreasing and that a - IIfilILip > w > 0. Then there exists ico > 0 such that the semigroup Pt, t _> 0, corresponding to the solution of (12.3.1) has a unique invariant measure p both on H = L2K and on H = L2.. Moreover there exists c > 0 such that for any bounded and Lipschitz function cp on H, all t > 0 and all x E H, IPtV(x)

- JH 4o(y)µ(dy)

< (c + 2IIzjI)e 'tIIPIILip-

Proof - As in the proof of Theorem 11.4.3 it is enough to check the assumptions of Theorem 6.3.3. We will show that

supE(IWA(t)IH) < +00,

(12.3.6)

supE(IF(WA(t))IH) < +0,

(12.3.7)

t>o

t>o

as the remaining conditions hold in an obvious way.

Spin systems

239

To prove (12.3.6) note that for t > 1, t

E I WA(t)I H

= E If S(t - s)dW(s))H

E ( Jot S(s)dW(s)

H

fk+1 S(s)dW(s)

x

k

+

lE

ft] S(s)dW(s) It

[t]-1

E IIS(k)IIHE I f 1 S(s)dw(s)I x+ sup

E 1JU

S(s)dW

k=0

[t]-1

< k= 0

<

e

1

e-akE I

2e'

a-

S(s)dW(s)I 0 JO

+ sup ]E u<1

H

H

j

S(s)dW(s)

(s)I

H

H

1JU 1 s up ]E

, t > 1.

S(s)dW (s)I H

To show (12.3.7) note first that H)1,2

>E I F(WA(t))I H

(E I F(WA(t))I

<- (I IN'A(t)IK)1,2 .

Since the distribution of WA(t) on K is Gaussian, there exists a constant c such that JIIWA(t)IIx Moreover there exists a constant a > 0 such that for the restriction S(t), t > 0, of the semigroup S(t), t > 0, to K, one has c(EIlWA(t)IIK)s.

IIs(t)IIK <_ e-at, t > 0.

It remains now to repeat the proof of (12.3.6).

Remark 12.3.4 Theorem 12.3.3, under slightly stronger conditions, was proved earlier in [4] using specific properties of the one dimensional heat equation. It is, however, clear that the dissipativity method allows us to treat spaces 7-( more general than L2(0,1). Existence of an invariant measure y for (12.3.1) was also obtained in S. Albeverio, Y. G. Kondratiev and T. V. Tsycalenko [4] with more stringent conditions imposed on f, p and (a.yj). Exponential estimates in Theorem 12.3.3 were obtained in G. Da Prato and J. Zabczyk [48].

Chapter 13

Systems perturbed through the boundary In some applications the noise can effect the evolution of a system only through the boundary of a region. For such systems conditions are given for existence and uniqueness of invariant measures. Here once more the dissipativity method is used.

13.1

Introduction

In several applications external forces enter the system through the boundary of a region where the system evolves. The same is true if the forces have a stochastic character. Consider for instance the

nonlinear heat equation in the region G = J0, i[d C Rd, with the noise affecting only a portion I'o

6d=0}, 241

Chapter 13

242

of the boundary F of G. Thus the state equation is of the form

X (t,

_ (0 - m)X(t, ) + f(X (t, e)), t > 0, (t, f)=0, REF\Fo, t>

EG, aax

(t,

aV

(t, f ),

E

I

ro, t > 0, (13.1.1)

where vXis the inner normal vector to the boundary at those points f E F where it is well defined. The process V(t), t > 0, is a Wiener process on U = L2(Fo) with the covariance operator Q not necessarily of trace class. The operator Q could be of integral type, Qu(a) =

fro

u E U,

E ro,

(13.1.2)

with the kernel g being the correlation function of the random field V(1, e), E Fo. Of special interest is the case when the field V (l, ), E Fo, is the restriction of a stationary random field on Rd-1. Then there exists an integrable function qo : Rd-1 -+ R such that rl) = qo( - 77),

, 77

E I'o.

(13.1.3)

Moreover qo is the Fourier transform of the spectral density of the random field. System (13.1.1) is our main example. We will establish existence and uniqueness of Markov solutions using the dissipativity method. We will show also that under additional conditions on f there exists a unique invariant measure for the solution. Asymptotic properties of the corresponding transition semigroup will be established as well.

We start from general information on parabolic systems with non-homogeneous boundary data. We formulate an integral version of a generalization of (13.1.1) and give conditions under which it has a solution. Then we apply the general theory for the specific model (13.1.1). We obtain in this way explicit conditions, in terms of the spectral density of the noise, implying the existence of solutions to (13.1.1) and the existence of a unique strongly mixing invariant measure.

Systems perturbed through the boundary

13.2

243

Equations with non-homogeneous boundary conditions

We start from the linear case. Let G C Rd be a fixed region, r = OG its boundary and let A be a linear second order differential operator and r a linear first order differential operator. A non-homogeneous boundary problem is of the form T (t,) = Ay(t, e), (t, e) E ]0, +oo [ x G, ry(t, e) = u(t, ), (t, e) E 10, +oo [ x r, y(0, e) = x(f ),

(13.2.1)

E I',

where u is a U-valued function and U = L2(r). Under some conditions on the data the solution to problem (13.2.1) is given by the formula t

y(t) = S(t)x + (A - A) J S(t - s)Du(s)ds, t > 0,

(13.2.2)

where \ is a real number, A the infinitesimal generator of a strongly

continuous semigroup S(t), t > 0, on a Hilbert space H = L2(G) and D a properly defined linear bounded operator from U into H. Assume for instance that the linear operator Ax = Ax, (13.2.3)

D(A) = {x E H2'2(G) : r = 0},

generates a strongly continuous semigroup S(t), t > 0. on H. Let A be a real number such that the problem

Aw - Aw = O, rw = u

(13.2.4)

has a unique solution in = Du E H for arbitrary u E U. If is a twice continuously differentiable function with values in U and x - Du(0) E D(A) then there exists a strong solution of the equation

z'(t) = Az(t) + .\Vu(t) - Du'(t), t > 0, (13.2.5)

z(0) = x - DU(O).

Chapter 13

244

In particular r(z(t)) = 0 for all t > 0. Define

y(t) = z(t) + Du(t), t > 0. Then

ry(t) = r(Du(t)) = u(t), t > 0, and

d (y(t) - Vu(t)) = (A - A)(y(t) - Du(t))

Ay(t) - Dot u(t).

Taking into account that (A - A)y(t) = 0, for all t > 0, we see that

at (t) = Ay(t), t > 0, 1

7y(t) = u(t), t > 0,

(13.2.6)

Y(O) = X.

Consequently the function y(t), t > 0, solves the problem (13.2.1) and is given by the formula

y(t) = S(t)(x -Du(0)) j S(t - s)[ADu(s) - Du'(s)]ds+Du(t), t > 0.

Proposition 13.2.1 If is a twice continuously differentiable function and x - Du(0) E D(A) then a solution y(t), t > 0, of the abstract boundary problem (13.2.6) is of the form

y(t) = S(t)x + (A - A)

J0 t S(t - s)Du(s)ds, t > 0.

The proof of the proposition is an immediate consequence of the following well known lemma.

Lemma 13.2.2 Assume that the operator A generates a strongly continuous semigroup S(t), t > 0, on a Banach space E and that p E W1"1(0, T; E) for some T > 0. If x E D(A) and

z(t) = S(t)x + f S(t - s)cp(u(s))ds, t E [0,T], t

(13.2.7)

Systems perturbed through the boundary

245

then z E C1([O,T); E) fl C([0, T]; D(A)). Moreover

f

Az(t) = S(t)x +

t

S(t - s)cp'(u(s))ds + S(t)cp(0) - cp(t), t E [0,T]. (13.2.8)

Proof -

For the completeness of the presentation we sketch a proof. One can assume that x = 0 and that t

cp(t) = c40) + f 0(r)dr, t E [0,T], where 0 E L1(0, T; E). Consequently

z(t) =

ft S(t - s)p(0)ds + f S(t - s)cp(0)ds +

ft

S(t - s) ( fS'(r)dr)

ds

f (jt S(t - s)b(r)ds) dr. t

For arbitrary r < t, f t S(t - s)O(r)ds E D(A) and A It t S(t - s)O(r)ds = (S(t - r) - I)0(r). r

Therefore

Az t = (5(t) - I cp(0) +

- r) - I t f(s(t

' r dr,

which is the required formula (13.2.8). Moreover z E C([0, T]; D(A)) and by direct differentiation t

z'(t) = S(t)y0(0) + f S(t - r),O(r)dr, t E [0,T]. The proof of the lemma is complete. The formula (13.2.2) can be extended by continuity to less regular functions u(.).

In particular assume that W(t), t > 0, is a cylindrical Wiener process defined in a probability space (St, F, IP) adapted to a filtration

Chapter 13

246

.Ft and with values in U. If Q is a nonnegative bounded operator on U such that the process

Z(t) = (A - A) f t S(t - s)DQ1/2dW(s), t > 0,

(13.2.9)

0

is a well defined H-valued process, then it will be accepted as a solution to the linear problem

aZ (t) = AZ(t), t > 0, rZ(t) = Q1/2 OW (t), t > 0,

(13.2.10)

It will be called an Ornstein-Uhlenbeck process in analogy to the classical noise case. A sufficient condition for (13.2.9) to be a well defined process is that t

I IIAS(r)DQ1/2IIL2(U;H)dr < +oo, t > 0,

(13.2.11)

where L2(U; H) stands for the space of Hilbert-Schmidt operators acting from U into H, see G. Da Prato and J. Zabczyk [44] and [45]. Let now F be a nonlinear, usually unbounded, operator on H. Ft-adapted H-valued and H-continuous process X is the mild An solution of the problem

jt-(t) = AX(t) + F(X(t)), t > 0, TX(t) = Q1/2

-W

(t), t > 0,

(13.2.12)

X(0) = x, if it is a solution of the following integral equation: t

X(t) = S(t)x + Z(t) + f S(t - s)F(X(s))ds, t > 0. In particular we have the following result.

(13.2.13)

Systems perturbed through the boundary

247

Proposition 13.2.3 Assume that the operators A and F satisfy Hypotheses 5.4 and 5.5. Then equation (13.2.13) has a unique Markovian generalized solution.

We finish this section by giving sufficient conditions for continuity of

the Ornstein-Uhlenbeck process Z(.) in some important subspaces of H. They will be used in § 13.4. To simplify our general considerations we assume from now on that the operator A is self-adjoint and that there exists a complete orthonormal basis of eigenvectors {gn} of A corresponding to the sequence {-An} 1 -oo of non positive eigenvalues. Let H2a = D((-A)a). If a stronger version of (13.2.13) holds, namely t

3 y> 0 f r--YIIAS(r)DQ1/2IIi2(U;H2a)dr < +oo, t> 0, (13.2.14) then the process Z has a version which is H2a-continuous. In particular we have the following result, see G. Da Prato and J. Zabczyk [45].

Proposition 13.2.4 If for some 'Y > 0 00

(13.2.15)

+00 n=1

then the process Z has an H2a-continuous version.

Proof - Note that

j

rIIAS(r)DQ1/2II(U;H2a)dr t

r--YII((-A)1+a)S(r)DQ1/2II2

L2 (U;H)

= J0

= Jo 'r-III _

Q112D*[((-A)l+a)S(r)]II

i2(U;H2a)dr

00

as+1IIQ1/2D*gn112 /' t

n-1

So the conclusion follows.

0

dr

rrye-2anrdr.

Chapter 13

248

Equations with Neumann boundary conditions

13.3

We will apply the general approach of §13.2 to problem (13.1.1) with

V(t) = Q112W(t), t > 0. For simplicity we set m = 0 and define

H = L2(G), U = L2(ro). Moreover the operator A is the Laplacian and r is the trace in F0 of the normal derivative operator aU and

D(A) = {x E H2(G) : a

0,

E F},

Ax=Ax. It is well known that A is a self-adjoint generator of a semigroup S(t), t > 0. Moreover the eigenvectors of A are of the form E G,

9nl,...,nd =

(13.3.1)

where nl,..., nd E N U {0} and go(d) _

1

2- cos ne,

,

E [0, 7r], n E N U {0},

(13.3.2)

and the corresponding eigenvalues are ^nl,...,nd =

-(ni + ... + nd), n1, ..., nd E N U {0}.

We will now find an explicit formula for the operator D. For this purpose it is convenient to consider elements u E U as functions in L2(F) equal to 0 outside of FO. We will solve (13.2.4) with A = 1. An arbitrary element u E U can be uniquely represented by the expansion with respect to the orthonormal and complete system {9n,,...,nd-1

:

nl, ..., nd_1 E N U {0}},

namely 00

U=

(u, 9n1 i...,nd_1)U 9n1,...,nd_1 nl,...,nd_1 =0

I

Systems perturbed through the boundary

249

and therefore the solution w = Du of (13.2.4) is of the form 00

w=

E

\u, gn1 i...,nd-I )U wn1,...,nd_1

nl ,...,nd_1=0

where wn1,...,nd-1 = Dgnl,...,rid_1, n1i ..., nd_1 E N U

{0}}.

By direct calculation we find that CC

[

[

wnl,...,nd_1 (S1, ..., Sd) = gn1,...,nd_1(S1, ..., bd-1 )hn1 .....nd-1 (Sd),

where hn1,...,nd-1 (ed) _

-

cosh[

+ ni +

+ na_l sink

[

+ n2d_1 ]

1 + ni

(e1, ,'d) E G and n1i...,nd_1 E N U {0}. It is clear that D is a bounded operator from U into H. By the very definition (D*9nl,...,nd)(S1I ..., Sd-1)

= gnl,...,nd-1 (b1, ..., ttbd-1)

[t

CC

10"

C

[

gnd(S )hnl,....nd-1 (S )dS

b(n d )

1 + ni + ... + n2 gnl,...,nd-1

d) E G, (13.3.3)

where

1 ifn=0, 2 if n E N.

Theorem 13.3.1 The Ornstein-Uhlenbeck process Z(t), t > 0, has a continuous version in the space H2a,2(G) provided there exists y > 0 such that

E J Q1/2gnl,...,nd-1 II nl,...,nd_I =0 L>

2

(\1 + n1 + ... + nd-1)

2a-{ ry- 1

< +00. (13.3.4)

,

Chapter 13

250

Proof- We can identify the spaces H",2 (G) with H2, = D((-A)"), a > 0. We will apply Proposition 13.2.4. Note that 1/2D* IIQ

gn1,...,ndll

2

-

b2(nd) 1/2 2 (1 + n2 + ... + n2d)2 IIQ gn1,...,nd_1 II '

nl,..., nd-1 E N U {0}. Consequently the sum 00

D* A2"+'r+l nl,...,nd Q1/2 E nl,...,nd-1=o II

gn1,...,nd

is finite if and only if 00

00

2

IIQ1/2 D*9n1..... ndl)

nl,...,nd-l =o

n=0

(n2 + ... + n2d-1 + n2)2a+'Y+l < +oo. (1 + n1 + ... + n2_1 + n2)2

However, there exist constants Cl > 0 and C2 > 0 such that for arbitrary a > 0 and 2a + < 1 2

00

2)+

C2(1 + a2)2a+'Y-2 < 1 ((1 + a2

)21

< Cl(1 + a2)2«+'Y-2,

n=O

and therefore the result follows.

Remark 13.3.2 If d = 1 then to = {0} and the condition (13.3.4) is automatically satisfied. We finish this section by giving conditions on the space homogeneous

noise implying continuity of the Ornstein-Uhlenbeck process in all LP ((0, ir)d) spaces. Thus we assume that the operator Q is of the form (13.1.2)-(13.1.3) with qo being the Fourier transform of the symmetric spectral density g, see j5.2: J d-1

eit(g(()d(, ( E R.

We will call g the spectral density of the noise process V. We need the following lemma.

Systems perturbed through the boundary

251

Lemma 13.3.3 The function TdT

hd,a(() =

(

11 (j=1

0.0

{(1

+ ni + ... + nd)2a-2

nl,...,nd=0

d

S E Rd

X 1 1+ ((j l+ nj)2 1+ ((jl- nj)211 is finite and continuous if and only if a < 1.

Proof - There exists a constant C > 0 such that d

1

(1+n2+...+nd)2a-2
j=1 Since d

00

11 j=1 [l + ((j + na )2][l + ((, - nj)2]

n1,.. .,nd=O

d7

2(2a-1

nj

0o

n2(2a- 2

- nj)2] - 11 j=1 n.?--O [1 + (() ' + n?)2][1 + ((j and 4 - 2(2a - 1/2) > 1 we get that if a < 1, the continuity of hd,a follows. If a > 1 and C. # 0, j = 1, ..., d, then hd,a((1, ..., (d) = +00. We have the following continuity result.

Theorem 13.3.4 Assume that a < 1 and for y E]O,1 - a[: fRd-I

(13.3.5)

g(()hd-l,a+-v(()d( < +oo.

Then the Ornstein-Uhlenbeck process Z(t), t > 0, has a continuous version in H2a+2,2(G).

Proof - We use Proposition 13.2.4 and check that the condition 00 2

n1,...,nd-1=0

IIQI129n...... nd-1 II2 \1 + n12

+ ... +

nd-1\ 2a+'

1

2 < +00

Chapter 13

252

holds.

In the present situation 2 !11/2 9n1...,nd-1 II IQ

\ = (Q9n1,...,nd-1,9n1,...,nd-i), n1,...,nd-i E NU {0}.

Let

9n1,...,nd-1 (C)

(t[ 9n1,...,nd-1(S) =

0

for e E ro,

170 .

Then (Q9n1 i...,nd_1 I gn1,...,nd-1 )U = (qo * 9n1 ,--d-1 , gni,...r'nd-1 / L2 (ad-1 )

and by Plancherel's theorem 12

Ild-1 9(O I9n......nd-1 \S) (2ir)d-1

(Q9n1,...,nd-1, gn1,...,nd-1 )1£.d-1

d(

V

where 1

9n1,...,nd-1 (0 _

(S)'

Rd-1

27r d d

1

,...,nd-1 (L ( )dS

C

(2i)d-1 [0"]d-1

Integrating by parts we obtain that 7r

e°C cos nCdC =

iC

n2

(el(C-n)" C2

- 1)

(13.3.6)

for allnENU{0} ands'ER;if(2=n2oneshould take in(13.3.6) the limiting value 2 if n E N and it if n = 0. Taking into account the

Systems perturbed through the boundary

253

definitions (13.3.1) and (13.3.2) one gets for a constant C1 > 0 that 00

E

nl,...,nd_1=0

Q II

12

gn1,...,nd-1 II

\I 2a+-y- 2 2((1 + n22 + ... + nd_1l 2

1 G)d-1

W

Ld1 g(() nl,...,nd-1=0 (1

n1 + ... + nd-1)

2

X Ignl,...,nd-1 (() d(

C1

f J

d-1

g(O [

(1 +

L00

n12

n1 ,...,nd-1=0

+ ... + nd-1) 2 2a+-Y-2

d

X

11 j=1

(J

d(. X

1

J

1

g(C)hd-1,a+2 (()d(

C1

Therefore the result follows.

Remark 13.3.5 Knowing the asymptotic property of hd_l,a+ry(C) as ISI -4 +oo one can replace (13.3.5) by more specific conditions. If for istance d = 2 then ((2 + n2)2a+2ry-1

h1,a+'Y C n=0

2

(n+ t) 2

< C(4(a+'Y),

for large ICI.

As a corollary we obtain sufficient conditions for continuity of Z in LP spaces.

Theorem 13.3.6 The Ornstein-Uhlenbeck process Z(t), t _> 0, has a continuous version in LP(G) provided that (13.3.5) and that one of the following conditions hold

W d = 1, 2, 3, p > 1, 4 < ca,

Chapter 13

254

(ii)d>4,1
C L°°(G) if d < 4a

and

H2a+2'2(G) C LP(G) if either d < 4a, or d > 4a and p<-

2d

d - 4a

So the result follows.

Remark 13.3.7 If d = 1, 2, 3, and 4 < a < 1 then condition (13.3.5) is sufficient for continuity of Z(t), t > 0, in all LP(G) spaces, p > 1.

13.4

Ergodic solutions

In a similar way as in Chapter 11 we will apply the dissipativity method developed in §5.5 and §6.3.

Theorem 13.4.1 Assume that the function f satisfies Hypothesis 11.2. If in addition for p = 2s2 the conditions of Theorem 13.3.6 are satisfied, then problem (13.1.1) has a generalized Markovian solution, in L2(G).

Proof - The operator A = A - m with the domain D(A) _ = 0} generates a dissipative semigroup in L2(G) TVif m > 0. This semigroup has dissipative restriction for all spaces K = L2s(G) and K = L2s2(G) Also the nonlinear operators F - 77, {x E H2'2(G) :

where

F(x)(e) = f(x(e)),

E G,

with natural domains, are dissipative in L2(G) and H if q is sufficiently small. Moreover by Theorem 13.3.6 the process F(Z(t)), t _> 0, has continuous versions in L2 (G). Consequently all the assumptions of Proposition 13.2.3 are satisfied and the proof of the theorem is complete.

In a similar way we can prove the following theorem on invariant measure.

Systems perturbed through the boundary

255

Theorem 13.4.2 In addition to the conditions of Theorem 13.4.1 assume that the function fo is decreasing and

m-IIf1IILip>W>0, Then the transition semigroup Pt, t > 0 corresponding to the solution of problem (13.1.1) has a unique invariant measure µ in L2(G). Moreover there exists c > 0 such that for any bounded Lipschitz function cp on H, all t > 0 and all x E H we have JH o(y)u(dy) < (c +

Proof - We apply Theorem 6.3.3 and argue in a similar way as in the proof of Theorem 12.3.3.

For other results on invariant measures for stochastic equations with boundary noise see B. Maslowski [118].

Chapter 14

Burgers equation This chapter is devoted to the Burgers equation on a finite interval, perturbed by the cylindrical Wiener process. Existence of a unique solution is established. It is also shown that there exists a unique invariant measure by proving a kind of boundedness in probability of some solutions and by establishing strong Feller and irreducibility properties of the corresponding transition semigroup.

14.1

Introduction

An important role in fluid dynamics is played by the following Burgers equation, see J.M. Burgers [15]. 2

[X2(t,)]

atX(t' ) = v C2X(t,e)+

,

(14.1.1)

2

with t > 0,

E R. It is known, however, that the equation is not a good model for turbulence. It does not display any chaotic phenomena; even when a force is added to the right hand side all solutions converge to a unique stationary solution as time goes to infinity. The situation is different when the force is random. Several authors have indeed suggested using the stochastic Burgers equation as a simple model for turbulence, D. H. Chambers et al. [21], H. Choi et al. [22], Dah-Teng Jeng [33].

The equation has also been proposed in M. Kardar, M. Parisi and J. C. Zhang [94] for studying the growing of a onedimensional 257

Chapter 14

258

interface. Let us recall their model. Let h(t, ) be the height of the interface at the point E R, then we have

a where

q2

02

2 F h(t,

h(t, E) = v a- h(t, e) +

W (t' f ),

+ 0

(14.1.2)

is the space-time white noise. Setting

X (t,) _

h(t,

we arrive, after differentiation with respect to , at the equation

aX(t,E) = v

X(t,6) +

aa

2

W(t, ), (14.1.3)

In § 14.2 we shall prove existence and uniqueness of a solution, following G. Da Prato, A. Debussche and It. Temam [36], of equation (14.1.3) supplemented by Dirichlet boundary conditions

X(t,0) = X(t, 1) = 0, t > 0,

(14.1.4)

and with the initial condition X (0, 6) = x( ),

E [0, 1].

(14.1.5)

Moreover in § 14.3 we prove that the corresponding transition semigroup is a strong Feller one and in §14.4 the existence and uniqueness of an invariant measure. Here we use methods in G. Da Prato and D. Ggtarek [38] (where a more general case with multiplicative noise is treated) and from Chapter 7. However, the proofs we present here are new. For a different approach on existence and uniqueness, based on the Feynman-Kac formula, see L. Bertini, N. Cancrini and G. JonaLasinio [8]. Equation (14.1.2) has also been studied in several space dimensions using the Hopf-Cole transform

h(t, f) = log

(.v(te))

,

which reduces equation (14.1.2) to heat equations with a random force. In this connection we quote H. Holden, T. Lindstrom , B.

Burgers equation

259

Oksendal, J. Uboe, & T. S. Zhang [87], A. Dermoune [54], Yu. Kifer

[97]. We mention finally the papers by L. Bertini and N. Cancrini [7], and G. Tessitore and J. Zabczyk [151], where equation (14.1.2) 82 W (t, ) is studied. with at W (t, ) replaced by In the sequel we will set v = A = 1 for simplicity. Then we write the problem in abstract form. We define

Au = u", V it E D(A) = H2(0,1) n Ho(0,1). A is a negative self-adjoint operator on L2(0, 1) and

D((-A)°) = H2-(0,1) if o E ]0, 1/4[. We denote by S(t), t > 0, the semigroup on L2(0, 1) generated by A, by {ek} the complete orthonormal system on L2(0,1) which diagonalizes A and by {Ak} the corresponding sequence of eigenvalues. We have

ek(e) =

sin

kE

k E N.

Setting X(t) = X(t, ), t > 0, we can write (14.1.3)-(14.1.5) as

dX(t)

AX(t) + 1 Dt(X2(t))J dt + dW(t), (14.1.6)

X(0) = x,

1

where W is defined by

W(t) = E Qheh,

(14.1.7)

h=1

and {Qh} is a sequence of mutually independent real Brownian motions in a fixed probability space (St, F, ]P) adapted to a filtration

Ft, t > 0.

Chapter 14

260

14.2

Existence of solutions

Here we are concerned with the problem

dX(t) _ [AX(t) + 1 De(X2(t))] dt + dW(t), (14.2.1)

X(0) = x, which we write in the mild form t

X(t) = S(t)x + 2 f S(t - s)D,(X2(s))ds + WA(t), where WA(t) is given by t

WA(t) = JO S (t - s)dW(s).

Now, setting Y(t) = X (t) - WA(t), t > 0, we obtain a deterministic equation, a.s. in 1,

Y(t) = S(t)x +

1f

S(t - s)D,[(Y(s) + WA(s))2] ds,

(14.2.2)

which can be considered as the mild form of the evolution equation

Y'(t) = AY(t) +

WA(t))21, t > 0,

Y(0) = X.

In order to give a precise meaning to equation (14.2.3), we introduce the mapping

F°(u)(t) = ft (-A)°S(t -

ds, t E [0,T],

(14.2.4)

0

for any o E [0,1[. If u E C([0, T]; Ho (0,1)) then F, (u) is certainly well defined and belongs to C([0, T]; L2(0,1)).

Lemma 14.2.1 The mapping FF defined by (14.2.4) can be extended to a bounded linear mapping from C([0, T]; L1(0,1)) into C([0, T]; H2o(0,1)), for arbitrary o E [0,14[.

Burgers equation

261

Proof - An essential tool for the proof is the following well known estimate:

I DeS(t)cpjj. < Clt-+

(14.2.5)

II(P112, cp E L2(0,1),

for some constant C1. Let us choose 0 E H2°(0,1) and consider the scalar product /

t

I = ((-A)°b, f S(t - s)Dgu(s)ds) , where u E C([0, T]; Ho (0,1)). Then

I=

=-

f

t

(S(t - s)(-A)°O, Dtu(s)) ds jt

s)(-A)°O), u(s)) ds.

Therefore III

Ilu(s)Illds.

J ot II

But

IIDe[S(r/2)(S(r/2)(-A)°V')]II.

< Cl (j) -3/4lIS(r/2)(-A)°V')II2 Cl

(2)-3/402 (2)-° II01l2

< C3r-(3/4+°)II0II2, r E [0, T],

where C2, C3 are constants. Consequently III < C3 J t(t

- s)-(3/4+°)Ilu(s)IIlds II'II2

(14.2.6)

Chapter 14

262

It follows from (14.2.6) that fo S(t - s)Dgu(s)ds E D((-A)") and I1(-A)O'

J t S(t - s)Dtu(s)dsll2

<

C3

< C3

j(t 4

1 - 4a

s)-(3/4+o,)IIu(s)Illds

t+

sup IIu(3)II1.

o<s
This easily implies the conclusion.

Remark 14.2.2 In particular we see that F = FO is well defined as a transformation from C([O,T]; L'(0,1)) into C([O,T]; L2(O,1)).

Remark 14.2.3 In the same way we can show that if 1 + o < 4 then for a constant C4 > 0 and all t > 0 t

0

ip

/

II(-A)° f S(t - s)Dtu(s)ds

< c4t4`4

p

p

(ft IIu(s)IIpds) o

(14.2.7)

Now we can rewrite equation (14.2.3), as

Y(t) = S(t)x + 2F ((Y(.) + WA(.))2) (t), t E [0,T].

(14.2.8)

We say that X E C([0,T]; L2(O,1)) is a mild solution of problem (14.2.1), if and only if X() - WA(.) is a solution of equation (14.2.8). Now we prove an existence and uniqueness result.

Theorem 14.2.4 For any x E L2(0,1), there exists a unique mild solution X of equation (1/.2.1), and for all T > 0 X E C([0, T]; L2(0,1)) n L2(0, T; C({0,1])), P-a.s.

Proof - We fix T > 0. The proof is divided into two steps. Step 1- Local existence It is not difficult to prove existence of a mild solution of equation (14.2.1), in a small interval, depending on w E St, by using the

Burgers equation

263

contraction principle on the Banach space C([O,T]; L2(0,1)). This follows from Step 1 and the fact that, WA() E C([0, T]; C([0,11)),

(14.2.9)

see G. Da Prato and J. Zabczyk [44, page 141].

Step 2- A priori bound replaced by a regWe first consider equation (14.2.8) with ular function cp E C([0, T]; Ho(0,1)) and x E D(A) :

Y'(t) = AY(t) + 2Dt [(Y (t) + 0)2], t > 0, (14.2.10)

Y(0) = X.

1

As is easily seen equation (14.2.10) can be solved by a fixed point argument in C([0, T]; Ho(0,1)), so it has a classical solution. We find now an a priori estimate on the solution Y which will be independent of regularity assumptions on cp and x. Multiplying both sides of equation (14.2.10) by Y and integrating with respect to e in [0, 1] we get lY(t)12de 2 dt

101

f

=I

It follows, integrating by parts, that

2 dt

11 IY(t)I2d +

2

11 y2(t)d

_

-2 j(Y(t) +

Jol Y2(t)Yt(t)d - J ol cp(t)Y(t)YY(t)d -

2 101

Noting that f Y2(t)YY(t)de

3

[Y3(t)] 1

0,

Chapter 14

264

we get 2 dt IIY(t)112 + Joy Yg (t)d

II o(t)II.IIYY(t)II211Y(t)II2 + 2IIp(t)II00IIYe(t)112

<_ 211YY(t)112 +

In conclusion we find dtIIY(t)112 +

211c(t)II0,

(14.2.11)

and thus IIY(t)II2

<

e8fo II (s)II

+ 2 f0

a

ds

IIxI12

(14.2.12)

e$f Ilw(s)Il. ds

dr,

and fJOT

IIY(t)II2dt

fT (14.2.13) 0

11 (t)II4dt.

Since the trajectories of WA(-) can be uniformly approximated, on any finite interval [0,T], by functions V from C([O,T]; Ho(0,1)) and since we have that D(A) is dense in C([O,1]) the estimates (14.2.12) and (14.2.13) hold true for any local solution Y of the equation with co replaced by WA. This shows that local solutions cannot explode in finite time and the proof of global existence of solutions to (14.2.1) is complete. Moreover, again by the estimates (14.2.12) and (14.2.13) one sees

that Y E C([O,T]; L2(0,1)) n L2(O,T; C([O,1])),

Burgers equation

265

which implies, in view of (14.2.9), that X E C([O, T]; L2(0,1)) n L2(0, T; C([O,1])),

as required.

14.3

Strong Feller property

Let X(-, x) be the mild solution to problem (14.2.1) whose existence

is proved in theorem 14.2.4. In this section we will show that the corresponding transition semigroup Pt, t > 0, has the strong Feller property on H. For this purpose we consider a modified Burgers equation. For any R > 0 let z/'R be a C' function on [0, +oo [ taking values in [0, 1] and such that

1 for r E [0, R], 0 for r E [R + 1, +oo[, and define a mollifier

MR(x) = x&(x), x E H.

(14.3.1)

Note that MR E Cy (H) and DXMR(x) = V(IIxII)I +

'OR(II xII )x

® x, x E H.

IIxil

Proposition 14.3.1 If MR is a mollifier given by (14.3.1) then the modified Burgers equation

dX =

!DC (MR(X))2] dt + dW(t), (14.3.2)

X(0)=x,

I

has a unique mild solution on a time interval [0,T]. Moreover the corresponding transition semigroup PR, t > 0 has the strong Feller property on H.

Chapter 14

266

Proof - Step 1-Existence and uniqueness The equation (14.3.2) can be rewritten in the form

Y(t) = S(t)x + 2 F(MR(Y + WA))2 (t), t E [0, T],

(14.3.3)

with X(t) = Y(t) + WA(t). The proof of existence and uniqueness of solutions to (14.3.3) is similar to that of Theorem 14.2.4 provided one can derive an apriori bound of the form (14.2.12). For this purpose we assume that Y satisfies the equation

Y'(t) = AY(t) +

2D (MR(Y + P))2 ,

t E [0, T], (14.3.4)

I with cP E C([O,T]; Ho(0,1)), x E D(A), and proceed as in step 2 of the proof of Theorem 14.2.4. In particular we obtain that 2 dt

IIY(t)IIZ +

-1

R(IIY(t) + sa(t)II) fo 1(Y(t) + V(t))2 YY(t)d 1

11 _ -'OR(IIY(t) + (P(t)II) 11 p(t)Y(t)YY(t)de + 2 P2(t)Yy(t)d6] 0 IIYY(t)II211Y(t)II2 + 2

IIYY(t)112.

Therefore the estimate (14.2.12) also holds in the modified case.

Step 2-Strong Feller property We proceed as in the proof of Theorem 7.1.1. We set

(Xx(t, x), h) = r7h(t, x), t > 0, x E H. It is easy to check that the solution X (t, x) of (14.3.2) is differentiable

with respect to x and that rjh(t, x) is a mild solution of the problem ii' = i?4 + Dg (MR(X)DxMR(X )I) , t E [0,T], (14.3.5)

,q(O) = h, P a.s.

Burgers equation

267

Note that by Holder's inequality, for arbitary x, z E H II MR(x)DSMR(x)zIIl < II MR(x)I12I1 D.MR(x)zII2 _< C1142,

(14.3.6) where C is a constant depending on the choice of the function MGR.

Moreover equation (14.3.5) can be rewritten as

rl(t) = S(t)h + 2F (MR(x)DxMR(x)11) (t), t E [0, T]. Taking into account Lemma 14.2.1 and estimate (14.3.6) we conclude that the solution to (14.3.5) exists in C([0,TJ;H) provided T is sufficiently small. Moreover, for a non-random constant CT, sup II /(t,x)II 5 CTIIh1I2, h,x E H. tE [O,T]

This implies, as in the proof of Theorem 7.1.1, the required strong Feller property on a small time interval, and then, by the semigroup property, on [0, +oo[.

Theorem 14.3.2 The transition semigroup Pt, t _> 0, corresponding to solutions of equation (14.2.1), has the strong Feller property on H.

Proof - For any R > 0 let PtR, t > 0, be the transition semigroup corresponding to solutions XR of equation (14.3.2). Let TR

= inf{t > 0 : IIXR(t,x)II > R}.

It is clear that

X(t,x) = XR(t,x) for all t < TR, x E H. Let now cP E B6(H), then PRV(x)

- Ptw(x)I

=

IE ((P(XR(t, x))

<

(t, x))) j

J]E

- P(X (t, x))) X,R«

2 sup I co(z)I]P(TR < t). zEH

Chapter 14

268

Since the functions PRcp are continuous for each R > 0 and t > 0 it is enough to show that for arbitrary M > 0 and t > 0 lim sup iP(TR
(14.3.7)

This follows, however, from the estimate (14.2.12) valid also for processes XR. In fact if IIxil < M then sup IIXR(s, x)II2 <- 2 sup IIYR(s, x)II2 + 2 sup II WA ('5' X) 11 2 s
s
s
dr < 2MZesfo IIN'A(3)Il1 ds +21 8 f ` IIWA(")II'd9II WA(r)I14 00

+2 sup IIWA (s,x)I12 s
Since the right hand side of the estimate is finite and independent of x, (14.3.7) holds.

14.4

Invariant measure

14.4.1

Existence

To prove existence of an invariant measure for equation (14.2.1) it is convenient to write the problem in a slightly different form. For any

a>0weset

Sa(t) = e-atS(t), t > 0.

Then the mild form of (14.2.1), with x = 0, is equivalent to the integral equation

X(t) = I j t Sa(t-s)D, (x2 (s)) ds+aJ t sa(t-s)X(s)ds+WA(t), where

WA' (t) _ f Sa(t - s)dW(s), 0t

and a will be chosen later. Finally, setting

Y(t) = X(t) - WA(t),

Burgers equation

269

we consider the initial value problem

Y'(t) = AY(t) + 1 Dt ((Y + WA)2) + aWA, (14.4.1)

Y(o) = 0.

J

Note that equation (14.4.1) defines a transition semigroup Pt, t > 0, on Bb(L2(0,1)) which is a Feller one (because the solutions to (14.4.1) depend continuously on the initial conditions). Therefore, by the general theory, see Chapter 6, to show existence of an invariant measure for (14.4.1) it is enough to prove that the family of measures 1 JT

is tight on L2(0,1). For this we need a lemma.

Lemma 14.4.1 For any e > 0 there exists ME > 0 such that, for all T > 0, T

I

TP

ME) ds < F.

(14.4.2)

Proof - Without any loss of generality we can assume that Y is a strict solution of (14.4.1). Multiplying both sides of this equation by Y(t) and integrating by parts in [0, 1], we find

I

2dt

d+

2

since

d

YWA

<,

1 j'y2ygd =0.

By the Holder inequality we find

+

J112 d < Ilu'A Il4 IIY114

IIW

[j'Y2d]"2

I1[Jo' (14.4.3)

Chapter 14

270

We want now to express 11Y114 in terms of 11Y112 and IIYIIH2(o,l) By

the Sobolev embedding theorem we have L4(0,1) C H1/4(0,1),

so that there exists a constant C > 0 such that IIYII4 5 C

IIYIIHI/4(0,1)-

It follows, from a well known interpolatory inequality, see e.g. J. L. Lions and E. Magenes [107], that f1

IIYII4 < C1 IIYI12/4 IIYIIH (O,l) =

C1

Y2de]1/s

.

IIYII2/4 [Jo

From (14.4.3) it follows that

+

2 dt IIYII2

I 1/2

+ 2 IIN'iII4 [j1 Y d

]

+ « IIWA-11211Y112.

(14.4.4)

Now recalling the inequality

x'

xy

and setting p = such that

p

y9

1

+ q,

1

p+q

=1, x,y?0,

3q = 5, we see that there exists a constant C2 > 0 5/8 IIYII2/4

C1 IIWA IJ4

1f l y2 d ]

< C211W 114/3IIYII2+

4

101

Y2 d

.

(14.4.5) Moreover

[f1]h/2

2 IIWA112 By substituting

2dtIIYI12 + 2

< 2 IIWAoIJ4 +

(14.4.5))

4

jy2d.

(14.4.6)

and (14.4.6) in (14.4.4) we find

y2<

<

C211wapI4/3IIYI12 +2IIu'AII4

+ «IIWAl12IIYII2. (14.4.7)

Burgers equation

271

By using the Poincare inequality 11 Y d > ,<2

1

and the following one 4

a IIWAII2 IIYII2 < 72

22

4

IIWiII2 +

we finally get dt

IIY(t)I12 +

IIWi(t)II413

2

IIY(t)I12

2C2

<_

872

+

IIY(t)II2 + 4 IIWi(t)II4

IIWa(t)I12, t > 0.

We now prove that there exists a > 0 such that t>O

E (IIW i(t)II4/3)

8C2

We have in fact

Jt

0 00

e-

en (e)

n=1 00 7r

E

E

n=1 a + 7r2n2 ,

Since WA(t, 6) is a Gaussian random variable, we have, for arbitrary p > 1 and a finite constant cp > 0,

]EIWA(t,f)Ip = ep (EIWA(t,e)12)p'2 <

cp

1

00

1

p/2

l

a + 2n2 /

, e E [0, 1].

Chapter 14

272

Taking p = 4 it follows that °°

1

Ta

E [11 WA(t)114] G C4

+ 7r

n=1

2

2n2

This implies (14.4.9) by the Holder estimate. Finally we use an argument due to D. Ggtarek, see G. Da Prato and D. G.tarek [38]. We fix M > 1 and set

((t) = log (IIy(t)II2 V M) . We have

d IIY(t)112. IlY(t)112 X{IIy(t)II2>M} dt

By multiplying both sides of (14.4.8) by 1

IIY(t)112 X{Ily(t)II2>M}'

we get 2

2

X{Ily(t)II2>M} < 2C2 2

M IIWA(t)Ii4 + 8 M IIWA(t)II2

Integrating from 0 to t and taking expectation, we find

E(((t) - ((0)) +

2

f

t

p (IlY(S)112 >_

M) ds

t

< 2C2

It [EllW2(s)114/3 + MEIIWA(s)114 +

2E11WA(s)112 ds.

But ((t) - ((0) = ((t) - M > 0, and so, taking into account (14.4.9), t

It

(IlY(s)1122 ? M) ds

f

t>EIIN'A(s)114ds

<

2 + Mtir2

+

M16a 4 EIIWA(s)112ds,

tX

4

Burgers equation

273

and the conclusion follows by choosing M sufficiently large.

We can now prove the result

Theorem 14.4.2 There exists an invariant measure for the transition semigroup Pt, t > 0, corresponding to solutions of equation (14.2.1).

Proof - We fix a > 0 such that the statement of Lemma 14.4.1 holds. Note that for arbitrary a > 0 the embedding

D((-A)°) C L2(0,1) is compact and therefore to prove the required tightness property it is enough to show that

For any e > 0 there exists M > 0 such that for all T > 1

IT T

P (II(-A)°X(t)112 > M) dt < E.

(14.4.10)

From now on we fix not only a but also a E ]0,1/4[. Repeating the calculations which led to the estimate (14.4.9) we easily deduce that K = supEIt(-A)°WA(t)I12 < +oo. t>1

Therefore by Chebyshev's inequality

IT = T

T

P (11(-A)' W(t)I12 > M) dt

IT TM Jl 1

C

(II(-A)°u'A(t)112) dt

-71) KM

So IT can be made small uniformly in T > 1 provided M is chosen sufficiently large. This way, since X = Y+WA, to show that (14.4.10) holds it is enough to prove it with the process X replaced by Y. This will be done in the same spirit as in the proof of Theorem 6.1.2 exploiting the regularizing effect of (14.1.3).

Chapter 14

274

For the mild solution Y of (14.4.1) and arbitrary t > 0 we have

that

(-A)O'Y(t + 1) = (-A)°S(1)Y(t) f t+1

+ (-A)° J

it

+ a(-A)" J

it

S(t + 1 - r)D (Y(r) + WA(r))2 dr

t+1

S(t + 1 - r)WA(r)dr.

Taking into account Lemma 14.4.1 and Remark 14.2.3 we have (for any p > 1 such that p + or < 1) II (-A)°Y(t + 1)112

< M1IIY(t)112 + M2 sup IIY(t +r)1122 o
+ M3 (ft+l

II WA(r)II Zdr )

,

for some constants Ml, M2, M3 and for all t > 0. Moreover from the estimate (14.4.8) IIY(t + r)II2

+

j

t+r e

2C2

-

ftt+r

e2C2

IIWA'(s)II4/3dxIIY(t)II2

f+rIIW.,(3)114/ads

a

2

[411h'vp114 +

8a2

a

2

IIWA (P) II2

dp,

and so Sup IIY(t + r)II2 <

e2C2 ftt}1 IIW. (x)II4/3dxIIY(t)I12

O
+

rt+l e2C2 fp+rllWa"(x)114/ads

{4IIV(P)II 4 + 8 2IIWa(P)I121 dp.

Burgers equation

275

Consequently

P(II (-A)°Y(t + 1)112 > M)!5 P (MiIIY(t)112 +P (M2

O<rpl

11Y(t + r)112 >_

)

1/P

(M3(ft+1

+P

s

Ilu'A(r)IlPdr)

M) >_

>

m 3

< I1(t) + I2(t) + I3(t)-

Moreover

12(t) < p

(Ivf2e2C2

ftt+1

IIK'A(s)114/3ds11y(t)II2

?

6)

+p (M2e2C2 ft"' IIWA_(.)1148,3 d. I'+1

[411WA(s)1124

X

+

s;

221IWA(S)1122

ds > 6M )

< I21(t) + I22(t)-

In addition 121(t) < P

(1W2e2C2ftt+1

I21(t) S ? (M2e2C2

+p

M/6)+r (IIYtII2

>

ftt+' Ilu'A(s)II413ds > M16) g

(f`+1

[411 u'A(S)114

+

2Z

llw

(S)I12,

ds >

M6)

6)

,

Chapter 14

276

Finally we see that

T f?(IR-AfY(t+ 1)112

+T

IT (e2f"' II'(s)II3ds > P

'T I

/)

>M/6\

I dt

P (IIY(t)112

+T I P

(fI+I

8a2II11'A(s)I121 1411WA(s)I14 +

(It+,

1

+T J T P

dT

> M) dt < T j?(MiIIY(t)II2> M) dt

(M3

i

IIWA(s)IIPds)

P

>

M

ds > M/6 I dt dt.

The first two terms on the right hand side can be made arbitrarily small uniformly in T if M is chosen sufficiently large (by Lemma 14.4.1). The same is true by Chebyshev's inequality with the remaining terms, because, as is easy to check, Sups>0E[IIWA(s)II4/3+ IIWA(s)II4 + IIWA(s) I2 + IIWA(S)II21 < +OC,

compare the derivation of the estimate (14.4.9).

14.4.2

Uniqueness

Finally we prove that the invariant measure for the equation (14.2.1) is unique. According to the general theory it is enough, see Chapter 7, to show that the corresponding transition semigroup is irreducible and has the strong Feller property. Since the latter was proved in Theorem 14.3.2, it remains to show irreducibility.

Proposition 14.4.3 The transition semigroup Pt, t

_>

0, on the

space Bb(L2(0,1)), corresponding to (14.2.1) is irreducible.

Proof-

Let us fix T > 0,a E L2(0,1) and b E Ho (0, 1). We show first that there exists v E L2(0, T; L2(0,1)) such that for the solution

Burgers equation

277

y(t), t E [0, T] of the problem yt(t, e)

= D2Y(t, ) + 2 Dg (y2 (t, e)) + v(t, t > 0,

y(t,0)

),

E [0,1],

(14.4.11)

= y(t,1) = 0, t > 0,

y(0, ) =

a(e), C E [0,1],

one has y(T, e) = b(e),

E [0,1].

Assume in addition that a

E H01(0, 1). Then there exists u E L2(0,T; L2(0,1)) such that for the solution z(t), t E [0,T],

of the linear problem

zt(t, ) = D2z(t, e) + u(t, e), t > 0, z(t, 0)

= z(t,1) = 0, t > 0,

z(0, e)

= a(e), e E [0,1],

E [0,1], (14.4.12)

one has z(T, e) = b(e), e E [0,1],

and one can easily check that z E C([0,T]; H(0,1)). Define

v(t, e) _ - 2 D2 z(t, ) + u(t, ), t > 0,

E [0,1].

Then v E L2(0, T; L2(0,1)) and by direct substitution one sees that z is the solution to problem (14.4.11). Now, from the well known identity for solutions of (14.4.11), 2 dt Ily(t)1122 + II

2

(y, v),

it follows that, if v = 0, then for almost all t E [0,T], y(t) E Ho(0,1). It is therefore enough to define v(s) = 0 in [0, ] where t E ]0, T[ is

Chapter 14

278

a moment such that y(t) E Ho(0,1) and for the remaining interval Ft, T] to use the first part of the proof. To prove the proposition we will estimate the L2 distance between

the solution X to equation (14.2.1), with xo = a, and the function y. Using our previous notation,

X (t) - y(t) = 2

F(X2

- y2)(t) + WA(t) - VA(t),

where

VA(t) = JO S (t - s)v(s)ds, t E [0,T]. t

From the proof of Lemma 14.2.1, see formula (14.2.6), it follows that IIF(X2

- y2)(t)112 < C ft(t - s) 411X2(s) - y2(s)Ilids. 0

But, for arbitrary s E [0,T],

IIX2(S)

- Y2(S)II,

<-

II X (S) - y(S)112 - IIX(S) + y(S)112,

and we see that t

II X (t) - y(t)112 <

2C f (t -

s)_

(IIX(S) - y(S)112 (14.4.13)

XIIX(S) +y(S)112)ds + IIWA(t) - VA(t)112.

Defining

Y(t) = X(t) - WA(t), t E [0,T], we have that IIX(t)112 < IIY(t)112 +

IIWA(t)112.

Taking into account the estimate (14.2.11), Wt IIY(t)112 <- IIWA(t)II00 + 411WA(t)II00,

derived at the end of the proof of Theorem 14.2.4, we have that if sup IIWA(t)II. <- y t
Burgers equation

279

then

IIY(t)112 < e"t + C2

f

t

ecl'ds, t E [0,T],

0

where

Consequently IIX(s) + y(s)112

< IIy(s)112 + IIY(s)I12 + IIWA(s)112 < IIy(s)II2 + e°'s + C2 sup

s
{iisii2

js

e"'dr + IIWA(s)112

is

+ e°'' + C2 f

ec'rdr + 7]

< C.

where the constant c depends only on 7.

From (14.4.13) we have that IIX(t) - y(t)I12

<-

2 cc J

t

(t - s) +IIX(s) - y(s)II2ds

+ IIWA(t) - VA(t)112. Still assuming sup IIWA(t)112 < 7, t
by a generalization of Gronwall's lemma sup II X (t) - y(t)II2 <- e sup II FVA(t) - VA(t)112, t
(14.4.14)

where the constant c depends only on T. Since the support of the distribution of the process WA(-) in C([O,T]; L2([O, 1])) is exactly the closure of the set of all functions

f

S(t - s)w(s)ds, t E [0, T], w

L2(0,T; L2(0,1)),

Chapter 14

280

we have for arbitrary e > 0 P (SUP II WA(t) - VA(t)112 <

> 0.

t
Let us fix

7 = e + Sup

IIVA(t)112.

t
Then we have P

(SUt
IIWA(t) - VA(t)112 < E

T P (SUt
II WA(t) -

VA(t)112 < E &

T

t
IIWA(t)I12 <_

7) > 0.

Therefore taking into account the estimate (14.4.14) P(SUP IIX(t)-y(t)IIP <ecfI t
1P (sup 11X(t)- y(t)II p < eC & sup IIWA(t)112 S 7 t
(SUP II WA(t) - VA(t)112 < E & SUP II WA(t)I12 <_ 7 1 >0. t
In particular

P(sup IIX(t)-b)11 <ei) >0. t
J

Since e > 0 can be arbitrarily chosen the result follows. We can now state our final result.

Theorem 14.4.4 There exists a unique invariant measure for the transition semigroup Pt, t > 0, corresponding to solutions of equation (1$.2.1). Moreover p is ergodic and strongly mixing.

Proof - By Theorem 14.4.2 there exists an invariant measure it. Since the semigroup Pt, t > 0, is a strong Feller one by Theorem 14.3.2 and irreducible by Proposition 14.4.3, the conclusion follows from Doob's theorem, 4.2.1.

Chapter 15

Navier-Stokes equations In this chapter we first establish existence of a solution to the stochastic Navier-Stokes equations on a bounded domain D C IR2 by a direct method. Then existence of an invariant measure is proved by a modification of the dissipativity method.

15.1

Preliminaries

Let D be an open subset of R2 with regular boundary OD. We are concerned with the problem

dtZ(t,.) = (vAZ(t, ) + ZZ(t, e) Z(t, ) + Vp(t, ))dt + dWQ(t), on [0, +oo[ xD,

0, on [0,+oc[ xOD, div Z(t, ) = 0, on [0, +oo[ x D, Z(0, e) =

on D. (15.1.1)

Here Z(t, ) is the velocity of an incompressible fluid, v is the viscosity parameter, and p(t, ) the pressure of the fluid. Moreover WQ is a Q-Wiener process in a probability space The distributional derivative of WQ(t), t > 0, represents external force acting on the fluid. 281

Chapter 15

282

In this chapter we first prove the existence of a solution to the stochastic Navier-Stokes equation on an open subset of R2. We use a simple and new fixed point argument instead of the more common Galerkin-Faedo approximation scheme. Then we show the existence of an invariant measure by adopting a method of F. Flandoli [66]. For the proof of uniqueness we refer to F. Flandoli and B. Maslowski [67]. These results can be generalized to the Benard problem, that is a Navier-Stokes equation perturbed by white noise coupled with the heat equation, see B. Ferrario [65]. We shall denote by V the set of all mappings u : D -> R2 of C°° class and having compact support in D, such that

div u(E) = Tr

E D.

0,

For all u = (ul, u2), v = (vl, v2) E V we set

lu(f)12 = ui(e) + u2O,

(u(0, v(0) =

E D,

u2(e)1)2(b), S E D.

For any p > 2, we shall denote by LP the closure of V with respect to the norm HUG = (JD

Iu(S(`)I2det)

1/p

and by V the closure of V with respect to the norm Ilully =

IDH

We

Obviously H is a closed subspace of L2(D) x L2(D) = L2(D;R2),

we shall denote by P the orthogonal projection of L2(D) x L2(D) onto H. We introduce finally the linear operator D(A) = [H2(D) x H2(D)] n v, (15.1.2)

Au = PAu, forall u E D(A). A is a self-adjoint strictly negative operator on H, and V = D((-A)1V2), see e.g. R. Temam [150]. We shall denote by {ek} an orthonormal

Navier-Stokes equations

283

complete system of eigenvectors of A and by {Ak} a sequence of pos-

itive numbers such that Aek = -Akek, k E N. We assume that {Ak} is not decreasing. A central role in our considerations will be played bythe space E = L4(O,T; f,4).

Since L4 is the closure of V with respect to II(xl, x2)114 =

Xl(e)14

(ID

1/4,

+ Ix2(e)I4) de) 1

(I

(x1, x2) E V,

we have 1/4

[IT IIUIIE

r JD

=

(Iul(t,e)I4 +

dd dt

1/4

(fT II u(t)II4dt) We will also need the following inclusion result.

Proposition 15.1.1 For any T > 0 we have L°°(O, T; H) n L2(0, T; V) C E,

and there exists a constant K, independent of T > 0, such that IIIIIIE C K(IIUIILoo(O,T;H) + IIUIIL2(0,T;V)), U E E.

(15.1.3)

Proof - We recall that due to the Sobolev embedding theorem H1/2(D) C L4(D)

and there exists Cl > 0 such that IIVIIL4(D)

C1IIvIIH1/2(D), V E H1/2(D).

(15.1.4)

Chapter 15

284

Define V1/2

= (H1/2(D) x H1/2(D)) n V.

Then V1/2 = D((-A)114). By a well known interpolatory inequality, see J. L. Lions and E. Magenes [107, pag.22], we have for some constant C2 > 0 and all t E [0, T] IIu(t)IIV1,2 s C2IIu(t)IIH2 Ilu(t)IIV2.

It therefore follows from (15.1.3) that

Ilu(t)114 < 2C Ilu(t)II i,2 < 2C C Ilu(t)IIH IIu(t)II i. Now integrating this inequality in [0,T] we easily get (15.1.4).

15.2

Local existence and uniqueness results

By applying the projection operator P to the first equation in (15.1.1)

and setting X = PZ we find the problem

dX(t) = [AX(t) + D4(X(t)) X(t)] dt + dW(t), T > 0, X (O) = x, (15.2.1)

where T > 0 is fixed and W is defined by

W(t) _

Qh/3hgh.

(15.2.2)

h=1

Here {/3h} is a sequence of mutually independent real Brownian mo-

tions in a fixed probability space (St,

F) adapted to a filtration

Moreover {Oh} is a sequence of positive numbers and {9h} is an orthonormal basis in H. We reduce problem (15.2.1) as usual to a deterministic one by setting Y(t) = X(t) - WA(t), t E [0,T], IP' a.s.

Navier-Stokes equations

285

We find

Y'(t) = AY(t) + Dt(Y(t) + WA(t)) (Y(t) + WA(t)) + f(t), t > 0, Y(0) = x, (15.2.3)

where WA(t) is the stochastic convolution

WA(t) = J S(t - s)dW(s), 0t

(15.2.4)

and S(t) = etA, t > 0. We now make the following basic assumption

Hypothesis 15.1 Process

has a version in L4 with trajecto-

ries integrable in the fourth power. Notice that an equivalent formulation of the above hypothesis is that WA(.) has an E-valued version. We will interpret equation (15.2.3) as an integral equation

Y(t) = S(t)x +

J0

t

S(t - s)De(Y(s) + WA(s)) . (Y(s) + WA(s))ds

= S(t)x + F(Y + WA)(t), t E [0, T], where F : E --+ E is a continuous extension of the operator

Fo:CI([0,T];V)-+E given by the formula (Fou)(t) =

J

t

S(t - s)(Gou)(s)ds, t E [0,T],

where

Go : Cl([O,T];V) -+ E, (Gou)(t) = Dtu(t) u(t), t E [0,T]. To show that the operators G and F are well defined it is useful to introduce a trilinear form on V x V x V by setting b(u, v, z)

=

(Dov(e) . u(e), z(e))d JD 2

i9Vh

Uk Zh d h,k-1 ID - k

Chapter 15

286

By the Holder inequality we have Ib(u,v,z)I S 41IvIIvIluII41IzII4,

(15.2.5)

so, by the inclusion (15.1.3), the definition of b is meaningful. We now list some useful properties of b.

Lemma 15.2.1 The following identities hold: b(u, v, z) = -b(u, z, v), V u, v, z E V,

(15.2.6)

b(u, v, v) = 0, V U, v E V.

(15.2.7)

and

Proof of (15.2.6) - We have 2

b(u, v, z)

=

OVh

h,k-1 ID 09G

Uk Zh d

2

auk

h ,k-1

dC 1f V h a-k zh D

2

OA vh uk E ak h,k-1 JD I/r

dC

div u (v, z) d4 - b(u, z, v),

and the conclusion follows since div u = 0.

Navier-Stokes equations

287

Proof of (15.2.7) - We have b(u, v, v)

=

l9yh ukVhd E h,k=1 ID ak

=

1.

2

a

1/

2 h,k=1 ID uk '9 k \vh) 1

2

8uk

2h,k=1 Dak

2

vh

<

-12IDdivulvl2de=0.

Lemma 15.2.2 The mapping Go can be continuously extended to G : E -* L2(O,T;V'). Moreover IIG(u) - G(v)II L2(o,T;Vl) 5 4(IItIIE + II'IIE) IIu - VIlE, u, v E E.

Proof-

Let u E C2([O,T];V), cp E L2(0,T;V). Then by Lemma 15.2.1, denoting by << >> the duality mapping between L2(O,T; V') and L2(O,T;V),

IT << Gou, co >>

=

jT

D

(D

u(t, ) . u(t, ), cp(t, )) dt dd

b(u(t), u(t), cp(t))dt = - 1T b(u(t), W(t), u(t))dt.

Chapter 15

288

In addition if u,v E C2([O,T];V) and So E L2(O,T;V) we have << Gou - Gov, cp >>

jTJ(Dut,f =

u(t, ) - Dfv(t,

v(t, e)) 4odt d

T

_ - f [b(u(t), u(t), p(t)) - b(v(t), v(t), p(t))Jdt

Tbut

0

t u(t))

b(v(t),

t v(t))]dt

0T

= J [b(u(t), cp(t), -u(t)) + b(v(t), cp(t), v(t))]dt

_

I

[b(u(t), o(t), v(t) - u(t)) + b(v(t) - u(t), (a(t), v(t))]dt.

It follows from (15.2.5) that << Gou - Gov,

>> 1< 4 f T [IIu(t) - v(t)II4 IIu(t)II4 IIW(t)IIV

+II u(t) - V(t)114 IIv(t)II4 II (t)IIVI dt T

< 4 (10 Iiu(t) - v(t)II4dt)

(fT

1/4

l1/4

(IT

(Ilu(t)II + Ilv(t)II4dt) I

II (t)II vdt)

< 4IIu - VII E (IIUIIE + II'IIE) IIWIIL2(o,T;V) .

Consequently IIGou - GovII L2(o,T;V) <- 4IIu - vIIE(IIuIIE + IItIIE),

and we see that the extension G exists and satisfies the required inequality.

Navier-Stokes equations

289

The following lemma is classical, see e.g. J.L. Lions and E. Magenes [107].

Lemma 15.2.3 Assume that A is a negative self-adjoint operator on H and V = D((-A)'12) C H C V'. Then A and S(t) = etA have continuous extensions from V to W. If t

y(t) = y(t;9) = 10 e(t- s)A g(s) ds, t E [0,T], 9 E L2(0,T;V'), then

y E L°°(0,T;H) n L2(0,T;V), and for a constant L > 0, independent of T > 0, II yII LO°(O,T;H) + II yII L2(0,T;V) < LIIgIIL2(O,T;V')

Lemma 15.2.4 The transformation F0 can be continuously extended to F : E - E. Moreover there exists a constant M > 0, independent of T > 0, such that IIF(u) - F(v)II E < M (IIuIIE + IIvIIE) In - vIIE, u, v E E.

Proof - Using the notation of the previous lemma we define the extension F by the formula

F(u)(t) = y(t,G(u)), t E [0,T], u E E. By Lemma 15.2.2 F E L°°(0,T;H)nL2(0,T;V). Moreover, for u,v E E,

IIF(u) - F(v)IIE <

Klly(.,G(u)) - y(-,G(v)))IIL-(O,T;H)

+ II y(', G(u) - G(v))II L2(o,T;v) Taking into account Lemma 15.2.3 again we get IIF(u) - F(v)II E < KL JIG(u) - G(v)II L2(O,T;vl), u, v E E.

Finally, by Lemma 15.2.2, II F(u) - F(v)IIE < 4KL In - VII E (IIuIIE + IIvIIE), u, v E E,

and it is enough to define M = 4KL.

Chapter 15

290

Theorem 15.2.5 Assume that Hypothesis 15.1 holds. Then for arbitrary x E H there exists a random variable r taking values P-a.s. in ]0,T] such that equation (15.2.1) has a unique solution on the interval [0, r).

Proof - We first remark that the function O (s) = S(s)x, s E [0, T], belongs to

L2(0,T; V) n C([0, T]; H).

By Proposition 15-1.10 E E. Moreover WA(-) E E, P-a.s.

Let us fix w E 12 such that WA(.,w) E E and write cp(s) _ WA(s, w), s E [0, T]. To prove the theorem it is enough to show that the fixed point problem

u = 'b + F(u + (p), u E E, has a unique solution in the space E = L4(0, T; L4) for T > 0 sufficiently small. We apply the following version of the contraction mapping theorem.

Lemma 15.2.6 Let F be a transformation from a Banach space E into E, a an element of E and a > 0 a positive number. If F(0) = 0, IlaiIE < Za and

II.F(zi) - F(z2)IIe

2

Ilzl - 2211E for lIziIIE < a, IIz211E < a,

then the equation

z = a + F(z), z E E, has a unique solution z E E satisfying IIzIIE < a.

Proof - Let :'(z) = .F(z) + a, z E E. By an induction argument II.k(o)II < a (1 - 2-k) , k E N. From the Lipschitz condition it follows that the sequence {.1' (O)} converges to a fixed point of F.

We now continue the proof of the theorem. Define E = E, a = + F(4p),

.1(u) = F(u + p) - F(4o), U E E.

Navier-Stokes equations

291

Then .F(O) = 0 and for u1, u2 E E and a = 44 IIJ'(ul) - F(u2)IIE < 2MaI ul

- U211 <- 2IIui - U21).

It is clear that we can find T > 0 such that in the space E L4(O,T; L4) II& + F(cp)II E < 11011E + IIF(c)IIE 1/4

<

II S(s)xII

+

ds)

a. (fT

The desired result follows from Lemma 15.2.6.

15.3 A priori estimates and global existence We now prove the main result of the present section.

Theorem 15.3.1 Under Hypothesis 15.1 for arbitrary x E H there exists ]P-a.s. a unique solution X) E E of the equation (15.2.5). The solution depends continuously on x E E.

Proof - The first part of the theorem is an immediate consequence of Theorem 15.2.5, and the a priori estimate of Proposition 15.3.2 below. Continuous dependence on initial data is clear for the local solutions, see the proof of Theorem 15.2.5, and therefore it holds also for global solutions.

Proposition 15.3.2 Assume that cp E E, x E E and that y E E is a solution to the equation y(t) = S(t)x + F(y + cp)(t), t E [0,T]. Then we have

Sup ily(t)IIH tE[O,T]

< eh' fT II

(r)II4dr

IIxIIH

(15.3.1)

+ K2

0 0

T

eKI fT

II'(s)II4ds,

Chapter 15

292

JoT IIy(t)IIVdt

<

IIxIIH + K1

+

K2 f IIW(s)I14ds,

ssup

IIy(S)II,10

T 0

(15.3.2)

where

Kl = 4(24)3C1, K2 = 32Ci ,

(15.3.3)

and C1 is the constant from (15.1.4).

Proof - It is enough to show (15.3.1) and (15.3.2) for x E D(A), regular cp and for the (strong) solution to the differential equation = Ay(t) + DE(y(t) + ap(t)) (y(t) + 4p(t)), t E [0,T], 1 dy(t) dt Y(O) = X.

(15.3.4)

We will show first that dIIy(t)IIH + IIy(t)IIv < K'IIy(t)IIH K2 are given by (15.3.3).

Multiplying (15.3.4) by y(t) and integrating in D we obtain by Lemma 15.2.2

2 dt II y(t)II H + II y(t)II V = b(y(t) + o(t), y(t) + p(t), y(t)) b(y(t) +
= -b(y(t), y(t), 4((t)) -

y(t), ap(t)).

We first estimate b(y(t), y(t), (p(t)). Since 2

ay!(t,e)

k,1=1

aek

b(y(t), y(t), p(t)) = E JD

yk(t, 6) yol (t, e),

we have 2

b(y(t), y(t), p(t))I <_ E Il yl(t)II HA(D) IIYk(t)II II(Pr(t)114. k,1=1

Navier-Stokes equations

293

By (15.1.4) it follows that 2

I b(y(t), y(t), p(t))I G_ C1 E II yl(t)IIH1(D) II Yk(t)II HI/2(D) II(PI(t)II4. k,1=1

Using the interpolatory estimate Ilyk(t)IIHI/2(D) < Ilyk(t)IIHI(D) Ilyk(t)112/2

we get

I b(y(t), y(t), '(t))I 2

G C1 E IIy!(t)IIHI(D) IIYk(t)IIH'(D) IIYk(t)112/2 II(P!(t)114 k,1=1 2

2

C1

2

yk(t) IIH1(D)

lyl(t)IIH1(D) I

k=1

1=1 2

XI

I

Ilym(t)112/2]

I

I

b(y(t), y(t), (p(t)) I 2

G C1

2

2

21/2

Ilyk(t)IIH 1(D)

!-1

k=1

I 2

x 23/4

II

l ym(t)112)

1/4

m=1

I1yl(t)IIH1(D)

2

ll(p.(t)112 E n=1

G 8C1 Ily(t)IIv2 Ily(t)IIH2 11'P(t)J14-

Finally this implies that I b(y(t), y(t), (p(t))I < 4IIy(t)IIv+2.243 Ci II y(t)II H

t E [0,T]. (15.3.6)

Chapter 15

294

To estimate b(cp(t), y(t), W(t)) notice that

I b(y(t), y(t),
2

h,k=1

D

ayh(t, l)

cph(t,e)de

,

2

< C1 >

Ilyh(t)IIHl(D) I1skk(t)I14

Ilkph(t)II4

h,k=1 \12

2

2

C1 E IIYh(t)IIHI(D) E II4Pk(t)II4J h

(k=1

1

2

C121/2

1/2

IIYh(t)IIHI(D) h=1

l1/4

22`

23/4

D

k(t)II4

(k=1

2

/

< 4C1

Therefore Ib(cp(t),y(t),cp(t))I

4IIy(t)IIv +16 Ci Ik (t)II4, t E [0, T]. (15.3.7)

Adding the inequalities (15.3.6) and (15.3.7) we get (15.3.5). Taking into account that (15.3.5) is a first order differential inequality for II y(t)II H, t E [0,T], we arrive at (15.3.1). Moreover, integrating (15.3.5) from 0 to T we obtain (15.3.2). With obvious changes in the proof we can deduce the following a priori estimate needed in the next section and generalizing (15.3.5).

Proposition 15.3.3 If for an a > 0, y is a mild solution of the equation

dt

Ay(t) + Dt(y(t) + co(t)) . (y(t) + cp(t)) + acp(t),

(15.3.8)

Navier-Stokes equations

295

then for arbitrary E > 0 d II y(t) II H + II y(t)II V

<

(e + Kl IIc (t)II4) IIy(t)II H

+

K2 Ilsv(t)114 4

+4a -. II(p(t)II2, t >- 0. 2

(15.3.9)

Existence of an invariant measure

15.4

We pass now to the existence of an invariant measure for equation (15.2.1) under the condition

Hypothesis 15.2 Process WA(-) has a continuous version in

D((-

A)1

+26

) for some 9 E ]0, 2 [.

We remark that since

D((-A)') C D((-A) 4) C L4, Hypothesis15.2 is stronger than Hypothesis 15.1. The main result of this section is the following theorem. We will adapt, following F. Flandoli, see [66], the technique which proved to be useful for dissipative systems in §6.3. For different ways of investigating asymptotic properties of the stochastic Navier-Stokes equation, we refer to A. B. Cruzeiro [31] and to S. Albeverio and A. B. Cruzeiro [3].

Theorem 15.4.1 Under Hypothesis 15.2 there exists at least one invariant measure for equation (15.2.1).

Let V(t), t > 0, be a Wiener process independent of V but having the same law as W(t), t > 0. We denote by W a Wiener process defined on the whole real line by the formula

W(t), if t > 0,

W(t) =

Chapter 15

296

and by Ft the filtration

Tt=a(W(s), s s, is the unique solution of the stochastic Navier-Stokes equation

dX(t) = AX(t) + D£X(t) X(t) + dW(t), t > s,

}

X(s) = 0.

Let moreover Z(t), t E R, be a solution of the linear equation,

D((A)_ 1 420

(15.4.1)

)-continuous, stationary

dZ(t) = AZ(t) + dW(t), t E R.

Since the semigroup S(t), t > 0, is exponentially stable on H we have

Z(t) =

_It

S(t - s)dW(s), t E R.

00

However, the explicit form of Z will not be needed in the proof.

Remark 15.4.2 Since

D((-A)

i+ze4

) C D((-A)°),

Hypothesis 15.2 implies that the process Z is also D((-A)°)-continuous.

Note that the space D((-A)°) is compactly embedded into H. Consequently if one proves that

the process X(t,0), t > 1, is bounded in probability as a process with values on D((-A)°), (15.4.2)

one gets immediately that the laws L(X(t, 0)), t > 1, are tight on H. This is certainly sufficient for the existence of an invariant measure. The proof of Theorem 15.4.1 will consist of two steps.

Step 1- For any a > 0 denote by Z. the stationary solution of the equation

dZa = (A - a)Zadt + dW(t).

Navier-Stokes equations Since

297

Za(t) = Z(t) +

-a

ft

e(A-a)(t-s)(Za(s)

- Z(s))

e(A-a)(t-o)Z(o)d, t > s,

one can assume, see Remark 15.4.2, that continuous process. Moreover let

is also a D((-A)')-

Ya(t, S) = X(t, S) - Za(t), t > S.

Then

Yc(t) = Ya(t, S), t > S, is the mild solution of the equation

d Y. (t)

= AYa(t) + Dt(Ya(t) + Z. (t)) . (Y. (t) + Za(t))

+ aZa(t), t > s,

= -Za(s) Y(s) The main aim of Step 1 is to show the following proposition.

)

Proposition 15.4.3 There exist a > 0 and a random variable such that P-a.s. IIYa(t,s)II2 <

for alit E [-1,0] and ails < -1,

ro

J

IIYa(t, s)IID((_A)1/2) ds <

(15.4.3)

for alit E [-1,0] and ails < -1. (15.4.4)

Proof - Taking into account Proposition 15.3.3 and the obvious estimate )1IIZIIH -< IIZIID((-A)1/2), Z E D((-A)'12),

one obtains that IIYa(t, s)IIH <

/t

+/

P

eftH-ai+e+K14]d°IIZa(s)II2

of _A1+e+K1IIZQ(C)114]dC

[K2I(fl 4 + 4a2] dQ.

(15.4.5)

Chapter 15

298

Let µa be the unique invariant (and Gaussian) measure for Z,,,(-). It

is supported by D((-A) 4 ) C L4(D) and therefore, by the strong law of large numbers, see Remark 3.3.2, P-a.s lim

1

s--00 t - s

f

t IIZ«(O,)114 do

= fV IIzI14 µa(dz).

It is easy to see that 0.

ali+mJV

Consequently there exist a > 0 and E > 0 such that -A1 + E + Kl

f

V

II zII

(dz) = -y < 0.

From now on we assume that a > 0 and E > 0 are fixed and that (15.4.5) holds. Then for a sufficiently large random so > 0 and all

s < -so, e-2(t-s).

(15.4.6)

We will need the following elementary lemma.

Lemma 15.4.4 Assume that Z is a continuous and stationary Gaussian process on a separable Banach space L. Then for arbitrary b > 0 there exist random variables j1 and 1:2 such that ]P-a.s. IIZ(t)IIL < 6 +' 21t15,

for all t < 0.

Proof - Write 77n =

sup

-n<s<-n+l

IIZ(S)IIL, n E N.

Then, by Gaussianity, for arbitrary p > 0,

lEr = E

sup

-n<s<-n+l

IIZ(s)II,.

Consequently P (7ln > ns) <

"P

nap .

(15.4.7)

Navier-Stokes equations

299

If by > 1 then 00

1: P (iin > nb) < +00 n=1

and by the Borel-Cantelli lemma, P-a.s., for all sufficiently large n 1n < nb.

This proves the lemma.

To finish the proof of Proposition 15.4.3 note that (15.4.3) is a direct consequence of estimates (15.4.5), (15.4.6) and the lemma. The estimate (15.4.4) follows similarly using in addition the a priori estimate (15.3.2). Step 2- We show that condition (15.4.3) holds using Proposition 15.4.3 and a regularizing property of the equation (15.3.8). The following result is a generalization of Proposition 15.3.3.

Proposition 15.4.5 For any 0 E [0, 1/2] there exists a constant C = C(9) such that for an arbitrary mild solution II(-A)By(t)II2 <

e° fo Ily(s)II II(-A)'11y(s)II ds

of (15.3.8)

II(-A)By(o)II2

+c I't ec fQ Ily(s)II II -A)1j2y(s)IIlds

da.

0

Proof - Multiplying equation (15.3.8) by (-A)By and integrating over D we find that 2

Wt II(-A)By(t)II2 + II(-A)2+By(t)I12

_ -b(y(t) + V(t), y(t) + p(t), (-A)2By(t)) = a((-A)By(t), (-A)°ca(t)), t > 0. We estimate first, in two steps, b(y(t) + p(t), y(t) + p(t),

(-A)21y(t)).

(15.4.8)

Chapter 15

300

Step 1- We show that there exists a constant C1 > 0 such that b(u,u,(-A)2ev)

(15.4.9)

II(-A)2VII2II(-A)12uII2.

< Cl

We have in fact, setting b(u, v, z) = (B(u, v), z), that I b(u, u, (-A)29v) I

< C21IB(u, u)II H2A-1(D) II

(-A)2BVII

H1-26(D)

< C2IIB(u,u)IIH2O-1(D)II(-A)B+2vII2. (15.4.10)

Since div u = 0 we have 2

(B (u, u))h =

k

uk =

k=1

(9

(uhuk)

k1

and so 2

IIB(u,u)IIH20-1(D) < C2 E

IIuhukIIH2O(D).

h,k=1

By a classical result on multipliers, see W. Sickel [140], IIVZIIH20(D) <

+0(D),

and therefore IIB(u,u)IIH20-1(D) < C4IIu21IH7(D).

(15.4.11)

Taking into account (15.4.9) we find I b(u, u, (-A)2BV)I

<

C2C4II u2IIHi+O(D)II (-A)B+2 VII2

< C2C4I1(-A)

2

uI12II(-A)e+2VII2,

which proves (15.4.9).

Step 2- We prove that for arbitrary e > 0 there exists a constant K(E) such that I b(v + u, v + u,

(-A)2Bv)I < 211(-A)'+12vII2

(15.4.12)

+K(e)

[IIvII2II(-A)2' VII2II(-A)°vII2 + 11(-A)+42 " UII2]

Navier-Stokes equations

301

From (15.4.9) I b(v + u, v + u,

C,I1(-A)B+2

(-A)2BV)I <

1+420

(v + u)II2

'+20 112 < 2C1 [II(-A)B+2vII2 II(-A) 1+22 vII2 + II(-A)0+2vII2 II(-A) 4 U j 4 2

<

2II(-A)B+2vII2 + 2

+2II(-A)B+2vII2

EII(-A)B+2vII2

' II(-A)

1+20 4

4

VII2

2

+ 2 e' II(-A)' +420 u112 1+22 + 2L1 II(-A) 4 VI12 + +2L' II(-A)

4

uII2 (15.4.13)

By interpolatory inequalities

II(-A)'4VII2

<

CsI (-A)+vII2II(-A)4+Bvll2

<

C61IVI12II(-A)2

vII2II(-A)BvII2II(-A)2+BVII2.

Therefore 2E111(-A)'

V114 4

<-

+

2II(-A)2+BVII2 C4C2 13 s E

IIVII22

II(-A)2 VII22 II(-A)BVI12 (15.4.14)

By substituting (15.4.14) into (15.4.13) we arrive at (15.4.12). From (15.4.8) and (15.4.12), 2 dt II(-A)By(t)I12 + II (-A)2+By(t)I12 <- 2EI1(-A)2+By(t)I12 +K(E) [IIy(t)1122 11(-A)

+aII

(-A)2By(t)II211

1,2y(t)II2II(-A)By(t)112 2

+

kv(t)II2 (15.4.15)

Chapter 15

302

Since 0 < 2, II(-A)2ez)112 <- II(-A)B+2 zll2. Consequently

II(-A)'+2y(t)II2 e

11(-A)'+12- y(t)II2

+

4, -

(15.4.16)

Combining (15.4.15) with (15.4.16) we see that Combining

2 a II(-A)By(t)112 + (1 - 3e)II(-A)2+'y(t)I12 < [K(e)Ily(t)1121I(-A)'/2y(t)II2] II (-A)'y(t)II2 +K(,-)11(-A)1+420 P(t)114

2

+a 2

Thus the proof of the proposition is complete. We are ready now to complete the proof of the theorem. It follows

from the proposition that for arbitrary t < -1 < r < 0 2 II (-A)°Y«(0, t)II2< e`fO IIY.(s,t)1122 1I(-A)1/2Y.(s,t)112ds II(-A)B1'«(r, t)I12

+e

<

f

°ecf,

IIYa(s,t)I1211(-A)1/2Ya(s,t)112d3 (IIZ«(a)II2

r

+ II(-A)

Za(a)II2) do,

II(-A)1"2Yo(s,t)112ds

eC[3UP-1<s
x [II(-A)'/2Y «(r, t)II2 + e f (IIZa(a)I12 + 11 (-A) 4 Za(a)II2) do,

.

Consequently, integrating the above inequality over the interval [-1, 0]

we get for t < -1 that II(-A)°Ya(0, t)112< e`[s tP_1<s
°1

It(-A)1"2Ya(s,t)I12ds

o

x [11(-A)l j2Y(r,t)112 + e

J_

(IIZ«(a)II22 + II(-A)

4

Z«(a)II2) da

.

Navier-Stokes equations

303

By Proposition 15.4.3 there exists a random variable e2 such that lP-a.s.

II(-A)BYa(O,t)II2 <- 6, for all t < -1. Moreover

II (-A)BX(O, t)II2 < II (-A)°Ya(O,t)II2 + II(-A)BZa(O)Ii2.

Since Za(0) takes values in D((-A)°) there exists a random variable S3 such that ]P-a.s.

II(-A)BX(O,t)II2 < 6 for all t < -1.

This proves the tightness of C(X(O,t)), t < -1.

Part IV

Appendices

305

Appendix A

Smoothing properties of convolutions A.1 Let A : D(A) C H --+ H be the infinitesimal generator of an analytic semigroup S(t), t > 0, in the Hilbert space H. We assume that, for some M > 0 and w > 0, we have IIS(t)II <- Me-Wt, t > 0.

For any a, y E ]0, 1[ and any p > 1, we define a linear bounded operator in LP(O,T; H) by setting

R«"'b(t) =

J0 t

(t - a)a-'(-A)"S(t - a)4o(a)da, 0 E LP(O,T; H). (A.1.1)

Proposition A.1.1 (i) If y > 0 and a > y + 1 then R,,.y is a bounded operator from LP(O,T;H) into Ca-'-P([O,T}; D((-A)')). 0 and a > I then for arbitrary b E 10, a - 1 [, Ra,.y is (ii) If a bounded operator from LP(O,T; H) into C6([0, T]; H). 307

Appendix A

308

Proof - Let cp E LP(O, T; H) and set 0 = Ra,., p. Then 0(t) - O(s) = ft(t - a)a-1(-A)1S(t - a)
+ f0 s[(t +J 0

8(s

a)a-1

- (s -

a)cp(a)da

- a)a-1[(-A)IS(t - a) - (-A)^'S(s - a)](p(o)do

= Il + I2 + I3. We will estimate each term 11, I2, I3 separately. ii ii

(jt((t

- a)II(-A)S(t - a)I )da)

1/p

11q

(jt I(s)Ipds)

,

where n + 9 = 1. Since the semigroup S(t), t > 0, is analytic there exists a constant M > 0 such that II(-A)"S(a)II <

M

Q

,

a E [0, T].

Consequently t

II1I < M

-

s a(a-1)qa-,rgda) 1/q

Ik IIP,

(10

where (Icpllp denotes the LP(O,T;H) norm. Thus for a constant M1 IIII

< M1(t - s)a-"-P

In a similar way 1121 < M (Jos[(s

- a)a-1- (t - a)a-1]q(t -

a)--ygda)l1q II
Since for arbitrary positive numbers a < b and q > 1 (b - a)q < bq - aq,

Smoothing properties of convolutions

309

we have

C

ry

(s

< MPIIoII [10

II2IP

s(s 110

-

a)(a-1)q-'Ygdo

MphvII p

(t -

J0 (t -

a)(a-1)q-'Ygda

- J s(t - 0,)(a-1)q-rygda 0

IS(a-1)q-'yq+1

(a-1)q-'Yq+l +

da -

(t - a) g

J

-

t(a-1)q-'Yq+1

S)(a-1)q-ryq+lI

MPII PII p

(a- 1)q-'Yq+1

(t

-

)(a-1)q--yq+1

Therefore, for a constant M2, s)a-

1121 <- M2(t -

r.

It remains to estimate II3I. Note that

a) = j t

a) -

-r(-A)1S(p)dp o

Therefore for a constant M3 > 0

t-o

f

0s

1131 <M3J (S

1

L_t7

dpI o(a)I da. p1+-r

If'y >0 we have 1131

-

M3

M3

J

S(s

(j8(s

-

- a)a-1 [(s - a)-'r - (t -

a)--rj

I o(a)I da 11(,

a)(a-1)q

[(S

- a)-'Yq - (t -

a)-'yg] d o ,

II(PIIP

3 (f(s - 0,)(,-1)q-'Ygda - j(s - 0,)(,-1)q(t - (7)-^,gdal11 q" (J s(S Ty

-

a)(a-1)q-'Ygda

- J 8(t -

a)(a-l)q(t

IIP

- a)-'Ygda11 q IIollp

Appendix A

310

By a similar argument as in the estimation of 1121 we arrive at the inequality S)"-ry-p

1131 < M4(t -

.

Ify=0then, forarbitrarybE]0,1[, 1131 <-M3

J3(s-Q)a-1 /

1dPIW(o,)jdi

s-v P

< M3

S

(S - )a1

.0

< M3 1(0' s(s

<

3 J S(s

-

-

6

1_6dpI P(a)I da

P P a)-6

ft-7

it

o P-

6dPI4o(a)1da

- a)a-1-6 [(t - o,)' - (s - c)"] cP(o)I da

- s)6 j(s - v)a-1-6y'(v)I dQ

<

<

ft -

b3(t

- s)6 (Js 0,(a-1-6)q do, 1I1q IIPIIp.

Therefore if b E JO, a - [ then n 1131 < M5(t -,S)6,

and the required result follows.

Appendix B

An estimate on modulus of continuity B.1 The following result, which goes back to A. M. Garsia, E. Rademlch

and H. Rumsey Jr. [76], is very useful in proving regularity of stochastic processes. It can be used to prove the Kolmogorov continuity theorem and also Sobolev embeddings theorem for spaces with fractional norms, see R. A. Adams [1]. The formulation below is a generalization of that given in D.W. Stroock and S. R. S. Varhadan [148, pp. 60-61]. The proof differs in a few minor points from that in [148].

Let 0 be an open bounded subset of Rd equipped with a norm Let

D = sup Ix - yI x,yEO be the diameter of 0. By A(.) we will denote the Lebesgue measure on Rd, although different choices are possible. Moreover we set

n = r.(0) = inf

inf

O
A(o fl B(x, p)) p

where

B(x,p)={yERd: Ix - yI < p}. 311

(B.1.1)

Appendix B

312

Theorem B.1.1 Let 0 and p be strongly increasing functions from, [0, +oo[ into ]0, +oo[ such that 0(0) = p(O) = 0 and limn ,o 1(r) _ +oo. For an arbitrary continuous function f : C7 -* E such that

r1

(11)

f(77)A(de)A(d,,) < B,

If (

LJ

If(x) - f(y)l provided x, y E 0 and

21x-yj

8

J

o-1

4d+1B

2u2d

J

p(du),

E 0.

Proof - We start from two lemmas. Lemma B.1.2 If cp > 0 on 0 then for arbitrary a > 0 A { rl E 0: 111

Proof - Let (5=

a.

J(

{iiE 0 :

J)(d)


(i )}

.

If A(CS) = 0 then the result is true. If )t(() > 0 then

< a J,3 W(rl)A(drl) <

a,[

o(rl)A(dr!)

Dividing both sides of the inequality by the positive number fo cp(e)A(de) we get the result.

Lemma B.1.3 Let

f , (If(e)-f(ii)I) .(dry),

I( ) = n

E

0.

For arbitrary a, b E 0 and for an arbitrary number r E ]0, D[ there exists c E 0 fl B(a, r) such that I(c) < 2

B r, + d

If (b) `

- f(c)l

p(b - c)

)

< 2 I(b) icrd

An estimate on modulus of continuity

313

Proof - Let 01 = S(E OnB(a,r) : I(() > 2 Bd}, ll

d

02 = (E o n B(a, r) : where If (b)

sa(C) _

2

co(1;) > I(b) }

,

- f(()I\ ,CEO\{b}.

p(Ib-(1)

l

It is enough to show that A (01 U 02) < A (0 n B(a, r)).

(B.1.2)

Since I(b) = fn cp(()d( we have, by Lemma B.1.2,

A(02) < a CEO:

r

0(() > I(b) <

Yrd 2

Moreover

A(O n B(a, r)) > Krd. So

A(02) < 1 A(0 n B(a, r)).

(B.1.3)

On the other hand if A(O1) > 0 then 2

B

A(01) < J I (()A(d() 1

Jo I (()\(d() <- B,

and consequently

A(O) <

2

d

< 1A(01 n B(a, r)).

(B.1.4)

The same inequality holds, of course, if .(O1) = 0. Taking into account (B.1.3) and (B.1.4) one gets (B.1.2).

Proof of Theorem B.1.1- Let x, y E 0, x y and x+y E 0. Write p = Ix - yl. From Lemma B.1.2 there exists xo E 0 such that

Appendix B

314

xo E B X+Y, 2) and I(xo) < 2d+1B, (one simply takes a = '+Y , r = 2). Let dn, n E {-1,0} U N, be a decreasing sequence such that d_1 = 2p and p(dn) = 2p (,1-(dn

}

do+1)) , n E {0} U N.

Such a sequence does exist by the continuity of the function p, since

p(0) < 2 p

(2) , p(a) > 2 p (22a)

From Lemma B.1.3 there exists a sequence {xn}, n E N U {0}, such

that xn E B (X, d 2

1n 0, I(xn)

d Bl

d, n E N,

(B.1.5)

2

< 2I(xn-1)

I.f(xn-1) - .f(xn)I p(Ixn-1 - xni)

(B.1.6)

E

) 2

(one takes a = x, b = xn-1i n E N). In a similar way one can costruct a sequence {yn}, n E N U {0}, such that yo = xo and

yf E B (y, d

2

dBl d, n E N. ( 2 )

1) n 0, I(Yn) <

Since do - 0, xn --+ x and yn -> y. From the inequalities (B.1.5) and (B.1.6) i,(if(xo)-f(xl)I)

2d+1 B2d+1

<

icdo Kdd

p(I xo - x1I )

4d+1

1

B K2dd d

0d

and consequently

(if(xn) - f(xn-1)I) < 2d+1I(xn-1) p(I xn

xn-1I)

IGdn-1

< 4d+1 dd

B '

n-1 dd n-2

and

if (Xn) - .f (xn-1)I

0-1

(4d+1

dB

dd-

-z ) p(I xn - xn-1I )

An estimate on modulus of continuity

315

Continuity of f implies that I f(x) - f(xo)I <- En '=1 -1

(+i

If(xn) - f(xn-1)1

Il. 2d n-1d) n-2p(I xn - xn-11) =

To estimate Il we need the following elementary lemma.

Lemma B.1.4 For n E N p(I xn - xn-1I) < 4 (p(dn-1) - p(dn))

Proof - From (B.1.5) (Ixn-xn-11) < p(Ixn-1 -x1+Ixn-x1) < p (12(dn-2 + do-1) , n E N, and from the definition of {dn} p

((d_2 + do-1))

= 2p(dn-1).

It is therefore enough to show that 2p(dn) < p(dn_1).

However,

p(dn) = 2p (2-(d. +dn-1)) < 2p(dn-1) and the result follows.

11

From Lemma B.1.4

Il < 4 n-1

0-1

B

4d+1

(p(dn-1 - p(dn)))

and therefore

Il <

4

r

J

'1

< 4 J 21x-y1

B

4d+1 2d2d

n-1

V-1 C4 d+1 B C2u2d

p(du)

) p(du).

Appendix B

316

Since

I f(x) - f(y)I < If (x) - f(xo)I + If (xO) - f(y)I the result follows.

Applying the above theorem to the functions

O(u) = U" p(u) = up/a a > 0, 3 > 0, one gets the following corollary.

Theorem B.1.5 Assume that ic(O) > 0. For arbitrary a > 0 and 0 > 2d there exists a constant c > 0 such that for an arbitrary continuous function f : D --- E one has If (X)

I f() - f(77)1' - f(y)I a < CI X - yla-2d 1010

Irll,3

(B.1.7)

Proof - We can assume that B

= JnJ0

If (

f(71)Ia

+oo.

I - nj"

From Theorem B.1.1 If (x) -

f(y)I S

8

f

B11a

21x-y1 4d

-ua_ du.

2d

au. a

K1

However, if 0 > 2d we have

J

2Ix-yI

ua

This completes the proof.

adu=lx - yl

0-2d CV

Appendix C

A result on implicit functions c.1 We are given two Banach spaces A and E and a mapping

F:Ax E - E, (A,x)-+.F(A,x). We assume that the following holds.

Hypothesis C.1 (i) F is continuous. (ii) There exists a E ]0,1 [ such that IIF(A, xi) - .F(A, x2)II E S allxl - x21IE, x1, x2 E E.

The following result is an immediate consequence of the contraction principle.

Proposition C.1.1 Under Hypothesis C.1 there exists a unique continuous mapping cp E C(A; E) such that F(A,,p(A)) = SP(A), A E A.

(C.1.1)

To study the differentiability of the mapping cP we need the following hypothesis. 317

Appendix C

318

Hypothesis C.2 (i) For all x E E, F(., x) is differentiable and the derivativeDA.F is continuous.

(ii) The mapping R -+ E, h -- .F(A,xl + hx2), is continuously differentiable for all X1, X2 E E, A E A. Moreover there exists a linear

operator DxF(A, xl + hx2) in L(E) such that

hF(A,x1 + hx2) = Dx.F(A,x1 + hx2) x2. We can now prove the following result.

Proposition C.1.2 Under Hypotheses C.1 and C.2, So is differentiable at any point \ E A in any direction µ E A, and Dcp(.A) µ = [I - D,,F(\, v(A))]-1 Da.F(A, 4o(A)) . p. .

(C.1.2)

Proof - We note that by C.1(ii) it follows I I Dx.F(A, x)II < a, x E E, A E A.

(C.1.3)

For A,p. EA and hE ]R, we have

9 (A + ha) - SP(A) = F(A + hy, c(A + hli)) - T(A, 4'(A))

= f0

1

Da.F(A + hµ, (A + hp)) - httd

1

+ Jo D..F(\, &(A + hµ) + (1 - f),p(A)) - (,p(A + hp) - P(A))d6. Setting G(A, h,

f0Dx.F(X, 1 (A + hfi) + (1 -

we have JIG(A,h,Y) 11,C(E) < a < 1, and so 1

p(A+hµ)-SP(A) = [I - G(A, h, µ)] -110 Da.F(A+ehu, o(.\+hµ)).hlude, and the conclusion follows by dividing by h and letting h tend to 0. To obtain the existence of the second derivative of cp(A) we need an additional assumption. We assume

A result on implicit functions

319

Hypothesis C.3 (i) There exists another Banach space G continuously and densely included in E such that the restriction of .F to A x G fulfils (C.1.1) with E replaced by G.

If (i) holds, we have, again by the Contraction Principle, that Dacp(A) µ E G for all A, µ E A. (ii) For all x E X, x) is twice differentiable and the derivative D2.F is continuous. (iii) The mapping R --+ E, h -+ DA.F(A,xl + hx2) µ, is continuously differentiable for all x1, x2 E E, A, µ E A. Moreover there exists a linear operator DSDA.F(A, xl + hx2) µ in L(E) such that dh DA.F(A,xl + hx2)

.

Dx(DA.F(A,xi + hx2) x2 µ) x2.

(iv) The mapping R -+ E, h -+ F(A, yj + hy2) µ, is twice continuously differentiable for all yi, y2 E G, A, µ E A. Moreover there exists a linear operator D2.F(A, yj + hy2) in L(E; L(E)) such that d2

dh2.F(A, yj + hy2) = D2Y(A, yj + hy2)(yi, y2) A E A, y1, y2 E G.

We can now prove the following result.

Proposition C.1.3 Under Hypotheses C.1, C.2, C.3, co is twice differentiable at any point A E A in any directions it, e E A, and we have

D2V(A)(µ, e) = [I - D..F(A, SP(A)))-1

{DaF(A,'p(A))(µ, ) + 2D.\

+D'F(A,

(C.1.4)

i, Dcp(A) . )}.

Proof - For any A, µ, E A and any h E R we have Dcp(A +

DW(A) . µ = J1(h, A) + J2(h, A) v(A)) [D4p(A + hE) . µ - DV(A) . µl ,

where

Ji(h, A) = DN.F(A +

p(A + hC)) . it - DAY(A, SP(A)) . µ,

320

Appendix C

J2(h, A) =

It follows that D ,F(A,
Now the conclusion follows by dividing by h and by letting h tend to 0.

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Index Genuinely dissipative systems,

Angle variable, 9 Approximately controllable, 141

114

Gibbs measure, 223 Global interactions, 223 Gradient systems, 160

Birkhoff's ergodic theorem, 6 Boundary noise, 239 Burgers equation, 255 existence of invariant measure, 266 strong Feller property, 263 uniqueness of invariant measure, 274

Invariant measure, 12 dissipative case, existence, 105, 117

absolutely continuous, 148, 158, 159, 162 density, 147 ergodic, 22 existence, 191 existence from boundedness,

Configuration, 223

Damped wave equation, 182 Dirichlet form, 19, 165 Dissipative mapping, 73 Doob's theorem, 43 Dynamical system, 3 angle variable, 9 canonical, 16 continuous, 4 ergodic, 4 infinitesimal generator, 5 invariant set, 9 strongly mixing, 5 weakly mixing, 4

89

existence in the linear case, 97

exponentially mixing, 40 recurrence, 102 regularity of density, 165, 168, 170

strongly mixing, 33 uniqueness, 191 weakly mixing, 33 Kolmogorov equation, 71, 135 mild, 136 Koopman-von Neumann theorem, 5

Factorization formula, 58 Feller semigroups, 19 Fortet-Mourier norm, 39 338

Index

339

Krylov-Bogoliubov theorem, 21

Local interactions, 223 Markovian semigroup, 12 Feller, 20 irreducible, 42 recurrent, 37 regular, 41 stochastically continuous, 12 strongly Feller, 42 symmetric, 18 Markovian transition function, 12

Navier-Stokes equation, 279 existence of invariant measure, 293 Null controllability, 130 Ornstein-Uhlenbeck process, 56,

dependence on initial conditions, 69 dissipative, 80, 87 linear, 96 mild solution, 66 second order dissipative, 181 wave type, 178 weak solution, 56 with delay, 197

Strong law of large numbers, 30

Strongly dissipative mapping, 73

Transition function, 12 Transition semigroup, 12, 56, 70, 97, 105, 122, 148 irreducible, 137, 141 strong Feller, 121, 129, 134, 153, 200

177

in chaotic environment, 186 in Finance, 184 wave type, 178

Random environment, 53 Reaction-diffusion equations, 210 Sobolev spaces, 149

compact embedding, 152 Spectral measure, 54 Spin systems, 223 classical, 225 quantum lattice, 233 Stationary measure, 12 Stochastic convolution, 56 smoothing properties, 305 Stochastic evolution equation, 65

Wiener process, 53 coloured, 53 cylindrical, 53