Lecture Notes in Physics Edited by H. Araki, Kyoto, J. Ehlers, MSnchen, K. Hepp, ZSrich R. Kippenhahn, MLinchen, H.A. Weidenm~iller, Heidelberg J. Wess, Karlsruhe and J. Zittartz, K61n Managing Editor: W. Beiglb6ck
281 Ph. Blanchard Ph. Combe W. Zheng
Mathematical and Physical Aspects of Stochastic Mechanics
Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo
Authors
Ph. Blanchard Theoretische Physik and BiBoS, Universit&tBielefeld D-4800 Bielefeld, FRG Ph. Combe BiBoS, D-4800 Bielefeld, FRG and Universit6 Aix-Marseille II and Centre de PhysiqueTh6orique C.N.R.S. - Luminy-Case 907 F-13288 Marseille, France W. Zheng BiBoS, D-4800 Bielefeld, FRG and East Normal University, Shanghai, China
ISBN 3-540-18036-2 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-38?-18036-2 Springer-Verlag NewYork Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. ,Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of / une 24, 1985, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1987 Printed in Germany Printing: Druckhaus Beltz, Hemsbach/Bergstr. Bookbinding: J. Sch&ffer GmbH & Co. KG., GrL~nstadt 2153/3140-543210
P R E F A C E
T hese
are
the r e v i s e d
Bielefeld present
Centre
D.
DHrr,
R.
H~egh-Krohn,
Meyer,
M. Fukushima,
H.
Rodriguez, O.
Zhanghi
for
Special
Rost,
Steinmann, helpful
thanks
at B i e l e f e l d yon Reder)
H.
E.
(Mrs.
has been from
We wish Carlen,
W.
basic
the a c t i v e
thanked
J.
mech-
collabora-
to the book.
of the
for s h a r i n g
reorganized
interaction
Golin, R.
G.F. F.
the
J.
to
A. Hilbert,
M. Mebkhout,
P.A.
Potthoff,
M. Sirugue,
J.C.
and
a number
Dell'Antonio,
Guerra,
Marra,
M. Serva, Stubbe,
with
our i n d e b t e d n e s s
Cohendet,
Nencka-Ficek,
Schneider,
is to
prerequisites)
of s t o c h a s t i c
thoroughly
close
S.
at
of two of the lecturers.
R. Jost, H.
L. Streit,
Zambrini
M. S i r u g u e and M.
and comments.
go to s e c r e t a r i e s , Jahns,
given
The p u r p o s e
substantially
to r e c o r d O.
Gandolfo,
Nelson,
advice
for e x p e r t
aspects
and we e n j o y e d
the i n v i t a t i o n
G. J o n a - L a s i n i o ,
Nagasawa,
Collin,
D.
1985.
inevitable
be e s p e c i a l l y
benefited
G. Bolz,- E,
course
"Bielefeld-Bochum-Stochastics"
should
and c o l l e a g u e s .
S. Albeverio,
the
and p h y s i c a l lively
of the l e c t u r e s
and we have
of f r i e n d s
from
who c o n t r i b u t e d
and for c o o r d i n a t i n g
The m a t e r i a l
of a g r a d u a t e
"Sommersemester"
BiBos
Volkswagenwerk
rewritten
R.
very
participants,
The R e s e a r c h Stiftung
was
Notes
(apart
of the m a t h e m a t i c a l The c o u r s e
tion of all
costs
in the
a self-contained
version anics.
University
Lecture
Mrs.
in p a r t i c u l a r
Jegerlehner,
Mrs.
to the s e c r e t a r i e s
Litchewski
and Mrs.
typing.
Bielefeld,
April
1987
C O N T E N T S
I.
INTRODUCTION
I.I
I.la
I.Ib
1.2
II.
Some Probabilistic Aspects in Classical and Quantum Physics ..............................................
I
Brownian Motion. Mechanics of Particles Submitted to Random Disturbation . .................................
I
From Feynman Formulations
4
Path Integral to Probabilistic of Quantum Physics ......................
Probabilistic Interpretation of Quantum Mechanics and Probability Theory ...................................
7
1.2a
Historical
Remarks
...................................
7
1.2b
The
Approach
..................................
10
1.2c
Probabilistic
1.3
Jacobi
1.3a
The
Hamilton-Jacobi-Fluid
1.3b
The
Madelung
KINEMATICS
Wigner
and
OF
Description
Madelung
Fluid
STOCHASTIC
of
Fluid
Commuting
Observables
...
12
............................
15
............................
16
...................................
DIFFUSION
19
PROCESSES
II.1
Brownian
II.2
Stochastic
II.3
Diffusion
II.4
Kinematics
II.5
The
II.5a
Brownian Motion with Lebesgue Measure as Initial Distribution .........................................
44
II.5b
Time-Reversed
46
II.6
Stochastic
Acceleration
II.7
Some
Examples
II.7a
The
Motion
......................................
Integration Process of
Basic Wiener
....................................
Diffusion
Time-Reversed
Processes
Diffusion
Diffusion
Process
...............................
Process
Process
22 31 34
....................
37
..................
44
......................
..............................
47
..................................
51
...................................
51
VI
II. 7 b
The
Brownian
II. 7c
The
Bessel
II. 7 d
The
Ornstein-Uhlenbeck
III.
NELSON
. . . . . ,. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Process
STOCHASTIC
Process
DYNAMICS
Newton
Law
III.2
Conservative
Newtonian
III.3
Mechanics
Conservative
III.4
Conservative Distribution
53
.......................
- NEWTONIAN
Stochastic
of
51
...................................
III.1
55
PROCESSES
................................ Diffusion
Processes
Newtonian
Process
60
...........
61
..........
62
Newtonian Processes with Stationary .........................................
67
Unattainability of the Nodes for Stationary Diffusion Processes ............................ ................
69
IIi°6
Diffusion
in
74
III.7
Newtonian
Diffusion
III.5
IV.
V.
Bridge
GLOBAL
EXISTENCE
an
External
FOR
on
Electromagnetic Riemannian
DIFFUSIONS
WITH
Field
Manifold
SINGULAR
.......
...........
75
DRIFTS
IV.I
Introduction
...........................................
82
IV.2
Existence
Nelson's
84
IV.2a
Heuristics
IV.2b
Unattainability
IV.3
Application
to
IV.4
Alternative
Methods
STOCHASTIC
of
Diffusion
Processes
............
84
........................................... of
the
Nodes
Stochastic
VARIATIONAL
to
and
Global
Mechanics
Construct
Existence
....
..................
Singular
Diffusions
85 92
.
94
PRINCIPLES
V.0
Introduction
V.I
The
V.2
Strongly
V.3
The
V.4
Construction of Diffusion Processes by a Forward Stochastic Variational Principle .....................
105
Other
110
V.5
Classes
Yasue
......................................... S(P)
Convex Action
Approaches
and
S(P,F)
Functionals
..........................
97
..........................
100
.....................................
to
97
Stochastic
Calculus
of
Variations.
101
VII
VI.
TWO
VIEWPOINTS
CONCERNING
AND
STOCHASTIC
MECHANICS
VI. ]
General
VI.2
Interference
VI.3
Observables
VI. 3a
Observables
VI.3b
Momentum
VI.3c
Repeated Measurements: A Case Against Stochastic Mechanics? ...........................................
119
VI.4
Indeterminacy
123
VI.5
Locality
VI.6
Scattering
VI.7
Spinning
VI.8
Pauli-Principle
VI.9
......................................
......................................... -
Measurement
............................
........................................... Process
.....................................
Relations
....................
~ .........
.............................................
The
VI.11
Bose
......................................
130
Quantum
VII.2
Trapping
VII.2a
A Model of Protosolar
VII.3
A
Limit
133
............................
135
Theory
AT
STOCHASTIC
Remarks
MECHANICS
......................................
Phenomena
and
Formation
of
Spatial
Patterns
the Formation of Jet-Streams in the Nebula ....................................
Covering Allen for
of
the
137 . 140
140
Planets
........................
142
Belts
........................
143
Radiation Transport
131
..............................
Field
LOOK
Model
126
129
Semiclassical
Van
117
....................................
General
The
116
Particle
VII.I
VII.2c
116
127
NON-QUANTAL
Cloud
115
....................................
A
VII.2b
114
Theory
• The Connection Between Stochastic Mechanics and Euclidean Quantum Mechanics ..........................
VI.10
VII.
Remarks
QUANTUM
in
a
Plasma
....................
145
APPENDIX
At.
Notations
A2.
Conditioning
A3.
Stochastic
A4.
Martingales
and
Conventions
............................
......................................... processes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
..........................................
147 148 149 150
VIII
A5.
Weak
A6.
Stochastic
A7.
Definition Variation
BIBLIOGRAPHY
Convergence Ito
and
Measures
Integrals
on
Metric
Spaces
......
.............................
and Characteristics of Quadratic ...........................................
..................................................
151 153
156
157
I. I N T R O D U C T I O N
1.1 Some P r o b a b i l i s t i c Aspects
in C l a s s i c a l and Q u a n t h m Physics
In physics p r o b a b i l i s t i c ideas and concepts o c c u r r e d for the first time in c o n n e c t i o n w i t h the statistical a p p r o a c h to thermodynamics,
the
s o - c a l l e d k i n e t i c theory of gases, during the second part of the last century and in the p u b l i c a t i o n of A l b e r t E i n s t e i n ' s paper on B r o w n i a n m o t i o n in 1905
[42]. E i n s t e i n ' s theory not only p r o v i d e d a d e c i s i v e
b r e a k t h r o u g h in the u n d e r s t a n d i n g of the p h e n o m e n a of B r o w n i a n motion; in the opinion of Max Born it also did "more than any other w o r k to convince p h y s i c i s t s of the reality of atoms and molecules, theory of heat,
of the kinetic
and of the f u n d a m e n t a l rSle of p r o b a b i l i t y
in the natu-
ral laws" I.la B r o w n i a n Motion. Mechanics of Particles S u b m i t t e d to Random Disturbation The m o t i v a t i o n tion was
for E i n s t e i n ' s w o r k on the theory of B r o w n i a n mo-
"to find facts w h i c h w o u l d guarantee as m u c h as possible the
e x i s t e n c e of atoms of d e f i n i t e finite size"
[95]. The e x i s t e n c e of atoms
and m o l e c u l e s was p o s t u l a t e d in the kinetic theory of gases some decades before by C. Maxwell,
R. Clausius and L. Boltzmann.
In his " A u t o b i o g r a p h i c a l Notes"
[95],Einstein indicated the rela-
tion of this w o r k to the state of physics at the b e g i n n i n g of this century:
"I d i s c o v e r e d that,
according to a t o m i s t i c theory,
there w o u l d
have to be a m o v e m e n t of s u s p e n d e d m i c r o s c o p i c p a r t i c l e s open to observation, w i t h o u t k n o w i n g that o b s e r v a t i o n c o n c e r n i n g the B r o w n i a n were already long familiar".
Indeed,
scribed by R. Brown in 1827
this p h y s i c a l p h e n o m e n o n was first de-
[19].
E i n s t e i n p r o p o s e d an e x p e r i m e n t b a s e d on a t h e o r e t i c a l model describing the e r r a t i c m o t i o n of a very small spherical p a r t i c l e t h r o u g h a viscous m e d i u m and s u b m i t t e d to the influence of thermal m o l e c u l e s of the bath,
for w h i c h he assumed a m o l e c u l a r structure as in the kine-
tic theory. Since the frequency of £he c o l l i s i o n is very high the v e l o c i t y of the test particle
changes very often and it is there-
fore i m p o s s i b l e to m e a s u r e the speed of p a r t i c l e s motion.
(IO21s-I),
submitted to B r o w n i a n
T a k i n g into account that because of their size the v e l o c i t y of
the B r o w n i a n p a r t i c l e s m u s t be m u c h smaller than those of the molecules, he c o n c l u d e d that the m e a n p o s i t i o n of the B r o w n i a n p a r t i c l e s is zero.
Hence E i n s t e i n p r o v e d that statistical fluctuations can produce sufficiently important effects to induce an erratic motion, w h i c h can be observed under a good microscope.
F r o m his model, E i n s t e i n c o n c l u d e d that
after a s u f f i c i e n t l y long time the r a n d o m m o t i o n of the spherical particles d o e s
generate a migration. Moreover,
menon is e s s e n t i a l l y a diffusion.
he p r o v e d that this pheno-
The basis of his d e s c r i p t i o n is the
notion that the suspended p a r t i c l e s are "diffusing" through the liquid in such a w a y that the dynamical e q u i l i b r i u m is m a i n t a i n e d between the osmotic force and the viscous force. For the p r o b a b i l i t y density of a B r o w n i a n p a r t i c l e to be in
x
at time
D(x,t)
t , E i n s t e i n obtains the
equation
~--~
where
~
v A p
(1.1)
is a p o s i t i v e constant called the d i f f u s i o n coefficient,
which
depends on the nature of the particle and of the p r o p e r t i e s of the liquid
(viscosity, temperature,
d i s p l a c e m e n t at time
t
...). This implies that the m e a n square
is given b y
<x2> t = 29t
(1.2)
assuming that the p a r t i c l e starts from the origin at time
t =0 .
This result is strongly r e m i n i s c e n t of the random w a l k process: the root mean square distance t r a v e l l e d is p r o p o r t i o n a l to Moreover,
V~ .
E i n s t e i n gives an explicit formula for the d i f f u s i o n
coefficient kT
= m-~ where
mB
is the r e s i s t a n c e due to the friction,
constant and
T
R
<x2>t
the B o l t z m a n n
then yields a k n o w l e d g e of
is the constant of perfect gas, and by
A v o g a d r o ' s number
k
the absolute temperature.
The m e a s u r e m e n t of where
(1.3)
(1.3)
N O . The first good d e t e r m i n a t i o n of
by J. Perrin in 1909 along this m e t h o d
R
k = ~--
one obt~in~ NO
was m a d e
[92].
The study of B r o w n i a n motion as a stochastic process was undertaken by N. Wiener in 1923 work
[110], p r e c e d e d by L. B a c h e l i e r ' s h e u r i s t i c
[10], and soon was d e v e l o p e d into its m o d e r n form by Paul Levy
and his followers, A. K o l m o g o r o v K.L. Chung
[25], M. Kac
[77], K. Ito [69], J.L. Doob
[75], P.A. Meyer
[41];
[85a,b]. T o g e t h e r with the
Poisson
process,it
r a n d o m processes, Einstein's motion, same
constitutes both
luchowsk±
attempted
the f r i c t i o n
should
acquire
pared w i t h
c o e f f i c i e n t , then
the
ian motion,
time
this
suggests
w here tion
Wt
mechanics)
Smo-
K
the p a r t i c l e (large com-
due to the Brown-
stochastic
= ~ dt + / ~
equation
dW t
process with
of S m o l u c h o w s k i ' s
by
will be a p p r o x i m a t e l y
fluctuations
the d i f f e r e n t i a l
point
on the
and d e n o t i n g
a long time
B -I ) the v e l o c i t y
is the s t a n d a r d W i e n e r
is the s t a r t i n g
is constant
K/8 . A f t e r
the r a n d o m
dx(t)
[101]
line of thought.
theory.
(in classical
limit v e l o c i t y
now into account
of
theory of B r o w n i a n
Smoluchowski
different
acting on a p a r t i c l e
the r e l a x a t i o n
K/8 . T a k i n g
a completely
a more d y n a m i c a l K
a dynamical
in a p a p e r by M.V.
but following
"species"
and applications.
theory does not p r o v i d e
If the force m8
in theory
this one a p p e a r e d
subject
one of the two f u n d a m e n t a l
(1.4)
covariance4
theory
. This
(expressed
rela-
in m o d e r n
terminology). This scribe
approach
of e x t e r n a l ing forces particle This scribe
to B r o w n i a n
the d y n a m i c s forces
suggests
moving
and on the other
due to the m o l e c u l e s
hand
submitted
that make
p o i n t of v i e w was p r o p o s e d
the erratic m o t i o n
mf
cies with the w e a k almost
to v i s c o u s
forces
and r a n d o m d2x
force
is a r a p i d l y
varying
the sense
same intensity). of the W i e n e r
two worlds:
to de-
the
Hence Langevin getting
world
described
(this ter-
all the
frequen-
is the d e r i v a t i v e
in
of B r o w n i a n m o t i o n
equation
represented
m
(1.5)
(1.5)
being
connects
by the drag
by the f l u c t u a t i n g
what we call today
Apply-
writes
"white noise" contains
(the paths
Langevin's
the m a c r o s c o p i c
wrote
called
to de-
of mass
dx v = ~ dt
This w h i t e noise
process
world
to a p a r t i c l e
fluctuations,he
its s p e c t r u m
non-differentiable).
and the m i c r o s c o p i c
equation,
force,
[78]
in a r a n d o m environment.
= mBv+mf,
is due to the fact that
surely
therefore
in w h i c h
by P. L a n g e v i n
mechanics
dt 2
minology
a model
to very rapidly vary-
up the m e d i u m
in 1911
of a p a r t i c l e
law of N e w t o n i a n
m
where
naturally
on one hand u n d e r the influence
is immersed.
ing the f u n d a m e n t a l submitted
motion
of a p a r t i c l e
a stochastic
force.
differential
dx(t) = v(t)dt
(I .6) dv(t)
= -Sv(t)dt + dW(t).
This d e s c r i p t i o n of Brownian motion has been a c h i e v e d by L.S. O r n s t e i n and G.E. U h l e n b e c k in the thirties
[106],
[108]. In some sense, this was
the first step toward a m e c h a n i c s of systems s u b m i t t e d to some random forces. Later developments led to stochastic m e c h a n i c s introduced in 1966 by E. Nelson
[90a] in connection w i t h q u a n t u m mechanics. N e l s o n
e x p l a i n e d the dynamical s i g n i f i c a t i o n of formal equations d i s c o v e r e d by I. F~nyes
[45] using a stochastic version of N e w t o n ' s equation,
in fact,
in this framework one is n a t u r a l l y led to the S c h r ~ d i n g e r equation, starting from a d e s c r i p t i o n of m i c r o p r o ~ e s s e s by means of diffusions.
I.Ib From Feynman Path Integral to P r o b a b i l i s t i c
Formulations of
Quantum Physics The founding of q u a n t u m m e c h a n i c s can be located b e t w e e n 1927. The m e a n i n g of the wave function p r o p o s e d by M. Born also
[71])
is that if a q u a n t u m m e c h a n i c a l
wave function
~
then
l~(x,t) 12
of finding the system in
x
1923 and
[18]
(see
system is d e s c r i b e d by the
represents the p r o b a b i l i t y density
at time
t . Despite the similarity between
S c h r ~ d i n g e r ' s wave m e c h a n i c s and d i f f u s i o n theory,
as we will see in
§I2, it was clear since the very b e g i n n i n g of q u a n t u m m e c h a n i c s
[74]
that w i t h i n q u a n t u m theory a new kind of p r o b a b i l i t y was involved. In 1948, R.P. F e y n m a n
[47,48] p r o p o s e d a global f o r m u l a t i o n of
q u a n t u m m e c h a n i c s based on p r o b a b i l i s t i c ideas,
in w h i c h the p r o p a g a t o r s
a s s o c i a t e d to time e v o l u t i o n are e x p r e s s e d as a functional integral,
the
so-called "Feynman path integral". Let us sketch F e y n m a n ' s idea briefly in the case of a p a r t i c l e under the influence of a p o t e n t i a l
V(x) . In classical m e c h a n i c s the
position of the p a r t i c l e is known at each time, and the p a r t i c l e describes a w e l l - d e f i n e d t r a j e c t o r y starting at time xa
and ending at time
tb
on the point
introduce a p r o b a b i l i t y amplitude
ta
from the point
x b . in q u a n t u m m e c h a n i c s we
K(xa, ta,xb,tb).
Feynman's idea was to
write this amplitude as the sum of the c o n t r i b u t i o n of all paths leaving xa
at time
ta
and ending in
xb
at time
K(Xa, ta; xb,t b) =
tb
X (t_) = x
T (t~) =x b
("sum over histories")
~(~).
(1.7)
Here is the m a i n d i f f e r e n c e to classical mechanics: p a r t i c l e there is only one possibility,
for the classical
the "classical path",
for w h i c h
the action functional tb S(y)
= I
L(y(t) ,~(t) ,t)dt
(1.8)
ta is stationary, Feynman's as
L
being the L a g r a n g i a n of the system.
"Ansatz"
y(t a) = x a
and
consists in assuming that all paths y(t b) = x b
such
give a c o n t r i b u t i o n w i t h the same
amplitude but w i t h d i f f e r e n t phases. function along the path,
y
This phase is given by the action
i.e.,
~
S (¥)
¢(y)
where
~
= Ae
(1.9)
is the P l a n c k constant divided by 2 # .
In F e y n m a n ' s
f o r m u l a t i o n of q u a n t u m m e c h a n i c s we can t h e r e f o r e
formally w r i t e i K ( x a , t a , X b , t b) = N I
e~S(Y)
~(y)
(1.10)
Y (ta) =x a Y (t b )=x b where
N
should be a suitable n o r m a l i z a t i o n
and
D(y)
is the "measure",
on the m a n i f o l d of paths, w h i c h can be formally w r i t t e n as a product of L e b e s g u e ' s measure,
e.g.
O(y)
The t r a n s i t i o n f u n c t i o n
=
R dy(t) t6[ta,t b ]
K
(1.11)
itself is not a p r o b a b i l i t y but the square
of its m o d u l u s defines a p r o b a b i l i t y d e n s i t y
P(Xa, ta,xb,tb)
=
IK(Xa,ta;xb,tb)i 2
One of the i n t e r e s t i n g aspects of F e y n m a n ' s
(1.12)
f o r m a l i s m is the fact
that it gives a c o m p l e t e l y i n d e p e n d e n t and s e l f - c o n t a i n e d f o r m u l a t i o n of q u a n t u m m e c h a n i c s and allows,
at least in principle,
direct e x t e n s i o n
of q u a n t u m field theory. A n o t h e r a p p e a l i n g feature of this f o r m a l i s m is its strong c o n n e c t i o n w i t h the L a g r a n g i a n f o r m u l a t i o n of classical mechanics,
allowing a mathematical
limit
~ ~ O, which,
control on the a p p r o a c h to the classical
from the o r i g i n a l ideas of Dirac
should be d e t e r m i n e d by the path w h i c h makes
S(y)
[38] and F e y n m a n ,
stationary,
i.e.
according to Hamilton's Unfortunately,
principle,
of the classical motion.
at this level the Feynman path integral
well-defined mathematical However,
the trajectory
object,
in particular
N(y)
under the influence of the work of Feynman,
is not a
is not a measure. M. Kac was able
to prove that the solution of the heat equation can be written
as an
integral with respect to the Wiener measure over the space of paths [75]. During the last four decades many works have been devoted to the mathematical
definition of this oscillatory
ilistic interpretation processes.
four classes.
say that the approaches which give a mathema-
the m a t h e m a t i c a l integral space)
can be roughly classified
The approach via a limiting procedure,
is useful
of t h e limit
[37,70b,76].
(Fresnel integral on an infinitely
and a method of stationary phase on Hilbert
adapted to investigate
asymptotical
The Euclidean
is obtained by analytical
strategy
exPansions
aginary time and in this way establishes ger's equation scription measure
and the heat equation,
Euclidean
As mentioned,
Schr~dinger
to study
The oscillatory dimensional
space are
around
Hilbert
well-
~ = 0
[3 a, 7b]
continuations
to im-
a connection between Schr~din-
furnishing
in the framework of functional
[57,100a].
in
which is the most
for explicit evaluation but it is difficult
properties
approach
and to its probab-
as expectation value with respect to stochastic
Let us briefly
tical meaning to Feynman's path integral popular,
integral
a probabilistic
integration w.r.t.
de-
Wiener
Kac has shown that the solution of the
equation = ~2
~ t f(t,x)
Af(t ,x) - V(x) f(t,x)
~
(I .13) f(O,x) can be written
(Feynman-Kac f(t,x)
where measure
~W
= ~(x) formula)
I = IEW [e -~ 5°t v (Y (r) +x) dT~ (Y (O) +x) ]i
denotes the expectation w.r.t.
can be expressed
Wiener measure.
Formally,
this
as -
1 .2
~y
dW (y) = e
(r)dT
~(y).
This formula has been used to obtain a definition tegral by analytical
(I .14)
continuation
(I .15) of Feynman's path in-
and has played an important rSle in
t a c k l i n g q u a n t u m field t h e o r y and the infinities of r e n o r m a l i z a t i o n [57]. Unfortunately,
this p r o b a b i l i s t i c
to q u a n t u m m e c h a n i c s
a p p r o a c h cannot be d i r e c t l y e x t e n d e d
in "real" time since the W i e n e r m e a s u r e w i t h a c o m -
plex c o v a r i a n c e is not a m e a s u r e
[20]. N e v e r t h e l e s s ,
g e n e r a l i z e d B r o w n i a n functionals
(e.g. d i s t r i b u t i o n s on W i e n e r space
of
[86]) a d e f i n i t i o n of F e y n m a n ' s p a t h integral,
as an a p p l i c a t i o n o f
in real time, has
been p r o p o s e d and d i s c u s s e d by L. Streit and T. Hida, and r e f e r e n c e s therein. and A.M.
Chebotarev
[24,28,84]
integral on phase space
see
[66]~and [102]
A n o t h e r a p p r o a c h was i n i t i a t e d by V.P. M a s l o v and can be g e n e r a l i z e d to F e y n m a n path
[27]. This p r o b a b i l i s t i c f r a m e w o r k differs from
the p r e v i o u s one in the sense that it works d i r e c t l y with real time and that the i n v o l v e d stochastic p r o c e s s e s are jump processes. If a s u c c e s s f u l t h e o r e t i c a l framework has been found,
it is in-
t e r e s t i n g to try to r e f o r m u l a t e its structure in d i f f e r e n t forms in order to isolate some aspects and perhaps to find new implications and developments.
The p u r p o s e of these lectures is to present p h y s i c a l and mathe-
m a t i c a l aspects of stochastic mechanics. ginning,
As we have s u g g e s t e d at the be-
the domain of p h y s i c a l a p p l i c a t i o n s of stochastic m e c h a n i c s
not r e s t r i c t e d to q u a n t u m mechanics. notions of s t o c h a s t i c m e c h a n i c s ,
is
But before i n t r o d u c i n g the basic
let us first discuss the rSle of prob-
ability theory on q u a n t u m physics.
1.2
P r o b a b i l i s t i c I n t e r p r e t a t i o n of Q u a n t u m M e c h a n i c s and P r o b a b i l i t y Theory
1.2a Historical Remarks F r o m the b e g i n n i n g of q u a n t u m m e c h a n i c s the search for a stochastic i n t e r p r e t a t i o n was m o t i v a t e d by the c o n s p i c u o u s similarity b e t w e e n the free S c h r ~ d i n g e r e q u a t i o n and the heat equation.
One of the first to
draw a t t e n t i o n to such similarity was E. S c h r 6 d i n g e r himself in 1931 [98b,c]. He c o m p a r e d the free e q u a t i o n for the q u a n t u m m e c h a n i c a l wave function in one d i m e n s i o n w i t h the d i f f u s i o n e q u a t i o n = ~t where
p(t,x)
efficient. at time
t
showed that
v ~2 ~x 2
(I .15)
is the density of the p a r t i c l e and
9
the d i f f u s i o n co-
D i s c u s s i n g the p r o b l e m of finding the p r o b a b i l i t y d i s t r i b u t i o n with
t 6 [to,t I]
p(t,x)
if
P(to,X)
and
P(tl,x)
is given as a p r o d u c t of two factors
striking analogy to the q u a n t u m m e c h a n i c a l e x p r e s s i o n the q u a n t u m p r o b a b i l i t y density.
But the d i f f e r e n c e s
~
are known, he plP2 , in =
(pl,p 2
I~l 2 for are posi-
tive a n d
@
is complex valued; moreover,
ability density amplitude
~
p
in d i f f u s i o n theory the prob-
itself and in q u a n t u m theory only the p r o b a b i l i t y
are solutions of a p a r t i c a l d i f f e r e n t i a l equation)
were,
from S c h r 6 d i n g e r ' s point of view, i m p o r t a n t enough to convince him not to try to adopt a stochastic i n t e r p r e t a t i o n of q u a n t u m mechanics. a d i f f e r e n t class of d i f f u s i o n processes, J. C. Zambrini
Using
the "Bernstein processes",
[114] has recently shown that the r e a l i z a t i o n of the pro-
gram initiated by S c h r S d i n g e r gives the genuine E u c l i d e a n v e r s i o n of stochastic mechanics. Not only the S c h r S d i n g e r e q u a t i o n w h i c h can be w r i t t e n for free particles
in one d i m e n s i o n ~ ~t
=
~2~
e
=
-
i
~X 2 '
~
(1.16)
2m
has a stochastic analogue, namely the d i f f u s i o n e q u a t i o n =
~ 32p 3x 2 '
~t
~ 6~+
but,as d e m o n s t r a t e d in 1933 by F~rth
(1.17)
[51b], see also
[71], an analogy
also exists for the H e i s e n b e r g u n c e r t a i n t y relations, w h i c h are often r e g a r d e d as the c h a r a c t e r i s t i c
feature of q u a n t u m mechanics.
In this d i s c u s s i o n of the stochastic analog of H e i s e n b e r g ' s relation, F~rth was the first to derive the u n c e r t a i n l y r e l a t i o n as follows. Let
xo
be the initial p o s i t i o n of a particle and
city, then its position at time x = x
The mean square
+ vt
o
~ = <x 2>
t
.
~ =
S~x21~(x,t)"
is therefore
II2dx and
using
one obtains after partial integrations i
=
2
dt 2 and d3~ -
dt 3
O
its initial velo-
(I .18)
= <x2> + 2<x v>t +
t 2 o o From
v
will be given by
(I .19)
(I .16) and its complex conjugate,
from which ficient
follows
that
of
~
t2
is a q u a d r a t i c
function
_
the o b v i o u s
it f o l l o w s
2
berg
setting
Ax = ~
uncertainty
the
_>o
that
>_ I
same
the d i f f u s i o n
and
~
-e 2
Ap = m V ~
, FOrth
obtains
the H e i s e n -
relation
h • Ap ~ 47
Ax
Along
(I .20)
(1.20) <x2>
Thus,
2
immediately 2
by
coef-
inegality
2 = I II~I12
and h e n c e
the
= 2 dg2
x I~-~ ~ +~'t
and
t ,
being
<2> From
of
-
lines
F~rth
process
by
-
defined
(1.21)
.
the u n c e r t a i n t y
6 = <x 2> = ~
of p o s i t i o n
for
x 2 p(t,x)dx
IR where
p
that
llpI[1 = I This
is a p o s i t i v e
solution
dB ~ = 2~
implies
in this
linearly given
i.e. glven
case,
in time,
as b e f o r e
of r a n d o m
(1.17)
such
o
and
+ 2~t
in c o n t r a s t
since
with
collisions
the a m o u n t
equation
I
= x
Hence
of the d i f f u s i o n
.
to
the m o t i o n
(1.19),
of e a c h p a r t i c l e ,
an i n i t i a l
velocity,
with
particles.
other
of p a r t i c l e s
through
by j =-
~ V p
the u n c e r t a i n t y
results
instead
now
from
The d i f f u s i o n
a fixed
unit
increases
area
of b e i n g
the a c t i o n
current
per
unit
j , time
10
was used by FHrth to define an osmotic v e l o c i t ~
u
I
u = -- j = - ~ V p = - ~) Vlog p . 0 P Using p,2 --~- dx
= f u 2 p ( t , x ) d x = ~ 2 f
(I .22)
and the obvious inequality p' (-~
F~rth obtains,
{Iplll
since
x 2 g) _> o
+
.
= 1 [
J
p,2
p
I dx >
-
B
IR
and hence by
(1.22) <x2> 2 ~ 2
Now, with
Ax
and
Au
defined as before, FHrth derives the following
d i f f u s i o n u n c e r t a i n t y relation
Ax • Au ~ v
in analogy to However,
(1.23)
(1.21). as he pointed out,
in contrast to
lower bound is given by the universal constant
(1.21), where the ~
originating
from
the d i s t u r b a n c e produced by the m e a s u r e m e n t process itself ~64], the lower bound in
(1.23) associated with the r a n d o m agitation of the sur-
rounding m e d i u m can be a r b i t r a r i l y small,
for example by lowering the
temperature since
T
~
is p r o p o r t i o n a l to
[42].
1.2b The Wigner Approach The search for an i n t e r p r e t a t i o n of q u a n t u m m e c h a n i c s
in the
framework of a classical p r o b a b i l i s t i c theory found support and encouragement between the thirties and the fourties thanks to a result obtained by E. W i g n e r
[111], w h i c h seemed to carry more weight than mere
analogy considerations. lating q u a n t u m m e c h a n i c s troduced a function
In fact,
it suggested the p o s s i b i l i t y of formu-
in terms of phase space ensembles.
W(q,p)
W i g n e r in-
of p o s i t i o n and m o m e n t u m variables,
con-
11
taining all information function
about the quantum state. For a pure state this
integrated over m o m e n t u m
probability
distribution
space, yields the quantum mechanical
of position,
namely
l~(q) 12 , and when inte-
grated over the position
space yields the corresponding
distribution
namely
of momenta,
transform of
~ . Moreover,
if
f(q,p)
i.e. a function on phase space, the observable
Q(f)
Wigner d i s t r i b u t i o n
W
~
~
probability
is the Fourier
is any classical
the quantum mechanical
associated t o
f
observable,
expectation
of
i n the state defined by the
is given by
w For pure state
l~(p) 12 , where
= ~ f(q,p)W(q,p) IR 2n
the Wigner distribution
W~
dqdp
.
(1.24)
is given by the explicit
formula
W~(q,p) and admits the marginal [
-
I (~)n
I
dx ~ ( q + x ) e IRn
2iP'.x M ~(q-x)
(1.25)
distributions
W~(q,p)dp
= l~(q)
f (q,p)dq l(p)12 .W
2
(I .26)
where I - (2~)n/2
~(p)
Hence the Wigner distribution space.
However,
although
W
probability
i p'x ~ ~ (x) dx
.
~n
is real,
it is not positive
of a statistical
and consequent-
distribution
in
It is natural to ask whether the fact that the joint
distribution
More precisly,
e
appears as a kind of density on the phase
ly has not quite the interpretation classical physics.
] f
for coordinates
two types of questions
(i) Is it possible
to describe
and momenta can be relaxed.
can be investigated. a quantum mechanical
state in
terms of an average over stochastic processes? (ii) Is it possible to reformulate probabilistic
framework
quantum mechanics
so that observables
in a purely
are represented
by random
variables? During the last five decades these questions attention of numerous m a t h e m a t i c i a n s question candidate
as well as physicists.
is closely connected with the mathematical
man's path integral.
Indeed,
the Wigner description
to give a functional
have attracted the The first
definition provides
integral representation
of the
of Feyn-
a good quantum
12
m e c h a n i c a l state in the sense that the time evolution of the Wigner f u n c t i o n can be described using stochastic p r o c e s s e s w i t h the product of the phase space and the torus
values in
[27]. This approach gives
a p r o b a b i l i s t i c d e s c r i p t i o n on an extended space. F r o m a m a t h e m a t i c a l point of v i e w the p r o p e r t y that the Wigner d i s t r i b u t i o n is given as the difference of two positive distributions
implies the existence of
stochastic processes valued in the product of the phase space and the torus
(or disc)[85c]~
The second question,
relative to the p o s s i b i l i t y
of r e p r e s e n t i n g q u a n t u m observables by random processes, was studied first by E. Moyal 1.2c P r o b a b i l i s t i c
[88]. D e s c r i p t i o n of C o m m u t i n g O b s e r v a b l e s
To be more precise, mechanics.
let us recall some basic facts about q u a n t u m
For more details see e.g.
[91]. The main m a t h e m a t i c a l struc-
ture introduced in q u a n t u m m e c h a n i c s is the s u p e r p o s i t i o n of states and algebraic operations on observables. system there exists a Hilbert space = {ei84, O ~ 8 < 2~}, a s e l f - a d j o i n t operator need an axiom stating
Aop
such as there is a unit ray
114111= 1 , c o r r e s p o n d i n g to each state Aop
that
As an extreme case we have: set of
H
To each quantum m e c h a n i c a l
c o r r e s p o n d i n g to each o b s e r v a b l e there are sufficiently many ~
~ H
and
A . We
and
exhausts all unit rays on
4
A
op and the
contains all p r o j e c t i o n operators. The e v o l u t i o n in time
of the system is d e s c r i b e d by a one p a r a m e t e r family of unitary operators
Ut
on
H
and can be achieved in two ways.
In the Schr6dinger
picture the state of the system evolves in time according to
4 (t) = where
40
U t 4o
is the initial state at time
O
and the observables do not
change w i t h time. It must be remarked that the Wigner d e s c r i p t i o n of the state gives rise to a Schr6dinger picture and the time evolution of the Wigner function obeys a Schr~dinger type equation.
In the Hei-
senberg picture the observables evolve in time according to
Aop(t)
= utl Aop U t
and the state does not change with time. If the system is in the state there exist
@
p r o j e c t i o n - v a l u e d measures
Aop
:f
]R
and if
A
{E l }
on
I dE l
is an observable, H
such as
13
Then
<4,El~>
determine result
is the p r o b a b i l i t y
the v a l u e
smaller
than or equal
~
is the e x p e c t a t i o n
value
The o b s e r v a b l e is an e i g e n v e c t o r
that,
of the o b s e r v a b l e
A
of
A
to
l
is the way
A
has
the value
in the state
associated
may be r e g a r d e d B
= 19
Hence given
as a r a n d o m v a r i a b l e
the usual Borel
field,
l
a
4
(1.27)
•
with probability
w{th
prohabilistic
theory.
to
4, we o b t a i n
since
of
oP
in w h i c h
quantum mechanical
an e x p e r i m e n t
in the state
= <~'A°P 9> = I I d <4,E19> IR
Aop9 This
if we p e r f o r m A
the e i g e n v a l u e
one,
if
rSle
in
l
• ideas play a central the state
4
(~,B,pA(dl))
the o b s e r v a b l e with
A
~ = IR,
and P9A = <9 'dEll>
Similarly, garded
any number
of c o m m u t i n g
as r a n d o m v a r i a b l e s
all o b s e r v a b l e s a theorem
on a a p p r o p r i a t e
of a q u a n t u m
of von N e u m a n n
state
cannot be r e g a r d e d
ility
space.
(1.3)
Let
operators
probability
system are c o m m u t i n g
that
the set of all o b s e r v a b l e s
we have
(Nelson-von
can be re-
space.
But not
and it follows
as a family of r a n d o m v a r i a b l e s
More precisely,
Theorem
self-adjoint
from
in a g i v e n on a probab-
the following
Neumann
[9Ob,91])
AI,...,A be n s e l f - a d j o i n t o p e r a t o r s on a Hilbert n H , such that for all x 6 IR n the o p e r a t o r x.A defined
space by
n ~xA. i=l l l
x.A=
is e s s e n t i a l l y there
exists
self-adjoint. 4
6 H
with
to find r a n d o m v a r i a b l e s (d,B,P)
with
~l,...,~n
the p r o p e r t y
pg(x.~>l)
where
that
either
the
Ai
commute
or
such that it is not p o s s i b l e on a p r o b a b i l i t y
for all
x 6 IRn
space
and all
i 6 IR
= <~,EI(x.A)9>
n Z x i ~i and {E l(x.A) } are the p r o j e c t i o n - v a l u e d i=l a s s o c i a t e d with the closure of x.A .
x-~ =
measures
Then, 11911 = I
14 Remark Translating this result into the setting of quantum mechanics one is led to say that
n
observables may be regarded as random variables
in all states if and only if they commute. Proof We don't distinguish notationwise between Suppose that for each unit vector of random variables,
and let
~ Z~
in
H
x.A
and its closure.
there is such an n-uple
be the probability distribution
of ~ 6 IRn. That is, for each Borel set
B
Let us compute the characteristic
in
IRn,
~(B)
= P~(~ 6 B) .
function of the measure
Z~
+¢O .
I
eiX'~d~* (~) : f
IRn
elldP~ (x" ~> I)
-oo
= ~+~ell<~,dEl(x.A)~> --co
= <~,elX'A~>.
Thus, the measure the polarization plex measure verifies
~
is the Fourier transform of
identity,
Z~
if
~
and
~
are in
<~,eiX'A~> . By H
there is a com-
which is the Fourier transform of
Z~9 = Z~ . For any Borel set
there exists a unique operator
D(B)
<~,eiX'A~ >
and
B c IR n by the Riesz theorem such that
<~,~(B)~>
= Z~(B).
Thus we have [
eiX~<~0,d~(~)~> = <£0,eiX'A~>
]IRn the operator sequently,
D(B)
is positive slnce
Z~
is a positive measure.
if we have a finite set of elements
ing points
x
3
~j 6 H
6 ~ n , then
i(xj-xk)'A l <@k ' e j,k
~J> [
=
X
i(xj-xk)'~ e
j,kJmn
=
if we define
[
<~(~),d~(~)~(~)>
,d~(~)
<~k
~j>
-> 0
Con-
and correspond-
15
ix..~ ~(~)
=
~
e
3
~.(~)
j Furthermore, Under
these
implies
1
conditions,
that there
representation orthogonal
=
eiO. A
3
and
e
i (-x) .A
the t h e o r e m
is a Hilbert
x ÷ U(x)
projection
of
of
on unitary
space
IRn
K
on
onto
IRn
and all
Since
e ix'A
is already
so that
ll~u(x)
so that
~U(x)~
U(x)
commute, Remark
= U(x)~
= e ix'A
tary r e p r e s e n t a t i o n
containing
H
K
such that,
if
of N a g y
[93]
and a unitary IP
is the
= eiX'A~
~ 6 H unitary,
llU(x)~ll =
Consequently,
dilation
K
H, then
~U(x)~ for all
(eiX.A) *
=
lleiX'A11
~It =
the
U(x)
~ 6 H .
of the c o m m u t a t i v e
and c o n s e q u e n t l y
It,ll
IIu(x)~l! •
and each
for all
=
A. 3
maps
Since
group
H
into itself,
x ÷ U(x)
TR n , the
e
is a uniix.A all
commute=
1 Nelson-von
having
Neumann's
theorem
a joint p r o b a b i l i t y
operators
but asserts
does
not c l a i m that there
distribution
only that
for n o n - c o m m u t i n g
such st0tes
are no state
self-adjoint
are exceptional.
Remark 2 This
result means
interpretation,
that if q u a n t u m m e c h a n i c s
it is not a p r o b a b i l i t y
matical
sense.
cerning
the i n t e r p r e t a t i o n
foundations, recent l y
space,
[67],
which
stochastic
In this
process
is a c a n d i d a t e and Fermi
natural way
of q u a n t u m m e c h a n i c s
[89d,e].
stochastic
tion of Bose
in a rather
a non-commutative
see
commutative
To solve
theory
framework
a Feynman
mathecon-
and its m a t h e m a t i c a l has been d e v e l o p e d
one c o n s i d e r s a
process)
field theory,
a probabilistic
som~ old problems
calculus
(or q u a n t u m
to obtain
quantum
has
in the classical
path
ncn-
with values integral
in Fock
descrip-
(see also P. G a r b a c z e n s k i
[54]).
1.3
Jacobi While
mechanics the
and M a d e l u n ~ Schr~dinger's
found
similarity
describing
Fluid attempt
its support
of the e q u a t i o n
hydrodynamical
to give
primarily
flows
an i n t e r p r e t a t i o n
in the analogy
for the wave formed
function
the basis
of q u a n t u m
of wave phenomena, with
the e q u a t i o n s
for another
early
at-
16
tempt
to a c c o u n t
the c l a s s i c a l
for q u a n t u m
physic
an h y d r o d y n a m i c a l
the g e n e r a l
short
review
of q u a n t u m
fluid.
this
The
Since
structures
fluid. F o r
Hamilton-Jacobi
We c o n s i d e r
a classical
and w i t h
phase
a point
in p h a s e
space
and by
If
H(x,y,t)
of time.
then
canonical
the
of
E~ M a d e l u n g
gave
mechanics
approach
details
very
will
[83]
reminiscent
be u s e f u l
mechanics,
description
more
in the f r a m e w o r k
of
to u n d e r -
let us g i v e
of c l a s s i c a l
a
mechanics
see [60a].
Fluid
freedom
function
In 1926
of s t o c h a s t i c
of the Hamilton-- J a c o b i
and of the M a d e l u n g
1.3a
phenomena
media.
description
the H a m i l t o n - J a c o b i stand
mechanical
of c o n t i n u o u s
space
Hamilton
dynamical
system
IR n x IR n.
We d e n o t e
(q(t),
p(t))
is the
equation
with
n
a path
(smooth)
of m o t i o n
degrees
by
(x,y) in p h a s e
Hamilton
space
=
~(t)
= -(VxH) (q(t),p(t) ,t) .
as
function,
can be w r i t t e n
~(t)
of
E IR n x IRn
as
(VyH) (q(t) ,p(t) ,t) (1.28)
Given
the
initial
values
at time
q(to) the i n t e g r a t i o n
of
q(t) In c l a s s i c a l
(1.28)
time
by
T(t,x,y)
=
(t',x',y')
that
the H a m i l t o n
Using
(1.28)
is a s o l u t i o n
and t h e i r
of H a m i l t o n ' s
(1.30)
an i m p o r t a n t
rSle.
Let
with
function
transform
space
÷ IR x IRn x IRn
x' = x, y'
H(x',y',t')
(1.29)
'
= P(Xo,Yo,t)
plays
IR x IR n x IRn
= Yo
in p h a s e
p(t)
reversal
t' = -t, and a s s u m e
' P(to)
the p a t h
= q(Xo,Yo,t),
mechanics
,
o
= Xo
gives
T:
be d e f i n e d
t
= -y
is i n v a r i a n t
under
time
reversal
= H(x,y,t)
under
equation,
T then
it f o l l o w s
that,
if
(1.30)
17
q' (t') = q(t) , p' (t') = -p(t)
is a solution of the e q u a t i o n of m o £ i o n
(t'
p l a y i n g now the rSle of
time) but w i t h the f o l l o w i n g initial c o n d i t i o n
q'(t&)
= x 0, p'(t&)
= -Yo
Now, let us i n t r o d u c e the H a m i l t o n - J a c o b i e q u a t i o n ~S ~-~ + H(x,VS,t)
= 0
(1.31)
s u p p l e m e n t e d by the initial c o n d i t i o n
S(x,t)
If some S s a t i s f y i n g
= S
o
x 6 IRn
(x)
(1.31) and (1.32) can be found, then it is w e l l - k n o w n
that it is p o s s i b l e to obtain a s o l u t i o n of Hamilton's only s p e c i f y i n g the initial p o s i t i o n view each g i v e n
S
(1.32)
q(to)
equations, by
= x ° . F r o m this point of
defines a w h o l e family of paths on phase space,
each of w h i c h c h a r a c t e r i z e d by the initial p o s i t i o n alone, as we shall d e m o n s t r a t e now. Starting w i t h a g i v e n
S
let us introduce the m o m e n t u m field
p
by
p(x,t)
and the v e l o c i t y field
v
= VS(x,t)
(1.33)
by
v(x,t)
=
(VyH) (x, p(x,t) ,t)
(I .34)
consider now the first order d i f f e r e n t i a l system { q(t)
= v(q(t),t) (I .35)
q (tO) = x o and suppose that
q(t)
is solution of
(1.35). Setting
p(t) = p(q(t) ,t) = VS(q(t) ,t) we see easily that
(q,(t),p(t))
w h i c h the H a m i l t o n equations
defines a path in phase space,
are satisfied.
(I .36) along
18
Instead of considering each single t r a j e c t o r y specified by q(t o) = x °
for some given
S , we can also consider a continuous dis-
t r i b u t i o n of paths, a s s o c i a t e d w i t h some density
p(x,t)
in configur-
ation space. In fact, ~(x,y,t)
starting from the L i o u v i l l e e q u a t i o n for the density
in phase space
~ w + {~,H} = 0 ~t where
{-,.}
(1.37)
denotes the Poisson brackets,
{a,b} = Vxa. Vyb - Vya. Vxb
i.e.
(I .38)
.
We can make the following Ansatz
(x,y,t) = p(x,t)~ (y-?S(x,t))
wich constrains the m o m e n t u m to verify easy to see that
~
satisfies
(1.39)
(1.33) for all times.
(1.37), if the density
p
It is now
is solution
of the continuity equation
~-£ ~t where
v
is defined by
+ V(pv) = 0
(1.40)
(1.34).
We define the H a m i l t o n - J a c o b i
fluid as a m e c h a n i c a l
a c o n f i g u r a t i o n space, d e s c r i b e d by two fields, function
S(x,t)
and the density
way that the H a m i l t o n i a n - J a c o b i
p(x,t)
system living
the H a m i l t o n - J a c o b i
and e v o l v i n g in time in such a
equation
~_~S + H(x,VS,t) ~t
= O
(1.41)
is satisfied. It must be r e m a r k e d that, vice-versa, ing
if we have a fluid verify-
(1.41), then the d e s c r i b e d procedure will provide a particle inter-
pretation. In the simplest case, the Hamilton function is of the form 2 H (x,y) = ~ Lm
+ V(x)
(1.42)
19
and the H a m i l t o n equations are
~(t) Therefore,
_ - p(t) m
,
the H a m i l t o n - J a c o b i ~S -~
+
1
~(t)
(1.43)
fluid is d e s c r i b e d by the system
(vs) 2
~-~
= - VV(q(t))
+
v(x)
=
o (1.44)
~0
~-5
+
V(pv)
V(x,t)
= 0
= 1
VS(x,t)
m
Let us e m p h a s i z e that the p o s s i b i l i t y of giving a p a r t i c l e picture for h y d r o d y n a m i c a l equations, paths!
as the previous one
based on d e t e r m i n i s t i c
comes from the fact that the e q u a t i o n for
S
is d e c o u p l e d from
the e q u a t i o n for the density
p . In fact, the equation for
simply a c o n t i n u i t y equation,
e x p r e s s i n g the local c o n s e r v a t i o n of
mass, while the e q u a t i o n for
S, w h i c h is of first order in time, al-
lows the i n t r o d u c t i o n of particle characteristic
p
is
trajectories i d e n t i f i e d w i t h its
lines, as w e l l - k n o w n from the general theory of first
order partial d i f f e r e n t i a l equation,
see e.g.
[9]
1.3b The M a d e l u n g Fluid AS r e m a r k e d by M a d e l u n g
[83] just at the b e g i n n i n g of wave mech-
anics,
it is also p o s s i b l e to r e f o r m u l a t e the standard S c h r ~ d i n g e r y2 H = y£ + v(x) e q u a t i o n a s s o c i a t e d w i t h the H a m i l t o n f u n c t i o n (1,42) w h i c h is namely iM ~ = ~t with
~ 6 L2(IRn,dx)
and
Let us separate modulus
A
the L a p l a c e - o p e r a t o r
R
and
S
(1.45) in
IRn .
and phase in the wave f u n c t i o n ~(x,t)
where
M2 -2-m A~ + V~
= eR(X't~
S(x't)
(1.46)
are r e a l - v a l u e d functions.
Substituting expression
(1.46)
for the S c h r ~ d i n g e r equation and separ-
ating real and imaginary parts, we obtain for the imaginary part ~R ~t
I 2m
(AS + 2VR • VS).
(1.47)
20 Introducing
the vector field
v(x,t)
and the probability
(1.47)
m
(1.48)
~
I~(x,t)12 = e 2R(x't)
(1.49)
density p(x,t)
expression
= ! VS(x,t)
can be put in the form of a continuity equation
~-£ + V ( ~ v )
~t
= 0
(1.50)
On the other hand, we obtain for the real part
The two equations Schr~dinger
~t3S _
~2 ~[AR+
(1.50)
and
equation.
(VR) 2- ( ~ ) 2 ] - V .
(1.51)
(1.51)
are strictly equivalent
to the
Using n o w V(e R) = (VR) e R Ae R
(1,51)
= [ A R + (?R) 2]e R
,
can be put in the form ~S + ~-[
1
2
~
(vs)
M2 + v
AeR
2m
R
0 =
(1.52) "
e We call the hydrodynamica ! system described by the Madelung striking,
{1.48),
(1.50)
fluid. The analogy with the Hamilt0n-JacObi
but now no immediate particle
This is due to the mysterious
interpretation
and
(1.52)
fluid is
is available.
nature of the "quantum mechanical
potential" ~2 2m
VQM which depends
on the density
In the next section, natural
the underlying
(1.52)
(e R = ~ )
particle
interpretation
of the Madelung
but only by allowing a random character
to
paths.
The classical zero in
(1.53)
following E. Nelson, we will show that a
and straightforward
fluid is indeed possible,
Ae R R e
approximation
consists
in which the randomness
in setting
disappears
M
equal to
and the trajectories
21
become t h o s e of the classical theory,
the M a d e l u n g fluid reducing to
the H a m i l t o n - J a c o b i fluid. We emphasize the fact that we have spoken about a p a r t i c l e picture and not claimed that particles and trajectories really exist in the p h y s i c a l realm. Remark In the classical a p p r o x i m a t i o n , a n optical analogy is even more suggestive than this h y d r o d y n a m i c a l analogy, solutions. VS ,
especially for stationary
Since the v e l o c i t i e s of the particles are p r o p o r t i o n a l to
the trajectories of these particles are orthogonal to the surfaces
of equal phase
S = const.
In the language of optics,
the latter are
the wave fronts and the trajectories of the particles are the rays. Hence,
the classical a p p r o x i m a t i o n is e q u i v a l e n t to the geometrical
optics approximation.
I I. KINEMATICS OF S T O C H A S T I C D I F F U S I O N PROCESSES
II.1Brownian
Motion
Let us c o n s i d e r a system such that its state at each time is completely specified by the p o s i t i o n of a point space
~d
~
in the c o n f i g u r a t i o n
and assume a d e t e r m i n i s t i c e v o l u t i o n e q u a t i o n
(t) = b(~(t) ,t)
where
b(x,t)
(2.1)
is some a s s i g n e d vector field in
~d
w h i c h may depend
on e x t e r n a l force fields acting on the system. Under a p p r o p r i a t e regularity conditions on
b
and for given initial c o n d i t i o n
(to) = ~O
(2.2)
(2.1) has, at least locally , a unique solution. An example of such determ i n i s t i c evolution, tions
of the previous form,
is given by the H a m i l t o n equa-
(1.28). In m a n y case there is no c o m p l e t e control on the external forces
or the e q u a t i o n of m o t i o n
(2.1)
is derived from an a p p r o x i m a t i o n in
w h i c h some degrees of freedom have b e e n neglected,
(e.g. the m o t i o n of
a small particle in a fluid where we neglect the interaction with the m o l e c u l e s of the fluid). A wide class of p h y s i c a l phenomena,
ranging
from statistical physics to a s t r o p h y s i c s and control theory,
shows that
very often a suitable m o d i f i c a t i o n of the dynamics gives good results, p r o v i d e d we take into account all effects coming from external fields and n e g l e c t e d degrees of freedom, by introducing some p h e n o m e n o l o g i c a l l y d e £ e r m i n e d r a n d o m d i s t u r b a n c e acting on the system during the evolution. To m a k e the previous d e s c r i p t i o n m o r e precise, the m o t i o n of a particle in
~3
let us consider
s u b m i t t e d to rapidly v a r y i n g forces
due to the collisions with m o l e c u l e s of the surrounding fluid, and let us m a k e the a s s u m p t i o n that the external g r a v i t a t i o n a l field is negligible.
The particle moves because it is c o n s t a n t l y b u f f e t e d by the mo-
lecules.
Taking some reasonable assumptions about the b e h a v i o u r of the
collisions,
one can deduce the p r o b a b i l i t y of finding the particles in
any given subset of x £ ~3
~3
at initial time
at time
t , k n o w i n g that it was located at
t = O . That is just w h a t was done by A. Ein-
23 stein
in h i s
section
description
be obtained
first
coordinate
unit
of time,
gain
the
model.
is t h e t i m e b e t w e e n
random
proportional
two
in
variable to
~-
shorten
Let
T
we have
per
and we
second,
the unit
be
in
suppose
now that
= - oV~]
on a
motion.
shorten
the
so as t o in o t h e r w o r d s ,
Thus
t/T
steps
are
s t e p is a s y m m e t r i c a l
qV~- , f o r e a c h
= P[~k
of c o u r -
of B r o w n i a n
we will
of l e n g t h
. Each
moves
can
random walk which
collisions.
t
process
its p r o j e c t i o n
the n e w t i m e - u n i t ,
time
, namely
of
the properties
successive
(old)
P [ ~ k = GVT]
the
step
step
Ber-
is of m a g n i t u d e
k
I
=
then IE [~k]
=
Var(~k)
where
seen
motion
in m o t i o n
a one dimensional
are received
but we must
by the particle
now
case possesses
impacts
correct
The particle
space but we can think
We describe
limiting
numerous
noulli
as w e h a v e
an idea on how the Brownian
dimensional
axis.
in a c e r t a i n
T
give
from a random walks.
se in a t h r e e
made
motion
(I.1.a).
We will
Since
of t h e B r o w n i a n
~
denotes
Let X O = 0
then
the the
0
= 2I (~V~)2
= ~E [~2]
expectation position
÷ ~I
(-~VT) 2 = o2 T
value.
at t i m e
t
(or a f t e r
t/T
steps)
is
just
tit Xt = If
T
is m u c h
Z k=1
smaller
as a n i n t e g e r .
~k
"
than
t
, t/T
is l a r g e
and may be thought
of
Hence we have
Ix t] = 0 Var
Furthermore, central Hence ± o~the
if
limit
if
is f i x e d
theorem
Xt
in o u r m o d e l ,
with
limiting
process).
t
t 2 [X t] = ? o T = o 2 t
equal
will
in w h i c h
probability
process
and
.
T ÷ O have the
the
normal
random
in t i m e
is t h e B r o w n i a n
, then by the De Moivre-Laplace
T
distribution
particle
jumps
, we perform
motion
Wt
the
N(O,t)
over
a distance
limit
T ÷ O,
(also c & l l e d W i e n e r
24
In this m o d e l we have used the fact that the d i s p l a c e m e n t is the sum of m a n y v e r y small independent contributions; good approximation.
Indeed,
this assumption is a
if r e l a t i v e l y h i g h v i s c o s i t y is assumed,
so that the v e l o c i t y of the particle is v e r y quickly damped, the disp l a c e m e n t s in n o n - o v e r l a p p i n g By s y m m e t r y
cal conditions have that
intervals of time should be independent.
(homogeneity and isotropy) (temperature, pressure,
~ [(~t+s - ~t )2] = f(s)
f
is continuous
etc.)
and if the physi-
remain constant, we should
is i n d e p e n d e n t of
tion w i t h the independence implies a s s u m p t i o n that
~ [~t ] = O
t . This condi-
f(s I + s 2) = f(s I) + leads to
f(s) = cs
(s 2)
. The
and the varian-
ce of increments is linear in time. If we denote by
dt
a strictly positive
interval of time
(small
c o m p a r e d to the time needed by the p a r t i c l e to cover a m a c r o s c o p i c distance and large with respect to the interaction time noting by
Wt
the c o r r e s p o n d i n g limiting process,
T ), and de-
in three dimensions
we have
E[dW t] = 0 ,
~ [dW~ dW ] = d2~iJdt
.
(2.3)
The t r a n s i t i o n p r o b a b i l i t y of a G a u s s i a n random v a r i a b l e is completely specified by its m e a n and covariance. us the t r a n s i t i o n function
p(x,t,y,s)
Therefore,
(2.3) gives
for the d - d i m e n s i o n a l B r o w n i a n
motion 2
p(x,t,y,s)
for any finite time
=
p
b a b i l i t y that the particle s
e
(2.4)
t < s
The p h y s i c a l m e a n i n g of
time
I (2z 2(s_t)d/2
ly-x 2~ ~ (s-t)
the Borel set
is that
p(x,t,y,s)dy
represents the pro-
(performing B r o w n i a n motion) will reach at
dy
around
y
if it started from
x
at time
t < s . Clearly, we have the initial c o n d i t i o n
lira p(x,t,y,s) s+t and
p
satisfies the parabolic e q u a t i o n 2 ~_£= ~s -~- Ayp
o
(2.5)
= ~ (x-y)
(Heat equation) (2.6)
25
The c o h e r e n c e of the p h y s i c a l i n t e r p r e t a t i o n of
is due to the nor-
p
malization condition
~IR p (x,t,y,s)dy 3
=
(2.7
I
and p(x,t,y,s)
= p(O,O,y-x,s-t)
(2.8
= D(y-x,s-t)
This p r o p e r t y is the c o n s e q u e n c e of the space-time t r a n s l a t i o n invarlance of our model. The C h a p m a n - K o l m o g o r o v equations
p(x,t,y,s)
for any
t'
such that
(compatibility conditions)
= [ p(x,t,x,,t')p(x',t',y,s)dx' J ]Rd
(2.9)
t < t' < s , takes care of the fact that at
each time the process d e v e l o p s w i t h o u t r e m e m b e r i n g the p o s i t i o n s occupied at an earlier time; this is the M a r k o v property. ficance of
(2.9) is obvious.
is located in
x
all the p o s s i b l e
at time
The physical
t
was in
intermediate state
y
at time
s
is the sum over
x'
at time
t'
, t < t' < s
the p r o d u c t of the p r o b a b i l i t y that the p a r t i c l e moves f r o m y'
at
move from
t
to
x'
Now, time
some
i n t e r m e d i a t e time
at time
signi-
The p r o b a b i l i t y that the p a r t i c l e w h i c h
t'
x
of
at time
and of the p r o b a b i l i t y to
t'
to
y
at time
s
if we denote by
Wt
the r a n d o m p o s i t i o n of the p a r t i c l e s at
t , then the k n o w l e d g e of the t r a n s i t i o n p r o b a b i l i t y
(2.4) allows
us to give m e a n i n g to all c o r r e l a t i o n functions of the process p r o v i d e d the initial d i s t r i b u t i o n
p(x,t o)
, at some initial time
to
for
Wt
o
is k n o w n
]E [F(Wo'
"''' wt )] = I F n
(Xo,
..., X n ) P ( X o , t o , X j , t I) (2.10)
......... P(Xn_ 1 , t n _ ] , X n , t n) P ( X o , t o ) d X o for
.....
dx n
t o < t I < t 2 ... < t n
In particular,
for the d e n s i t y at time
p(x,t) so that
p(x,t)
t
we have
= I P(Xo'to'X't)p(Xo'to)dXo
satisfies also the d i f f u s i o n e q u a t i o n
(2.11)
26 ~_~
= ~2
Dt
(2.12)
-2- ~x p
but w i t h initial c o n d i t i o n given by
lim t+t
p(x,t) = p(x,to) o
A l t h o u g h the t r a n s i t i o n p r o b a b i l i t y the same e q u a t i o n
(2.6)
and
the physical m e a n i n g of given through particular,
(2.73)
p
and the density
and
p . The link b e t w e e n
(2.11) and the initial c o n d i t i o n
determine the density
p
satisfy
(2.12) there is a deep d i f f e r e n c e b e t w e e n
p
the k n o w l e d g e
p
p
(2.5) and
and
p
is
(2.~3). In
of the t r a n s i t i o n p r o b a b i l i t y
p
does not
w h i c h in fact depends on some specified ini-
tial condition. If we are i n t e r e s t e d in the b e h a v i o u r of the r a n d o m trajectories t ~ Wt
of the B r o w n i a n particle, we need to k n o w m o r e than the transi-
tion function
(and c o r r e l a t i o n functions). We have to prove the exis-
tence of a p r o b a b i l i t y m e a s u r e on the space of trajectories c o n s i s t e n t w i t h the B r o w n i a n t r a n s i t i o n function in the following sense: finite c o l l e c t i o n of m e a s u r a b l e 0 < t I < t 2 ... < t n
;(~tl
= I
sets
A I ,... ,A n
in
]Rd
for any
and any
we have
£ A]
dxl
.....
....
AI
I A
Wtn 6 A n ) dx n P ( X , O , X l , t l ) p ( x l , t l , x 2 , t 2)
......
n
....... P(Xn_ 1 ,tn_ I ,Xn,tn) where we have assumed that the path starts at the point
(2.14) x
at time
zero. This c o n s t r u c t i o n was first a c h i e v e d by N. W i e n e r in 1923 and the p r o b a b i l i t y measure, (started at point
PW,x
[111]
' is the so-called Wiener m e a s u r e
x )
Let us indicate briefly the main steps of the c o n s t r u c t i o n of the p r o b a b i l i t y space. Let
T = [O,t]
be a finite interval of time and c o n s i d e r
the one point c o m p a c t i f i c a t i o n of is the space of all
~d
functions from T
~d
. The space of t r a j e c t o r i e s into
~d
,
27
= {~[T ~ ~ d } = [jRd ]T
By the T y c h o n o v Let by
C (~)
Cc(O)
belongs from
theorem,
it is a c o m p a c t
be the s p a c e of c o n t i n u o u s
the s u b s e t of c y l i n d r i c a l
to
= f
, and seperate
for the u n i f o r m
a positive
form
Ix
= I dxl
~(tk)),
functions
the p o i n t s
on
~
A function
topology. and d e n o t e
F:O ÷ f
(2.15)
V~ 6
is a stth-*-algebra of the alge-
of
~ , thus
Cc(~)
is d e n s e
in
topology.
U s i n g the t r a n s i t i o n
Ix(F)
functions
functions.
(e(tl) . . . . . .
The set of c y l i n d r i c a l C(~)
C(~)
in the p r o d u c t
C (~) if t h e r e e x i s t s k > O , a continuous function k c , ~ and O < t I < t 2 ... < t k < t such that
[~d]
F(~)
bra
space
on
probability
Cc(O)
(2.4)
and
(2.9), we can d e f i n e
by
"'" I dxk P ( x 1 ' t 1 ' x 2 ' t 2 ) " ' "
P ( X k ' t k ' x ' t ) f ( x l ..... Xk) (2.16)
where
f
is a f u n c t i o n
ded linear
functional
associated
on
with
I
x
n o r m as
c a n be e x t e n d e d I x ) on
by the Riesz Pw,x(d~)
C(~)
sup ~6~
(2.15).
It is a b o u n -
IF(e) ], F 6 Cc(~)
to a b o u n d e d
linear
, by the t h e o r e m
representation
theorem
x
form ~ (with the same x of S t o n e - W e i e r s t r a s s . F i n a l l y , defines
a probability
measure
on
~(~) = I Hence,
along
Cc(Q)
fix(F) L _< [IFit = then
F
the r o u g h
Pw,x (d~lF(~)
statement
(2.3)
takes
(2.17)
a precise mathematical
con-
tent. Let conditional
~
and
~ [o[-]
expectation;
denote
respectively
the e x p e c t a t i o n
and
we h a v e
]E [dWtIWt]
=
0 (2.18)
]E [dW t dWJ,wt]
= g2~iJdt
28
The e x p e c t a t i o n Higher
order
being
taken with
conditional
moments
We have e s t a b l i s h e d it lives
of
~d -valued
ly and a p r o b a b i l i t y is n a t u r a l have
constructed
tinuous the
PW,x
real-valued
ly that B r o w n i a n the support ries,
x = 0
on
is of m o r e
have
PW = PW,O
suggested
fami-
Hence
its support?
it We
on the set of all con-
(we denotes
functions was
one
PW,x
such that
paths are continuous.
on this
on this collection.
[O,t]
~(0)
= x
, in
)" The limita-
since w e b e l i e v e
intuitive-
If one reason to i n v e s t i g a t e
is tO find the a n a l y t i c a l
another
o-algebra
it is a m e a s u r e ~
but
one needs a "nice"
a "large"
concentrated
so that
f o l l o w i n g we assume
tion to c o n t i n u o u s
of the p r o b a b i l i t y m e a s u r e
does the m e a s u r e
functions
PW,x
o(dt)
In application,
functions,
density
to ask: w h e r e
to the W i e n e r m e a s u r e
are of o r d e r
the e x i s t e n c e
in a rather big space.
collection
respect
properties
technical
of B r o w n i a n
nature but very
trajecto-
important.
Sup-
pose that we set
M t(~)
We are i n t e r e s t e d probability time
t
tion. able
in
sup IW T(~)I T6[O,t]
Mt(e)
because
that the path will
, which
is that there
it is the s u p r e m u m
and in e l e m e n t a r y
measure
set in
functions.
the p r o b a b i l i t y
measure
that
assures
= 0
Now, what
that
for
functions
~ before
Mt(o)
mo-
to be m e a s u r -
many random variables that the sup-
is m e a s u r a b l e .
of
An e x a m p l e
for some
Therefore, PW
the p r e v i o u s
as the set of c o n t i n u o u s
the m e a s u r a b i l i t y
should
W(t)
choice
On the
that
functions
that paths
if
[t-sl
should be c o n c e n t r a t e d
of such a sufficient
of support
for
such
Mt •
should g u a r a n t e e
say r o u g h l y
- W(s)
of
condition
are con-
is small
then
close to the
is the r e q u i r e m e n t
~,8 > O
(IWt - WslB ) < Mlt-sl e
which
us the
than
of B r o w n i a n
t h e o r y we are only g u a r a n t e e d
sort of c o n d i t i o n
The c o n d i t i o n
the d i s t r i b u t i o n origin.
"kinematics"
is no reason
gives
larger
M~(e) = sup IW t ( ~ ) I w h e r e {t n} is a c o u n t a b l e dense u tn n , then Mt(m) is m e a s u r a b l e and Mt and M t agree on
the c o n t i n u o u s
tinuous?
~ 6)
let
[O,t]
f(O)
the
over u n c o u n t a b l y
r emum of c o u n t a b l y m a n y m e a s u r a b l e other hand,
PW(~IMt(~)
not be at a d i s t a n c e
is a way to describe
The trouble since
=
is a H ~ i d e r
(2.19)
condition.
Let us consider,
in the one d i m e n s i o n a l
case,
the joint distri-
29
bution
of
and
Wt
Ws
which
is e x p l i c i t l y 2 x -~-~-2~ e
I ~ Introducing
(I-x) 2 -2~zlt_sl e
dx dy
.
V2~o21t-sl
the n e w v a r i a b l e s = x+y 2
we
given by
find that
'
q=y-x
the distribution
of
Wt - Ws
is
q2 1
e
2~Zlt-sl
dq
V2~Zlt-sl from which
follows
that ~2
+co
_
1 ~2~21t-sl
]E (IW t - W s 14)
I
e 2oZlt-s
4
dq = 3o41t-si 2. (2.20)
-co
More
generally,
because
Wt - Ws
is a G a u s s i a n
v&riable,
we have
(IW t - Wsl 2n ) = C n i t _ s i n
(2.21)
where Cn =
and
Hence
(2n-
the
condition
Moreover,
by the
see e.g.
[IOO,a].
Lemma
(2n - I)!!
(2.1): Let
I)!!
= 1.3.5 ..... (2n - I)
is s a t i s f i e d
Kolmogorov
lemma
and almost Wt
has
all paths a HSlder
are continuous.
continuous
version;
[Kolmogorov]
{~t}o
be a stochastic
(i~t+ h - ~ t i P ) <
for some
2n
K
, some
For
O < q < r/p
and
s
r < p
, we have
process
obeying
Klhl 1+r
a n d all
t
then
all d y a d i c
I~ t - ~s i ~ C(~) it-sl q
for
with
O < t < t+h < T rational
t
.
30
where
C(~)
is f i n i t e
In p a r t i c u l a r , find
~t
defined
and
for w h i c h
and
from
king
n
(2.21) large,
for
Let us
(2.2):
Lemma
for all p o i n t s
with
the p r e v i o u s
first
q's
of the W i e n e r much
more
all
nowhere
prove
joint
holds a.e.
one can distribu-
for all
q
the
sample
close
arbitrarily
so b y ta-
to ! . Hence 2 c o n t i n u o u s of o r d e r
the B r o w n i a n
following
paths
nI
close
are H ~ i d e r
about
2I
to
paths,
see e.g.
theorem.
~(t)
= Wt(~)
of a W i e n e r
pro-
differentiable.
that
at time
t + ~(t)
is n o n - d i f f e r e n t i a b l e
with
one.
t
be a f i x e d
D t = {~l~(t) Then
continuous
real
number
in the
is d i f f e r e n t i a b l e
the p r o b a b i l i t y
The m e a s u r a b i l i t y
of
Dt
function
f
for r a t i o n a l
h
at t i m e
follows
f r o m the
is d i f f e r e n t i a b l e
such
If(t+h)
Dt =
interval
of t i m e
t }, w h e r e
[0,1]
~(t)
and
= Wt(~)-
P(D t) = O .
l i m f (t+h)h -f (t) , for r a t i o n a l n u m b e r h h~o t i a b l e at p o i n t t there exists e > 0
Then
the c o r r e c t
(i.e.
(2.3) : Let
that
~ersion
inequality
process
is k n o w n
we h a v e
Almost
are
probability
continuous
n > I , choosing
. In p a r t i c u l a r ,
cess
a H~ider
we can o b t a i n
trajectories I < ~ . In fact,
Theorem
everywhere.
s ).
the
[68]
has
tions t Then
~
almost
that
O <
lhl
following at
t
, exists. and
observation.
if a n d o n l y If
f
an i n t e g e r
A
if
is d i f f e r e n M > I
such
< e
- f(t) I < M lhl .
U M>
D Mt I
where ~tM = {e tHE > O s.t.
e (t+h) - ~ ( t ) l<Mlhl ,O < lhl< e ,Vh :6~} 2
MVlhl
and P(DtM)
< 2 inf -h
I o
Hence P(~t )
=
0
.
I ( 2 ~ 2 ) I/2
e
2 ~ - dy
=
0
.
31
The
lemma
shows
The p r o o f probability [0,1]
i+2
P(D t) = 0
of t h e o r e m
P(
U
v a n i s h e s ~ 6[0'I]
t £ [0,1] where
that
. For
[n t]
, i+3
2.2 n e e d s
~(t)
Let
~(t)
be d i f f e r e n t i a b l e
large
integer
successively,
i~(~
~(
Let DM'~ be the set of DM'~ 3'1 is a measurable set 1,3 D =
integer
part
of
I <
-n-
n
n t
namely
a derivative
the in
at some p o i n t
, set
, and
that
somewhere
i =
let
j
[n t] + 1 , run
over
i+I,
then
j•
-
has
result
that
sufficiently
is the
a stronger
~t )
)
~
7M
,
j
satisfying
and considers
U
=
i+i
this the
,
i+2
,
property.
i+3
.
Obviously,
measurable
set
DM
M>
I
with DM =
U m >_ I
=
D
is the
event
sufficiently Hence
that
large
lim
U
A
0 < i < n
i < j < I + 3
II.2
assumed
< lirn n I[P (
= 0
Dt
M
such
holds
that
at some
for all
point
n
i/n
.
13 =
0
.
.
is m e a s u r a b l e
a complete with
probability
probability
space
(see
zero.
Integration
L e t us b r i e f l y (see A p p e n d i x ) .
inequality
that we w o r k w i t h
U t£[O,I~)
Stochastic
an i n t e g e r
DM, ." n. . 1,3
. Now
P(D)
If we h a v e
exists
the p r e v i o u s
p(D M)
Appendix),
_ M,n n u i < j < i + 3 l,J
U 0 < i <_ n
--
there
U Dt C D t6[0,I]
Hence
0 n >_ m
discuss
The p r o b l e m
some b a s i c is to d e f i n e
facts
about
stochastic
an i n t e g r a l
of the
integral
form
t I
]
F(WT)
dw T
o The will
following make
integral.
theorem
clear
the
about
property
difficulty
of the p a t h
we e n c o u n t e r
of the W i e n e r
by trying
to d e f i n e
process such
an
32
Theorem for
(3.4):
e > O
Let
, and
~
n
be a d-dimensional
positive
g(n,~)
2n Z k=1
=
Wiener
[ W
- W
i)
if
~ < 2 ,
then
g(n,e)
~ ~
as
n ~
ii)
if
~ = 2 ,
then
g(n,e)
= d
as
n ~
iii)
if
e > 2 ,
then
g(n,~)
= O
as
n ~
LP
one w e h a v e
the c o n v e r g e n c e
in ii)
a n d iii)
is v a l i d
space
p <
Proof:
Using
and the
estimate
the
independence ~ [Ix - ~
of t h e
increments
(x) i2] < ~ [ x 2 ] w e n(1-~)
[[g(n,e)
where Let ly
in a n y
(k-l) -n
probability
Moreover
and define
[~
k2 -n
then with
with
process
integer.
m s = IE [IWl ~] e < 2
large.
then For
and
for any
such
- 2
m
l] < 2
in p a r t i c u l a r fixed
process
n (I -~ ) m2~
m2 = d
I 2n(1' 2
~
of the Wiener obtain
e
. >
for
n
sufficient-
n
P(g(n,~)<
~) <_ P ( ( g ( n , ~ )
- 2
< P(Ig(n,e) -
- 2
n(1- ~Y m s)
I n ( 1 - 5) m) < - ~ 2 e
]E (Ig(n,~) < --
- 2
2
m
[)>
½
2
ms)
n ( 1 - ~) m e i 2 )
(~
1 2n(1-~) m 4 c~ by the Doob's:inequality
then
P(gln,e)
< I) < 4 m 2 e -
Thus
using
since of
ii)
1
the
and
respectively. on
first
Borel
is a r b i t r a r y
~ [Kg(n,2)
iii) The
Cantelli
and part
is s i m i l a r . LP
- dl 2m]
i)
lemma
g(n,~)
of t h e t h e o r e m
We have
convergence in c a s e
2 -n
m2
to estimate
can be proved
ii)
> I
for l a r g e
is p r o v e d .
n
The proof
P ( I g ( n , e ) I > E)
by direct
computation
a n d b y u s i n g of t h e t r i a n g u l a r
in-
33
e q u a l i t y in the last case. F r o m this t h e o r e m it follows that the typical paths of W i e n e r process are not of b o u n d e d variation.
On the other side,
if we consi-
der for s y m p l i c i t y the case of B r o w n i a n m o t i o n in d i m e n s i o n one, the W i e n e r m e a s u r e is c o n c e n t r a t e d on c o n t i n u o u s interval
[O,1]
function h a v i n g on each
a q u a d r a t i c v a r i a t i o n given by
PW - lim
(t2 n) Z k=1
I W k 2-n - W ( k _ 1 ) 2 - n I 2 = 2 t .
For each f u n c t i o n in this class ItS's formula is valid i.e. for each C 2 - f u n c t i o n
F
(see Appendix)
we have t
r
F(W t) - F(W o) = I o
F' (Ws + -~~2
I o
F"
(Ws) ds
(2.22)
w h e n the first integral in the right h a n d side of the above e x p r e s s i o n is d e f i n e d as the limit for
n ~ ~
of the following e x p r e s s i o n
[t2 n ] Z F'(W k=l (k-1)2 -n)
- W [Wk2-n
(2.23) (k-I)2 -n]
ItS's formula is the basis of the s t o c h a s t i c calculus based on W i e n e r ' s process. The B r o w n i a n m o t i o n is a continuous m a r t i n g a l e and plays a cen ~ tral role in the class of continuous martingales. algebra g e n e r a t e d by
Let
Pt
{Ws}o<s< t . A stochastic process
w i t h respect to f i l t r a t i o n rable. An adapted process
{Ft}t61 Mt
if
Mt
is for each
is called a m a r t i n g a l e
be the o-
Mt
is adapted t
Ft-measu-
(Ft-martingale)
if
[M t - M s I F s] = 0
V s < t .
F r o m the m a r t i n g a l e p r o p e r t y of the W i e n e r process and from the special form of the a p p r o x i m a t i o n
(2.23)
i n t e g r a b l e o c c u r i n g in ItS's formula are Appendix).
it follows that the stochastic (local) m a r t i n g a l e s
(see
E s p e c i a l l y the stochastic process defined as t f 2 ~2 t = 2 J W s - dW Wt s o
is a m a r t i n g a l e and this p r o p e r t y c h a r a c t e r i z e s the W i e n e r process in the class of c o n t i n u o u s martingales. Since each typical path of the W i e n e r process is not of b o u n d e d
34
v a r i a t i o n it is clearlYr impossible, define the integral integration.
| J
H t dW t
Ht
being an adapted process,
to
using m e t h o d s of the usual theory of
I
A!
For the special integrands for w h i c h Ito s formula is app-
licable it is possible to us the pathwise c o n s t r u c t i o n discussed above. 2 For more general integrands the L - c o n s t r u c t i o n of It8 must be used. Setting
I
H(t)
dW t = E h i [Wt.+i - Wt. ] i l i
I
(2.24)
for e l e m e n t a r y integrand of the form
H(t,~)
I[ti,t]
= E h.(~) i 1
I[
] (t) ti,ti+ I _
(2.25)
b e i n g the c h a r a c t e r i s t i c function of the interval
[ti,t j]
and using the m a r t i n g a l e p r o p e r t y of the B r o w n i a n m o t i o n and the independence of the increments we have the following isometry
:"
I
Htdwt2]: rl ~
Using ~bw this isometry the stochastic more general class of integrands
I
integral can be extended to a
(for m o r e detail see e.g.
[68]).
II.3 D i f f u s i o n Process The Wiener process is the p r o t o t y p e of a diffusion. the a s s u m p t i o n of h o m o g e n e i t y and isotropy
(e.g. if some external force
is present), we can no longer expect to have ther
~ [d~tl~ t] = f(~t,t,dt)
If we drop
~ [d~t[~t] = O , but ra-
for some function
f , if this expecta-
tion exists. In v i e w of the d e s c r i p t i o n given at the b e g i n n i n g
(2.1) we will
assume that (2.26)
]E [d~tl~ t] = b ( ~ t , t ) d t + o(dt)
This a s s u m p t i o n is both intuitively reasonable and t e c h n i c a l l y powerful. Next,
note that for the W i e n e r process
all the random variables
in the m o r e general situation. if
d~ t
function
is i n d e p e n d e n t of b(x,t)
dw t
is independent of
Ws,S < t , but we cannot expect this p r o p e r t y Indeed,
~t ' then
in the situation discusses above, ~ (d~tl~t) = ~ (d~t)
, and the
w o u l d have to be spacially constant. We can, however,
35 assume that where the particle moves
to between
pends only of the position
of the particle
of the rest of the history
of the particle.
and the future of the particle present,
its past and future
property. verifies
More generally, sition and time,
assume
exept mentioned
This suggests vian diffusion
moments
to develop
process
~ (t)
d~t = bi(~t,t)dt
~d
where now
W ti
are
d
independent
depends
on the po-
we can have + o(dt)
of mechanics
~d +
~ (t)
(2.27)
order still equal to
a model in
the past
but given the
otherwise.
in
of higher
de-
This is the Markov
~E[d~ t d~ t t ~t ] = ~iJ(~t,t)dt with the conditional
t+dt
that the process
of the process
for a process
and
In other words,
are independent.
the covariance
namely
t
t and not on any
may not be independent,
In the following we always this property,
time
at time
o(dt)
based on the Marko-
given by
d ~ ~iJ(~t,t)dWJ j=1 standard
(2.28)
Brownian motion,
wio = o
[dW t I ~t = x] = 0 [dW~ dW~ or, in integral
I ~t = x] = ~iJdt
form
t i x i ~t (s,x) = + ~ b±(~T(s,x),T)dT
t n
~
,
+ j_Z_1 ! olJ~ T(s, x) ,T)dW 3,0<s
S where the stochastic
(2.29) integral
is taken in the sense of It8 [69].
To this process we can associate which
Atf(x)
on
o
C ~
(]Rd)
which takes Lemma
a semigroup,
the generator
of
is defined by
u
IEx,t[f(~t+h(x,t)] h
~ f(x)] (2.30)
and can easily be computed
the following
(2. 5) :
Let
function
= lira h+O
form for a diffusion
using
ItS's
formula,
process:
(ItS's formula) be a function
defined
for
t 6 [O,T],x
6 ]Rd , continuous
38 with
first o r d e r d e r i v a t i v e s
t
in
and
x
and second
derivatives
in
x , then
du(~t,t)
{~u
=
~-~ +
d
Z i=I
b i ~u
I
d
ijojk
. + y Z ~x I i,j,k=1
~u (~t,t) dt ~xi~x k ]
d +
Then
At
Z i,j=1
(oiJ ~u ) (~t t)dWJ ~x I '
is r e p r e s e n t e d d Z bi(x,t) i=I
At =
This g e n e r a t o r
on
p(x,t,B,s) B
a Borel
Kolmogorov
starting
point)
~d
and
p
forward
~t
process,
"
we have
as for the
(2.33)
p
function
is r e g a r d e d
verifies
as a f u n c t i o n
= - AtP(x,t,B,s)
the
of the
(2.34)
is absolute'ly c o n t i n u o u s
with
respect
to
measure p(x,t,dy,s)
the d e n s i t y
of
t < s . This
(i.e.
t p(x,t,B,s)
the L e b e s g u e
generator
22 (2.32) ~xi~x k "
probability
equation
In the case w h e r e
~Jk(x,t)
= P[~s 6 B I ~t = x]
set of
backward
.. ~13(x,t)
in the M a r k o v
a transition
(2.31)
by
the f o r w a r d
Since we are i n t e r e s t e d
for
(~d)
I d ~ + ~ Z ~x i i,j,k=1
is c a l l e d
Brownian motion
C ~o
(t)
of p r o b a b i l i t y
Kolmogorov
= p(x,t,y,s)dy p
equation
verifies
the F o k k e r - P l a n c k
(differentiation
equation
now w i t h r e s p e c t
or
to the
final point) ~--{ p(x,t,y,s) where
At
iS
the
formal
L2-adjoint
_~
~s p(x,t,y,s)
(2.35)
= AsP(X,t,y,s)
= -
of d Z i=1
At
, that
~
i ~y
is
•
(bl(y's)p(x't'y's)
(2.36) I
d
+ 2 i,j,k=l
~2 ~yl~yl
ik(y,s
) kj
(y,s) p(x,t,y,s) .
37
In the
case
reduces
to
where
oi~(x,t) ~
-~p(x,t,y,S)~s
= q~i~&
= - Vy
with
0
constant,
this
formula
a2 + -~- Ay p ( x , t , y , s )
(b(y,s)p(x,t,y,s))
(2.37) II.4
Kinematics
of D i f f u s i o n
As b e f o r e , where
I
let
c a n be
~
IR,
ped with
the t o p o l o g y
topology
induced
all c o m p a c t complete space
given
separable
space
of
we d e n o t e
their
f 6 LI(~,P) given
Z
family
time
{-t}t6i past
called
In v i e w
become
they
are
t
•
the
equip-
is,
~
a
as a m e a -
suppose
we
a probability
are space
variables,
integrable,
conditional
for ~
is a B a n a c h
at
random
the
and
as p r e v i o u s l y
of
B
, and
expectation
if
of
f ,
of the
process)which 6 I}
, the
family
consider
, called
is at e a c h o-algebra
sub-o-algebras _
of
B
{Pt}t6i
calculation
the P - c o m p l e t i o n
where of
B
N
a B-measurable
of s p e c i a l
~t
' is g i v e n
IV)
coordinate
the
interest.
an i n c r e a s i n g
of the
(see s e c t i o n
B = a{B,N}
time
are
provides
filtration
is d e s c r i b e d
the c o n f i g u r a t i o n
is g e n e r a t e d ~ y
' for the p r o c e s s
or s t a n d a r d
a-field
system we
= ~t(~)
by filtra-
process.
it is c o n v e n i e n t
is the
, and a new
family
to
of all P-
filtration
Pt
Pt =
n o { P t + ~ , N } . The n e w f a m i l y is i n c r e a s i n g , r i g h t c o n ~>O Po(P_~) c o n t a i n s all P - n u l l sets, w e r e f e r to t h e s e con-
tTnuous
and
ditions
as "the is t h a t it o n l y
The
state
t , Pt
. The
natural
sets,~called
trations
obtain
is a s u b - a - a l g e b r a
: e ÷ ~(t)
Certain _ -~_
at time
a new
verifying
Z
denote
the ~t
of m a r t i n g a l e
introduce
ly w h e n
~
B = o{~t,t
Pt = ~{~s 's ~ t} tion,
~
on
provided
, we
which
look
B . Now,
on
coordinate
Hence
The
field
f
function
(or the
function.
can
P
. If
That
d
~ [fIZ]
At e a c h by a r a n d o m
space), we
measure
P(d~)
of time,
= suplf(t) I t6K trajectory'space
functions
, t h e n we w i l l
, by
process
to the B o r e l
expectation
~ [f] = f~ f(~)
. Hence
[O,T]
~:I~
~K(f)
the
(Polish
[O,T]
functions
on c o m p a c t s .
of s e m i - n o r m s
space
I =
respect
. The Borel
interval
convergence
I . This m a k e s
metrizable
with
of c o n t i n u o u s
or a f i n i t e
family
K
a B-probability
(~,B,P)
null
~+
space
of u n i f o r m
in the c a s e w h e r e
surable
by
be the
b y the
subsets
Processes
future
Hence
{Ft}t61
before
we
usual
conditions".
a Borel depends
define the
The on
intuitive ~
on c o n f i g u r a t i o n s
at t i m e
introduce
function
t
, Ft
is
up to time
, is g i v e n
a decreasing P-completion
by
filtration. Ft
meaning
of t h e s e
Pt(Pt)-measurable t
fil-
precise-
.
F t = a { ~ u , U > t} If in a s i m i l a r
is a d e c r e a s i n g ,
left
. way
as
contin-
38
uous family such that present
at time
Ft(T+~)
t , N t , is given by
its P-completed
i Wt
~t
type in
are n-independent
is a constant
some smoothness
C(l+Ixl) b+
the past at time time
and we denote by
the
~t
in processes
standard Wiener processes we assume
on the drift
in
]Rd). Moreover,
b+(x,t)
(2.32)
: b+
coeffi-
we impose
is a smooth
func-
C . The forward drift based on information
of how the configuration At
with covariance
that the diffusion
for some constant
t . The forward generator
which are
(2.38)
gives the best prediction t
~t
+ odW t
(hence the identity
condition
tion b o u n d e d b y (or velocity)
sets. Finally,
~d
and for the sake of simplicity
cient
o{~t }
we are interested
and of diffusion
d~t = b+(~t,t)dt where
all P-null
version.
In the following, Markovian
contains
will change
in
just after
for this process
takes now
the form ~2 • 7 + -~- ~ .
A t = b+(x,t) In order to describe define
a reasonable
neither sense.
~t
sists
E.Nelson
too strong
f(~t )
[90] we introduce
in the use of conditioning
Hence we define f(x,t)
6 C2(~dx
a forward
From previous
is differentiable
irregularities
king the mean over all possible
Let
of such a process we have to
notion of derivatives.
nor any function
Following
to eliminate
the kinematics
(2.39)
a regularization
of trajectories.
with respect values
of
derivative
D+
procedure
The method
to the past
~t+dt'
Ft
con-
and ta-
dt > O.
in the following
way.
~)
[f(~t+h't+h)D+f(~t,t ) = lira ~ h h¢O As the considered
properties in the usual
process
is Markovian,
f(~t't)
the previous
1 ] Pt "
definition
(2.40)
reduces
to [ f(~t+h't+h)-h D+f(~t,t ) = lim IE h¢O This implies
the following D+f(x,t)
f(~t't)
I
] Nt .
(2.41)
-
formula lira h¢O
[ f(~t+h,t+h) I~ h
- f(x,t) (~t = X].(2.42)
39
From this def£n±t~pn,
taking into account
(2.39)
it follows
D+~ t = b+(~t,t)
(2.43)
and using ItS's calculus one obtains the explicit
~f = ~--{ (x,t) + b+(x,t)oVf(x,t)
D+f(x,t)
Now let us assume that @(x,t)
~t
has at each time
with respect to the Lebesgue measure,
A
in
Hence for all integrable
t
D+f
d2
+ -~- Af(x,t).(2.44)
a smooth density
that is
p(x,t)dx
P[~t 6 A] = I A for all Borel sets
formula for
(2.45)
~d functions
f
the expectation
r = ] ~d
p(x,t)dx
of
f
is given
by ]E [f(~t)]
Assuming rator
At
that
f
f(x)
(2.46)
.
is smooth and using the definition
of the gene-
of the process we obtain
d-T ]E [f(~t)]
=
f(x)
p(x,t)dx
d =
~IRd(Atf) (x) p ( x , t ) d x u2
= [
(b+.V
+
7
A) f(x,t)
O(x,t)dx
and integrating by part --= ~P ~t The density
p
_
~2
V. (b+p) + -~- AQ
verifies
.
(2.47)
(in the weak sense)
(forward Kolmogorov equation).
If
b+
the F o k k e r - P l a n c k
a smooth density at initial time, this property holds the considered
time interval.
equation
is smooth and if the process has
In chapter
for all times in
IV we will discuss
such a con-
dition. Let us also notice that the following property holds d d-~ ~
[f(~t 't)] = ~
[D+f(~t't)]
(2.48)
40 N o w let us assume that we are able to define the following
oper-
ation (2.49)
called the backward derivative
which we can rewrite
for Markov proces-
ses D_f(~t,t)
= lira IE h¢O
[f(~t,t)-f(~t-h't-h) h
I
] (2.5O)
Nt
and [ f (x,t)-f (~t_h,t h) = lim ]E h h¢O -
D_f(x,t)
] ~t=x
(2.51)
To derive the explicit form of D_f , let us establish for smooth functions f and g the following useful derivation formula d d-T XE [f (~t,t)g(~t,t)] = ]E [ (D+f) (~t,t)g(~t,t) ]+-]E[f(~t,t)D_g(~t,t) ] (2.52) Hence for
a < b
in
I
we have to prove
]E [f(~b,b)g(~b,b) b = I
- f(~a,a)g(~a,a) ]
]E [(D+f) (~t't)g(~t't)
+ f(~t't)n-g(~t't)]dt
a Let us assume G(t)=g(~t,t)
f
and/or
g
in
C2(]RdxI) and define F(t)=f(~t rt) t o and make a partition of the time interval (a,b) t. = a + Jib-a) ] n
j = 0,1 ..... n
then ]E [F(b)G(b) n-1 = lim E n+~ j=1
- F(a)G(a)]
d-1 = n-~olimj=IX ]E [F(tj+ 1)G(tj)
(F(tj_1)-F(tj))
- F(tj)G(tj_l ) ]
G(tj)-G(tj_ I) + F (tj+ I )+F(tj) ] -(G(tj)-G(tj j)) j 2 2 -
n-1 = lira Z ]E [ (D+F) (tj)G(tj) + F(tj)D_G(tj )] n+~ j=1 b = I ] E [(D+F) (t)G(t) + F(t)D_G(t)]dt . o
b-a n
41
Notice that relation (2.52) gives an integration hy part formula. Moreover, the following derivation formula holds d d-{ ]E [f(~t,t)] = ]E [D±f(~t,t)] Now, for
f
and
g
in
C2(]Rdx I) o
(2.53)
we have
__datI ]E [f(~t,t)g(~t,t) ]dt = O I hence by formula (2.52) O =
I
]E [ (D+f) (~t,t)g(~t,t) + f(~t,t)D_g(~t,t) ~dt
I
then
I
f
[( + b+. V + -~ A) f] (x,t)g(x,t)p(x,t) dx dt ~t ~2
i
= - I~IRId f(x,t) (D_g) (x,t)p(x,t) dx dt .
Integrating by part the left hand side we obtain df(x,t)[( -
- b+oq - Vob+ + -~ ~)gp~ (x,t) dx dt
= - [ [ f(x,t)(D_g)(x,t) JI ~ d
p(x,t) dx dt .
Hence f(x,t)[(I
- b+.V +
d
-7 + P
A)g] (x,t) p(x,t) dx dt -2-
+ I ~IR f(x,t) (D_g) (x,t)p(x,t)dx dt = - I ][]R f(x,t) (D+g)(x,t)p(x,t)dx dt I d I d using the Fokker-Planck equation (2.47), the second integral of the left hand side vanishes, since f and g are in C 2 ( ~ n x I) o then if we define b_ by b (x,t) = b+(x t) - ~2 Ap(x,t) -
'
(2.54)
p (x,t)
we obtain D_g(x,t) = ~t (x,t) + b_(x,t).Vg(x,t)
~2 Ag(x,t) - -~-
(2.55)
42
In particular,
for processes
for which
(2.54)
is m e a n i n g f u l
D_~ t = b (~t,t) b (x,t)
is called the backward
The physical following: time
interpretation b+(x,t)
t
and
at time
t .
sion
P
the
(velocity).
of the forward
is the mean velocity
b_(x,t)
and backward of particles
is the mean velocity
More generally, bability
drift
in the following
configuration
with coefficient
~t
b~(x,t) ~
is the
leaving
x
of particles
we will
process
diJ(x,t), =
drift
is
and
at
entering
x
say that under the proa
(Nelson)
smooth
diffu-
hi(x_ ,t) , I_< i , j <_ n
in c a s e i) ii)
the configura£ion the
~ z3
all smooth
of a positive functions
The following
h+o+lim
is Markovian;
are all smooth bounded
are entries
iii)
process
definite matrix.
bounded by
limits exist
C(I + IxL)
for any
~ [ f ( ~ t + h ) - f(~t ) , L , h Nt] =
lira ]E[ f(~t)- f(%t-h) h~O+ h
functions
'' 22 ~j [½ ~13 --+~xl~x 3
Nt] =
~j [_ ~ ~ 3
we will
Let us remark that the Fokker-Planck b
I
D = 2
(D+ + D_)
-
a notion
in the next section.
equation
can he reformula-
,
~7 • (b_p) - -~- A p .
to define other
] f(~t )
(forward and backward).
(52
~P _ ~t It is convenient
drift
i b+ ~
we consider we can define
look at this problem
ted in terms of the backward
C .
_ _ ~xi~xJ
.
that for the processes
of time reversal,
for some constant
f 6 C~(~ d ) o
Hence we have two kinds of velocity This suggests
where the ~Z3(x,t) i The b+ and b _i are
(2.56)
"time-derivations" (2.57)
and ~D = ~I (D+ - D_)
(2.58)
43
so t h a t
D
We
introduce
= D e 6 D
two other
velocities I
v(x,t)
the
current
velocity,
= ~
I
velocity.
= ~
is a l w a y s
with
D
and
~D
(b+(x,t)
+ b_(x,t))
(2.60)
(b+(x,t)
- b_(x,t))
(2.61
We e m p h a s i z e
u(x,t)
u
associated
and
u(x,t)
the o s m o t i c
(2.59)
~2 V p ( x , t ) 2 p(x,t)
-
that _ o2 2 ?log
(2.62
p(x,t)
a gradient.
Hence b ± (x,t) and the sion
action
in t e r m s
of
D
of
v
= v(x,t)
and and
6D
on s m o o t h
(2.63)
function
f
have
an expres-
u ~f = ~
Df(x,t)
± u(x,t)
(x,t)
+ v(x,t)
• qf(x,t)
(2.64)
+ -~ Af(x,t).
i2.65)
and 6Df(x,t)
Notice case
that
for
defined (2.47)
D
D± by
and
is a d e r i v a t i o n and
the
= u(x,t)
@D
. D
field
(2.57)
~-~P +
a n d of the o s m o t i c
(2.66), law
can be seen
v(x,t)
?
•
(pv)
which
as t h e d e r i v a t i v e
. Moreover, in the
sense)
is n o t
along
the Fokker-Planck
the
the
flow
equations
f o r m of a c o n t i n u i t y
equation
= O
(2.66)
equation
u2 -~- aP = ?
conservation
(in t h e u s u a l
can now be put
~t
Equation
• Vf(x,t)
which
o (PU)
(2.67)
c a n be d i r e c t l y
(1.40),
(1.50),
compared
is s t r i c t l y
with
the h y d r o d y n a m i c a l
connected
with
the
continu-
44
ity in time of the random trajectories. Finally,
let us remark that for smooth density the time deriva-
tive of the osmotic v e l o c i t y exists and the continuity e q u a t i o n and the osmotic e q u a t i o n ~U ~t
(2.67)
(2.66)
implies
O 2
_
2 ?(V.v)
- V(u-v)
(2.68)
II.5 The T i m e - R e v e r s e d D i f f u s i o n Process
II.5a Brownian Motion with Lebesgue Measure as Initial D i s t r i b u t i o n As we h a v e seen before,
the W i e n e r process is a M a r k o v process,
hence its p r o b a b i l i s t i c b e h a v i o u r in the future is c o m p l e t e l y determ i n e d by its state at the present time. This b e h a v i o u r is d e s c r i b e d by the t r a n s i t i o n p r o b a b i l i t y density _
p(y,s,x,t)
I
=
(x-y)
2
2~ z (t-s) e
= ~(x-y,t-s) (2.69)
(2~(t-s)°2)d/2 for
t > s , and
o2
a p o s i t i v e constant; with
The d i s t r i b u t i o n of the W i e n e r process cient
~
defined in
(2.8).
(with d i f f u s i o n coeffi-
q 2 ) is d e t e r m i n e d by the t r a n s i t i o n p r o h a b i l i t y and its initial
distribution.
Let us know c o n s i d e r the case where the initial d i s t r i b u t i -
on is the L e b e s g u e m e a s u r e on p r o b a b i l i t y measure!
~ d , hence the initial d i s t r i b u t i o n is not a
Nevertheless,
m e a s u r e for the W i e n e r process,
the L e b e s g u e m e a s u r e is an invariant
that is, if at an initial time the pro-
cess is given with a L e b e s g u e distribution,
the same occurs for any
time. Because of the Lebesgue m e a s u r e we h a v e not an u n d e r l y i n g probab i l i t y space but we can introduce a m e a s u r e space o-finite m e a s u r e
~
and define the notion of martingale.
{F=t}t6[O,T]' ~t_ c F= A map
ii)
Xt For
is
(t,~) ~ Xt(~)
: ~ × [O,T] ~ IRd {t
{t-measurable
is called a m a r t i n g a l e D)
if
V t E [O,T].
t > s , we have I iXtldP < ~
Let
and the o-finite m e a s u r e
Xsd~ = I A
I
with a
an i n c r e a s i n ~ family of sub-a-algebras.
(with respect to the family i)
(~,~,~)
Xtd~ V A c ~s
such that
A
.
A Hence the Wiener process (~,B,~)
where
~
Wt
is defined on the m e a s u r e space
is the space of continuous trajectories,
B
the
45
Borel ~-algehra distribution
and
D
p[W] t
the natural
is a
P~W~-martingaleNow,
the measure
induced by the initial Lebesgue
and the transition p r o b a b i l i t y filtration
(2.69).
If we denote by
of the Wiener process
Wt
then
Wt
in the above sense. v Wt
let us consider the time reversal process
defined by
v W t = WT_ t where
(2.70)
Wt
is the Wiener process with initial Lebesgue distribution. v The time-reversal process W(t) is also a Markov process adapv ted to the fiItration ~[W] = ~[W] t£[O,T] For all continuous func~t -T-t ' " tions f(x), g(x) with compact support in ~ d we have the following
I f(Ws)g(Wt)dD
= I~dl]Rdf(x)
p(x,s,y,t)g(y)dx
_ ~_x) =
I
I
(2~ (t-s) o2) d/2
[
f (x)e
dy
2
(2.71)
2~ ~
g(y)dx dy
iRd]l~ d
but
I
Since
v Wt
i Consequently, (2.69)
and v
is" a
from
f(WT-s)g(WT-t)d~
i
{2o2 [ (T-S)- (T-t) ] } d/2
have
the
(2.75)
and
(2.72)
•
(2.72) we have
=
2 (2.73) Ix-y[ 2~ z [ (T-s) - (T-t) ] f (x) e g (y) dx dy .
r
]Rd ] ]Rd
the transition p r o b a b i l i t y v W~ ~
v
f(Wt_s)g(WT_t)dD
is also M a r k o v process,
a
by
v
f(Ws)g(Wt)d ~ =
same
density of
distribution
as
Wt
v Wt
is also given v
, moreover,
Wt
-tP[W]-martingale In particular,
the time reversed Wiener process
cess, when the initial d i s t r i b u t i o n
mark that in general the time reversed Wiener process process.
is a W i e n e r pro-
is a Lebesgue measure.
Let us re-
is not a Wiener
46
II.5b Time-Reversed As previously from
[O,t]
to
Diffusion let
~d
solutely continuous
~
Process
be the space of all continuous
. Suppose that under probability
~t
(P
the configuration
is a Markov diffusion process of the following type
d~t = b+(~t,t)dt + ~dW t where b+(o,t)
is a vector field,
(2.74)
sufficiently
regular.
to prove that ~t = ~T-t is also a M a r k o v d i f f u s i o n p r o c e s s type. For sake of simplicity we choose The following consideration ron-Martin
ab-
probability measure with respect to the measure
of the Wiener process with initial Lebesgue measure) process
functions
P << ~
[21]-formula.
We are qoin~ of the previous
~ = ~.
is based on the Girsanov
[56]-
Came-
Let us give a simplified version corresponding
to the case where the Wiener process has a Lebesgue
initial distribu-
tion Theorem
(2.6):
Let
~
be a measure on the filtration
under
~ , ~t(~), t £ [O,T]
measure
dx
negative
function such that
space
(~,~t,~)
such that
is a Wiener process with Lebesgue's
as initial distribution,
and let
O(x,O)
S p(x,O)dx = I
[S~ b+(~s,S)d~s
be a non-
If
I t - ~ ~o I b+(~s,S)l 2 ds]
Qt = p (x,O) e is a martingale, dP = QTd~
then under the probability
P
defined by
the process t Wt = ~t - I
b+(~s'S)ds o
is also a Wiener process with the same diffusion coefficient ~t
and with initial distribution
Now,
let us suppose that
p(x,O)dx
~ = C(O,T)
as
.
is the space of continuous
real
[O,T] and {Ft[~t]},t6[O,T] the natural filtration of the function on configuration process ~t satisfying (2.74) under the measure P such that =
dP = QTd~
log QT
as defined in the previous
=
log p(~o,O)
fT
theorem.
+ Jo b+(~s'S)d~s
-
Then,
I IT
2
o Ib÷(~s,S) [2ds
(2 75)
47
by It6's
formula T log p(~o,O)
= log O(~T,T)
- I
V log p ( ~ s , S ) d ~ s o
IT
1
JO
-
( 7 S + g A) l o g
p(~s,S)ds
u s i n g the r e l a t i o n
b+ = b
+ V__p_p -
and the F o k k e r - P l a n c k
p
equation
log QT = log p(~T,T)
in b a c k w a r d
+
form
(2.56) we o b t a i n
b_(~s,S)d~ s - ~
[b_(~s,S)12ds
o
+
Vob_ ( ~ s , S ) d s
o (2.76)
.
o B u t the s e c o n d and last t e r m s of the r i g h t h a n d side c o m b i n e d become a backward stochastic integral v ~ = ~T-t we can r e w r i t e
(see A p p e n d i x ) ,
together
t h e n if we
denote
v log QT = log p(~o,T)
-
IT o
v v i IT v b_(~s,T-s)d~ s - ~ Ib_(~s,T-s) 12ds o (2.77)
Then define
v
v
I
v v - [S~ b _ ( ~ s , T - s ) d ~ s + ~ Qt = P(~o 'T) e
~lb
v
_(~s,T-s) [2ds] (2.78)
and v v dP = QT d~
.
(2.79)
V
By the p r e v i o u s
theorem
~t
V
is a M a r k o v i a n
v v d~ t = - b _ ( ~ t , T - t ) d t where
Bt
from
(2.78)
diffusion
II.6
is
a Wiener
and
under
P
(2.80)
process
v F [~] . B u t =t v the ~t is a
the time r e v e r s e d p r o c e s s .
Acceleration
To i n t r o d u c e we h a v e
+ dB t
adapted to the filtration v (2.79) we h a v e P = P . Hence under P
process,
Stochastic
diffusion process
to d e f i n e
a n o t i o n of a c c e l e r a t i o n
in s t o c h a s t i c
a s e c o n d o r d e r time d e r i v a t i v e .
A priori
kinematics there exist
48
four b a s i c twice
candidates:
differentiable
D+D+~,
D+D_~,
as a f u n c t i o n
d 2 ~ / d t 2 . H e n c e the m o s t g e n e r a l acceleration
on s m o o t h
D _ D+~
and
D_D_~.
If
~
of time e a c h of t h e m r e d u c e
acceleration
functions
which
c a n be w r i t t e n
reduces
as a
four
is to
to the usual parameters
family aKl~
= [KD+D+ + ID+D_ + ~ D ~ D + + ~D_D_]
(2.81)
w i t h the c o n s t r a i n t ~ + ~ + ~ + 9 = I
Under time reversal seen before,
from
(2.80)
v D+~ = - b_
= - b_
(t)
becomes
v ~t = ~T-t
and as we have
follows v (~t'- T - t)
(~T-t'
T-
t)
and m o r e g e n e r a l l y v D±~ t = - D, ~T-t If we ask for i n v a r i a n c e tions reduces
which
tion.
Indeed
a < = a <, y1- < ,
y1- < ,
<
a K = [ (D+D+ + D_D_)
+
the f a m i l y of a c c e l e r a
(I/2 - K) (D+D_ + D _ D + ) ] ~ t
(2.83)
as
a<
=
aN
=
[ (D+ + D_\2 (D+ - D_)2] 2 j + 8 ~
~-~ +
(2.85)
(v.D)v
coincide
in
(1.52)
if we c h o o s e
(2.85)
of the M a d e l u n g
fluid equa-
and t a k i n g the g r a d i e n t
/ ~ ~I/2 ~ = k-~-/
is a m o t i v a t i o n
acceleration
~t' 6 = 4K - I (2.84)
+ 8 ( ( u - V ) u + -~- ~u).
is v e r y r e m a i n d i n g VS v = -~-
T h e n this c o n s i d e r a t i o n the s t o c h a s t i c
under time reversal, family
setting
two e q u a t i o n s
(2.82)
to a one p a r a m e t e r
can be w r i t t e n
The e x p r e s s i o n
"
and
the
6 = - I
for the c h o i c e
8 = - I
and
t a k e s the f o r m
a = ~I (D+D_ + D_D+) ~
(2.86)
49
Let
us n o w c o n s i d e r
the
following
d~ t = - m ~ t d t
where
Wt
is a W i e n e r
as i n i t i a l
Markov
process
in one d i m e n s i o n
+ odW t
process,
(2.87)
and with
the
invariant
Gaussian
measure
distribution ~X 2
I ~j~o2 e
Po(X)
o2 (2.88)
then
D+~t
=
- ~t
(2.89)
D_~ t = ~
(2.90)
Hence v(x)
with
the
choice
= 0
(2.86)
a = -
which
quantum
tion
u(x)
for t h e
III.3.
[62,b]
(2.92)
of t h e p o s s i b l e Then, E. N e l s o n
oscillator
Newton
[112]
anociated
according
the
and chapter
choice
of
8
with
the
V,
and
see
also
(2.57)
-
state
of
for a discussion
o consideration,
acceleration
by the
we define
symmetric
following
expression
(2.93)
(2.58)
a = ~D 2 -
and by
fundamental
[30,a]
a = 21 (D+D_ + D _ D + ) ~
using
oscillator.
o 2 = -- , as w e w i l l s e e in s e c m u s i n g v a r i a t i o n a l p r i n c i p l e s see e.g.
to the previous
stochastic
law for the harmonic
when
For other motivations
[90,a,e]
(2.91)
~2~ t
is n a t u r a l l y
harmonic
= - ~x
acceleration
is j u s t t h e d e t e r m i n i s t i c This process
the
and
(~D)2]~
(2.94)
(2.60) ~v
a = %T +
(v-V)v -
o2
( u . V ) u - ~-- A u (2.95)
~v
- ~t +
( v o T ) V - V(
1
u
2
o2 - -~- VoU)
50
Remark By trying showed
to g e n e r a l i z e
that the diffusion
but rather
c a n be a n y s t r i c t l y
(length) 2 d i v i d e d
The the
value
positive
As d i s c u s s e d
acceleration
a n d the a s s u m p t i o n
of t i m e
moreover
stochastic
considered
reversal
that the
of dimension
related above
invariance
coefficents reduces
D+ + D_
the general
yields
satisfy
t o the
shows
the
following
_
for
Let X D be the diffusion As a side remark
in
Newton
is e q u i v a l e n t
choice
~ + I + ~ + ~ = I
2
choice
stochastic
which
K = I a n d ~ = ~.
+ B
(Nelson's
that the
form
0 >]
e
2
to an e q u a t i o n
of
family of acceleration
)
aD
B = 4 ~ - I.
to a c h o i c e
namely
[( then
[30 a]
determined,
+ ~ D+D_ + ~ D_D+
acceleration
by D a v i d s o n
a Where
Davidson
is
a = ~ D+D+ + I D_D_
Assumming
constant
of v is o f c o u r s e
acceleration.
stochastic
then the
mechanics
v is n o t u n i q u e l y
b y time.
specific
stochastic
of the
stochastic
constant
[90 b] w a s law with
B = - I). D a v i d s o n
this acceleration
to the S c h r 6 d i n g e r
equation
leads
provided
B
process
we point
out
corresponding
that Ehrenfest's
to the D a v i d s o n
model.
theorem
d2 E
[X D]
= E
[a D]
dt 2 follows
f r o m the
choice
is j u s t a c o n s e q u e n c e f o r m of t h e d y n a m i c s no
significance
cannot
choice
the N e l s o n ' s relation
In o t h e r w o r d s
in t h i s r e s p e c t .
of t h e
choice
[58 a.c.].
Moreover
uncertainty stochastic
leads
to t h e
Ehrenfest's
invariance.
and the way how it enters
use H e i s e n b e r g ' s
specific
of a D.
of t i m e r e v e r s a l
in the t h e o r y
are of
it is e a s y to s h o w t h a t o n e
relations
acceleration simplest
theorem
The particular
in o r d e r to (see VI.
single out
4). H o w e v e r
f o r m for the u n c e r t a i n t y
a
51
II.7
Some Basic
II.7a
time at
Examples
The
Wiener
Let
us c o n s i d e r
t x°
Process
In t h i s
o , Bo(X)
bability
the Wiener
case,
the
= ~ ( x - x o)
density
. The
p(x,t)
process
initial
in
]Rn
distribution
with
starting
distribution
respect
at
x°
at
is t h e D i r a c m e a s u r e t
admits
to the Lebesgue
at t i m e
measure
a pro-
(x_xo) 2 =
I (2~2(t_to~n/2
p(x,t) where
is the v a r i a n c e
~2 Hence
Vlogp(x,t)
e
2oz (t-to)
(2.96)
of t h e p r o c e s s . I X-Xo ~2 t - t o and the osmotic
=
velocity
takes
the
form x-x
I
As
the
forward
u~x,uj
=
drift
b+ = 0
b+(x,t)
and the
current
2 t-t
= 0
velocity
o
(2.97)
o
we have
,
x-x o = t-t o
b_(x,t)
assumes
the
(2.98)
form
I X-Xo v(x,t)
(2.99)
- 2 t-t o
The
stochastic
acceleration
I = y D+b_
a(x,t) We can
interpret
"prepare" on the
"free"
release
this
a Brownian
is
II.7.b
"free"
Brownian
I 2
to
X-Xo (t~to) 2
heuristically
(starting)
particle
Wt
condition
that
in t h e
The Brownian Let
tial
formula motion
reduces
a very
(2.100) in the
at p o i n t strong
following
xO
force
, we have
way.
To
to a p p l y
and gradually
to
it.
L e t us r e m a r k sure
(2.94)
the B r o w n i a n
sense
that the
motion
with
acceleration
initial
Lebesgue
mea-
vanishes.
Brid~e
be a one N° = O
dimensional and variance
Wiener o2t
process
on
. The process
[O,T]
with
ini-
52
~t
is a G a u s s i a n
= T -Tt
process
BO =
a ÷ Wt
such
~
,
+ Tt
,
[b.WT)
t 6 [O,T~
(2. 101)
that
BT = 8
(2.102)
with mean re(t) = IE [Bt]
= T__tt T a + ~t B
(2.103)
and c o v a r i a n c e r(s,t)
Then
= ]E [ (Bt-m(t)) (Bs,m(s)) ] = ~2s ~ ;
the d i s t r i b u t i o n
at time
t
is g i v e n
by the
= ~
Pc8 The p r o c e s s Now, variance.
Bt
2~2t
is the
Since
=
Bt
is a G a u s s i a n N
.
that
Brownian
Bt
process
the d i s t r i b u t i o n
Wiener
using
process
Hida's Wt
to h a v e
~ - ~ ~
"
with mean
zero
covariance
of
Bt
the p r o c e s s
Bt
takes
T
(2.106)
(2. 107
i
[65]
there
(2. IO8
exists
a standard
that
T-t Bt -
and
is g i v e n b y 2 x I e 2
Bt = ~(T-t) it d w s t ] o T-s Then
and unit
oz t(T-t) T m
representation
such
zero mean
~
I to)'x,-" ~(x,t) = %0
Now,
Bridge.
Bt
is
Bt It)
.(2.105)
N N s (T-t) = TR __[BtBs] = t(T-s) ; S _< t
rts,t)
Notice
of
T 2
Then
e
the p r o c e s s
the v a r i a n c e
2
(x-m(t))
(T-t)
so-called
let us n o r m a l i z e
T
t(T-t)
T
(2.104)
density
I
(x,t)
s
the
form
t ~
+ ¥
(2.109)
ft d W s ~ +
(x-t)
]
~-s
o
(2.110)
53
or, in differential
form,
dBt = (6T~ • which
can be rewritten
(2.111)we
obtain
It 6 - a s ds + W t T-s o
(2.105)
_
follows
Hence the backward b
~-x
(2.112)
T-s for the osmotic
~ + = ~I (E
u(x,t)
(2.311)
for the forward velocity
b+ ( x , t ) and from
+ dW t
also as
Bt = ~ + From
I t ~-~)dt tins\
velocity
8 T-t
velocity
T t (T-t)
(2.113)
x)
is given by
(x,t) = x-~ -
(2.114)
t
We have all the elements
to compute
the stochastic
acceleration
obtaining 6-x + x-~ (T_t) 2 t2
a (x,t) -
This result
extends
ian bridge,
if we define
mensional
Brownian
II.7c The Bessel Let
Wt
immediately
(2.115)
to the case of an n-dimensional
it as the n-tuple
of
n
independent
Brownone-di-
bridges.
Process
be a n-dimensional
Wiener
process
of covariance
Define Rt
IWtl =
as the radial
part of ~2 dR t = -~
=
( n 2\I/2/ X WII \i=I
W t ; by It6's
formula
(2.116) (2.27) we obtain
dt n W ti dW~ (n-l) R~ + i~ I Rt
Using now the fact that a stochastic
It8 integral
(2.117) of the form
54
~t
Yt (~) =
JO a ( T , ~ ) d W T ( m )
mensional
the t r a n s p o s e d
a(T,~)
if and o n l y
m a t r i x of
M
, = ;t
W~ d W i s
Wt is
,where
Brownian motion
n
O i~I
a one d i m e n s i o n a l
is a m x n if
(see e.g.
MM t = [82,b]
matrix,
I , where
is a m - d i Mt
denotes
[114] , it f o l l o w s
Rt
that
(2.118)
Wiener process with variance
~z
. Then
(2.117)
rewrites 2
dt (n-l) ~ + dW' t
dR t = ~
which
is a d i f f u s i o n
equation.
The a s s o c i a t e d
I 3z n-1 A = ~ (o 2 ~ r 2 + r where
r = /x~
+ ... + x~
cess the B e s s e l
(2.119)
generator
)
(2.120)
, it is t h e r e f o r e
natural
f u n c t i o n of the n - d i m e n s i o n a l
g i v e n by
=
2C (2~oz(tnto))n/2_
the Cn'S d e p e n d on the p a r i t y of
C2p+I
2p = 1.3 ..... (2p+I)
C2p
-
process
is
- 2o2(t-to) e
(2.121)
n
2 (p+1)!
that this p r o c e s s
measure.
Bessel (r_ro) 2
P(ro,to,r,t)
Notice
to call this pro-
process.
The t r a n s i t i o n
where
is given by
has the m e a s u r e
In this p a r t i c u l a r
dD(r)
case w e can c o m p u t e
= rn-ldr
as i n v a r i a n t
e a s i l y the k i n e m a t i c a l
quantities b+(r,t)
= o2 n-1 2
1 r
u(r,t)
= o2 n-1 2
I r (2.122)
b
(r,t) = _ o2 n-1 -
v(r,t)
2
= 0
I r
55
and
the
stochastic
acceleration
a(r)
Notice is
that
"free"
in
in
If w e the
Dirac
the
the
= _ ~2
case
sense
suppose
measures
n =
that
hy
(2.123)
3
we
the
the
r°
given
n-3 3 r
that
at
is
obtain
an
stochastic
process
, then
has
the
example
of
a process
acceleration initial
distribution
distribution
at
which
vanishes.
time
given
by
t > to
is
2 (r-r O) 2C d~t(r)
Notice any
that
time
o
= O,
the and
= O
o
log
where
Yn
is
that
the
e
r
particle
reaches
n-1
the
dr
.
(2.124)
origin
at
vanishes.
following r
- 2~2(t-to)
n (2~a2(t_to))n/2
probability
t > O
In t
the
=
computation,
let
us
assume
for
simplicity
that
.
p(x,t)
a constant
with
u(r,t)
= -
b_(r,t)
=
2 r 2~2t
= Yn
+
respect
r + 2--t r (~-
to
o2(n-I) 2 ~2
n-1 2
(n-l)
r
log
2
. Then
I r
(2.125)
1 r)
(2.126)
and a(r,t)
For
a more
II.7d
detailed
=
study
The
Ornstein-Uhlenbeck
Let
us,
described
in
as
a last
(1.6)
in
1 r 2 t2
~2 n - 3 r3
of
Bessel
the
Process
example,
the
case
(2.127)
process
[82,b].
[106]
consider where
see
the
the
Ornstein-Uhlenbeck
configuration
space
in
process ~
.
dx t = v t dt (2.128) dv t = - ~vtdt
+
Bo
dW t
56
where
~
is a c o n s t a n t
t i n g f r o m the origin. ]R x ~
and
it is a M a r k o v i a n For i n i t i a l
equation
(2.128)
Wt
is a s t a n d a r d B r o w n i a n m o t i o n
Considered
diffusion
conditions rewrites
as a p r o c e s s
x o , v°
It
VsdS
;t
o v t = e -St v O + BO
star-
space
p r o c ess. the s t o c h a s t i c
in i n t e g r a l
xt = XO +
on the p h a s e
differential
form
e-~(t-S)dw s
(2.129)
o the s o l u t i o n
vt
beck velocity
process)
of the s e c o n d e q u a t i o n with
initial
p r o c e s s w i t h m e a n and c o v a r i a n c e
m(t)
r(t,s)
=
]E
the g e n e r a t o r
in
(the O r n s t e i n
(2.18)
condition
v
o
g i v e n by
(2.130)
= IE (v t) = e - S t V o
[(vt-m(t)) (Vs-m(s))]
of the p r o c e s s
=
02
[e-8 It-s! -e -B(t+s) ]
(2.131)
can be w r i t t e n
d 2 02 d 2 A = - 8v ~-~ + 8 2 dv 2 and t h e t r a n s i t i o n
Uhlen-
is a M a r k o v - G a u s s i a n
probability
(2.132)
density verifies
the F o k k e r P l a n c k equa-
tion 22
~-{- p ( v ' , t ' , v , t )
= ~-~ (vp(v',t',v,t))
+ 8202
p(v',t',v,t) ~v 2 2
[v-v
w h i c h can be e a s i l y s o l v e d
O
B°2 (1-e-2B(t-t°)
I
P(Vo,to,V,t ) =
Notice
bability
Uhlenbeck velocity
distribution du = p (v) dv -
w i t h the o s c i l l a t o r
(2.134)
process with
invariant
pro-
2
-
coincides
e
[7802 (1,e-28(t-to) ]I/2
that the O r n s t e i n
e -8 (t-to) ]
I 'FB o 2 process
V BO~
e
dv
defined
has the t e n d a n c y to go t o w a r d the origin.
in
Indeed
(2.135)
(II.6.).
This p r o c e s s
57
E[v t]
E[vt
=
[2.136)
o
Vt']
= il v v ' p ( v ' , t ' , v , t )
p(v')dv'dv (2.~37)
= 8~2e-~(t-t')
More
generally
for a r b i t r a r y
(v t vt,)
The
configuration
mean
~(t)
m(t)
Xt
is a
= x O + 1-e-Bt
= o2min(t,s)
differentiable
Gaussian
process
of
r(t,s)
~2 r(t,s)
(2.138)
= B ~2 e - ~ ! t - t ' [
process
and c o v a r i a n c e
t and t'
v°
(2.J39)
(-2 + 2e -St + 2e ~Ss-
+ ~
e-BIt-sI_
e-B (t+s) (2.140)
then
the
a(t)
and
variance
= ~
is g i v e n
by
((x t - Nmt) 2) = ~o2
the p r o b a b i l i t y
density
(28t - 3 + 4e_~ t - e "2~t)
takes
the
form
(x - x o
~(Xo,to,X,t)
B u t X t is not does
not
=
I 2~a (t-t o)
a Markov
factorize
The
invariant
measure
Let
us r e m a r k
that
converge ~2
. Then
to the the
to the W i e n e r Let
2a(t-to)
as a p r o c e s s
# f(t)g(s)
for this
if w e
transition
us n o w
compute
Ornstein-Uhlenbeck
process at
x
o
-
D+x t = v t
with
1 ~-Ba2 ,
xt
with
covariance
measure
on
~ .
~(Xo,to,X,t)
converges
with variance
in d i s t r i b u t i o n
o2 associated
with
the
measure
dv dx
D + v t = - Bv t
the
[IOO,a]).
process
variance
quantities
invariant v2 e ~2
fixed
of a W i e n e r
the k i n e m a t i c a l
process
d~(x,v)
being
probability
starting
[~46] and
is the L e b e s g u e
B ~ ~ , ~2
Ornstein-Uhlenbeck process
(2.142)'
in JR. I n d e e d
(see e.g.
process
let
(1-e - 8(t-to) ) 2
- T
e
process,
~(t,s)
v°
(2.141)
(2.143)
(2.144)
58
I ~(D+
- D_)x t = uX(x,v,t)
I = O, ~ ( D +
D_x t = v t
D _ v t = Bv t
and the stochastic
acceleration
If an e x t e r n a l stein-Uhlenbeck
,
force
process
- D_)v t =
+ D_D+)x t
on t h e p a r t i c l e ,
(2.128)
(x,v,t)
= - 8v
(2.145)
a = @(D+D_
acts
uv
the
vanishes. associated
Orn-
is g i v e n b y
dx t = v t dt (2.146) d v t = - 8v t d t + f ( x t ) d t
where
f ( x t)
is a f o r c e
that
f(x)
self.
Moreover,
t h a t w e c a n no l o n g e r
transition
explicit
for g e n e r a l
as a p r o b a b i l i t y Indeed,
f
probability.
expression
tribution
is let e q u a l
to one)
and we
assume
= - ~V(x)
Notice
the
(the m a s s
+ ~dW t
invariant 2V
if
- - Bo2 e
the Fokker-Planck
p(x,v)
assumes
the
the velocity
it is p o s s i b l e
measure
is in
equation
process
easy to compute
Nevertheless,
for t h e
measure
consider
it is n o t
which
b y it-
explicitly to exhibit
can be
an
interpreted
LI(xR d ,dx) (2.43)
for t h e
invariant
dis-
form
~202 2
and admits
AvP(X,V)
the
+ V.VxQ(X,V)
following
p(x,v)
where
N
solution - B 72 [I~ v 2
case,
D+x t = v t
,
uX(x,v)
O
stochastic
(2.147)
= 0
+ v] (2.148)
constant
the k i n e m a t i c a l
=
D_x t = v t and the
(x,v)]
= N e
is a n o r m a l i z a t i o n
In t h i s
- V v" [ (Bv-f)p
,
which
depends
quantities
on
V
.
are given by
D + v t = _ ~v t + f ( x t)
,
uV(x,v)
=
-
Bv
D _ v t = ~v t + f ( x t)
acceleration
is just
(2.149)
59
a
Finally,
x
I
= [
let us r e m a r k
dYt -
is a M a r k o v i a n Uhlenbeck
(D+D_ + D _ D + ) x t = f(x t)
f (Yt) 8
the
in t h e
b(x,t)
Smoluchowski
(2.151)
in c o n f i g u r a t i o n
following
- f(x) ~
process
dt + o d W t
approximation
process
Define
that
(2.150)
, let
sense x,v
space
[106], be the
of t h e O r n s t e i n -
[90.b]: solution
of t h e
coupled
equations
dx t = vtdt (2.152) dv t = - Bvtdt + ~b(xt)dt
with
initial
conditions
Xo,V °
dY t = b(Yt)dt
For
all
v°
, with
lim
uniformly
for
probability
.
Let
+ odW t
+ ~adW t
Y
be the
solution
(2.153)
.
one holds
(2.154)
Xt = Yt
t 6 [O,T]
, for
of
b
and
o
fixed
.
III.
III.1
NELSON
Stochastic
STOCHASTIC
the d y n a m i c a l ternal
through
The dynamics
process,
principle.
In this
chapter, being
u
coupled
the d i f f u s i o n
V(
velocity
Now,
(3.1)
law
+
(u-V)u
F = ma m
(v.V)v
+
-~-
as we will
discuss
~t
be the p o s i t i o n
approach
the second N e w t o n
states
F(x,t)
IV, d e t e r -
we introduce
of mass
dynamics
m
Following
at time
is N e w t o n ' s
is the product
of the particle.
[90,a,b]
diffusion
m
in chapter
of
of the
Nelson's
a stochastic
origi-
analogue
of
law.
In the case w h e r e
law in mean)
characteristics
of a p a r t i c l e
law of the n o n - r e l a t i v i s t i c
: the force acting on a p a r t i cl e
by a M a r k o v i a n
(3.2)
Au
of the i n f i n i t e s i m a l
by the a c c e l e r a t i o n
nal d y n a m i c a l
where
-
[115,a,b].
t. The fundamental
mass
for
,
V.v + v o u )
and therefore,
let
v
of a
equations
g2 a
it [22,a,b],
the
in Chapter V.
the stochastic a c c e l e r a t i o n
and the current
us the time e v o l u t i o n
m ines
or
let us c o n s i d e r
studied
system of n o n - l i n e a r
~v _
gives
law,
2
~u _ ~t
~t
and
of an ex-
can be g i v e n by
the N e w t o n
principle
law specifies
developed
"moving",
"influence
mechanics
of the diffusion,
then t h e
velocity
and dynamics.
part of our theory,
of stochastic
the v a r i a t i o n a l
If some d y n a m i c a l
the osmotic
kinematics
is to e x p l a i n what we m e a n by
a variational
diffusion
PROCESSES
is to explain what we m e a n by
the a c c e l e r a t i o n
first approach,
of two parts:
the k i n e m a t i c a l
chapter,
part
force".
specifying
consists
at hand,
in the p r e v i o u s
- NEWTONIAN
N e w t o n Law
Each m e c h a n i c s In the c o n t e x t
DYNAMICS
the p o s i t i o n process,
~t
of the p a r t i c l e
then the N e l s o n - N e w t o n
is d e s c r i b e d law
(Newton
that
(3.3)
(D+D_ + D _ D + ) ~ t = F(~t,t)
is an e x t e r n a l
(deterministic)
force
field acting on
the particle. As starting Newton
law
babiliStic
(3.3)
point
for the
is not e n t i r e l y
meaning
stochastic
mechanics,
satisfactory,
of the stochastic
the stochastic
since the direct
acceleration
pro-
is not yet w e l l - u n -
61
derstood. Moreover,
the stochastic N e w t o n l a ~ appears m o r e as a con-
straint on the drift than as a f u n d a m e n t a l law. We come back to this p r o b l e m in the study of stochastic v a r i a t i o n a l p r i n c i p l e
(Chapter V).
In the following, we call N e w t o n i a n d i f f u s i o n a M a r k o v i a n diffusion process for w h i c h the drift is d e t e r m i n e d by the stochastic Newton law
III.2
(3.3).
Conservative Newtonian Diffusion Processes As in classical N e w t o n i a n mechanics,
the one of c o n s e r v a t i v e potential
an important special case is
forces, where the force
F
derives
from a
V, w h i c h depends only on the p o s i t i o n
F(x) = - W C x ) .
Hence the N e w t o n law
(3.4)
(3.3) r e w r i t e s
m_2 (D+D_ + D_D+)~t = - .~(~t ). Moreover, (2.57)
(3.5)
if we assume that the current v e l o c i t y defined by
is also a g r a d i e n t ~
v(x,t)
the right hand side of
= ~.S(x,t)
(3.2) becomes also a g r a d i e n t
Bv--V(½
v2
~t
I u2- c2 V . u + ~ V ) - ~
(We have used the fact that if Conversely,
(3.6)
~
v
if the v e l o c i t y
is a g r a d i e n t v
(3.7)
" (v-v)v
I
= ~(Vv)
2
.)
is solution of 2
8v ~-~ +
(voq)v = V(~I u2+ ~
w i t h initial c o n d i t i o n v
of
(3.8)
v°
such that
q-u - m1 V)
(3.8)
Vo(X) = V S o ( X )
then the s o l u t i o n
is always a gradient.
A M a r k o v i a n process
such that
(3.5)
and
(3.6)
are satisfied will
be called a c o n s e r v a t i v e M a r k o v i a n process. Notice that the "conservative"
Now
S
p r o c e s s e s defined here are qualita-
has the d i m e n s i o n of an action by unit mass.
62
tively d i f f e r e n t from the "dissipative" d i f f u s i o n processes such those studied in Section II, §5, as Wiener process, B r o w n i a n Bridge, The p r o b l e m of c o n s t u c t i n g a ~ a r k o v i a n )
....
d i f f u s i o n process with
a given initial density and given g e n e r a t o r is w e l l - k n o w n in the case where the drift
b+
is s u f f i c i e n t l y smooth
is w h e n the o p e r a t o r the Laplacian.
b÷.V
[22,a,b],[68],[90,c].
That
can be seen as a "small perturbation" of
In this case the m e a s u r e on path space is given by a
C a m e r o n - M a r t i n [21] Girsanov
[56]
formula,
and is an a b s o l u t e l y con-
tinuous t r a n s f o r m a t i o n of the Wiener m e a s u r e with the given initial density. In the case of c o n s e r v a t i v e N e w t o n i a n d i f f u s i o n process the drift b+
is d e t e r m i n e d t h r o u g h the stochastic N e w t o n
longer a "small perturbation".
law and
b+oV
is no
Before d i s c u s s i n g the c o n s t r u c t i o n of a
d i f f u s i o n process a s s o c i a t e d with a singular drift
(Chapter
V), let us
discuss the p r o p e r t i e s of conservative N e w t o n i a n diffusions.
III.3 M e c h a n i c s of C o n s e r v a t i v e N e w t o n i a n Process In this section, we always assume the existence of a conservative N e w t o n i a n process lity
St
with a sufficiently smooth density of probabi-
p(x,t). For such a process we can "linearize" the dynamical equations
(3.1) and
(3.2) by introducing the complex function i 2 S (x,t) ~(x,t)
where
S
is given by
First, Log p
= ~p(x,t)
e
~
defined as
(3.9)
(3.6).
let us suppose that the density
is well-defined,
p
is strictly positive.
c o n s e q u e n t l y the osmotic v e l o c i t y is finite.
Setting R = ~I !og p
(3.10)
rewrites R(x,t) + ~ o
S(x,t) (3.11)
(x,t) = e
and 2R(x,t) J~(x,t) J2 = e
= p(x,t).
(3.12)
63
Taking into account
(3.4),
(3.5)
and the fact that the osmotic velo-
city is given hy
u=
equation
(3.7)
o
2
(3.13)
VR
rewrites ~S _ 04 ~t 2
~ R 21 I (~) ~ - ~
[AR +
and the c o n t i n u i t y e q u a t i o n
(3.1)
(VS)
2
I - -m V
(3.14)
takes the form
DR 1 ~--~ + AS + 7R-US = O.
(3.15)
We recognize the M a d e l u n g fluid e q u a t i o n in Section
(I, 3,2),
if we identify
v e r i f i e s the "Schr~dinger-like"
since
S
(3.14) and and
R
[60,a], [83]
with
~ m
discussed
. Hence the function
equation
2 o2 ~ ( x , t )
i ~(x,t)~t -
In fact,
02
+ ~ mo
V(x)~(x,t) .
(3.16)
(3.15) are d e t e r m i n e d up to a function of time,
are defined through a gradient,
and
(3.16) m u s t have
the general form i ~ (x,t)~t -
2 ~_ A~(x,t) ?
+
V(x)~(x,t)
where
m(t)
is real at least if
ty of
m(t)
can be deduced from the fact that
Moreover,
~
S , m(t)
S~#dt
is independent
can be taken zero.
for a c o n s e r v a t i v e N e w t o n i a n process the d i s t r i b u t i o n p(x,t)
p(x,t)
where
(3.17)
(3.17) holds for all times. The reali-
of time. By an a p p r o p r i a t e choice of
of the process
+ ia (t) ~ (x,t)
at time
t
is given by
(3.18)
= l~(x,t)I 2
is a solution of the Schr~dinger e q u a t i o n
(3.16) with initial
condition
(x,o) such that
2 l~o(X) I
= ~o(X)
is the initial d i s t r i b u t i o n of the process.
(3.19)
64
Conversely, if ~ is a solution ~ithout nodes, that is 2 t~(x,t) I > O for any ti~e, of the Schr~dinger equation (3.16) with initial condition
(3.19) R+
~(x,t)
with
S
= e
2
~
S
= V~ e
(3.20)
2
P = I~ I
Introducing
now
u = o
u 2
and
v
by (3.21)
VR
and (3.22)
v=?S u
and
v
verify equations
(3.14) and
(2.63) the forward and backward drift conservative
diffusion process
the initial distribution Moreover,
ma (~t) To illustrate
=
b+
and
~t
w i t h density 2 l~(x,O) l
being
the process
(3.15). Then we can define by b_
and consequently a 2 = [0(x,t) I ,
p(x,t)
satisfies the stochastic Newton law
- W(~t).
(3.23)
the construction
of a diffusion
associated with quan-
tum e v o l u t i o n , we consider the simple case of the one dimensional monic oscillator.
The evolution ~2
i~
~ =
22
2m ~x2
Let us consider the solutions
~q0,Po
is described by the Schr6dinger
~ (x,t) = (m-~)~-I/4 exp [ _m~
1
0 + ~ m~2x20
given by the coherent
har-
equation (3.24)
states
~ t] (x_a(t))2 + ~ xp(t)- ~i p ( t ) q ( t ) - 1.~ i3.25)
associated with the classical {q(t),p(t)}
solution
{q(t),p(t)}.
More precisely,
is the solution of the Hamilton equation
(1.28)
for the
classical Hamiltonian 2 H =
1 2 2 + ~ m ~ q
(3,26)
65
Hence
m The c l a s s i c a l time
t = O
'
solution
-
q
associated
with
(3.27)
" initial
condition
(qo,Po)
at
t a k e s the f o r m Po = qo c o s ~ t + ~w - - sin~t
q(t)
(3.28) p(t)
The s t o c h a s t i c 9qo,Po(X,t)
= -rnmqosin~t + Po c o s m t
process
~t
to the c o h e r e n t
state
, has the d e n s i t y p(x,t)
and the
associated
.
function
S
~ -~/2 e x p [ - ~ (~-~)
=
is g i v e n b y the = I (xp(t)
S(x,t)
(x-q(t)) 27]
following
- ~I p(t)q(t)
(3.29)
expression - I ~t)
(3.30)
where we have chosen 2
U s i n g eqs.
(3.21)
=
and
~_ m
(3.31)
(3.22), we d e d u c e
u(x,t)
= -~(x-q(t))
v(x,t)
= ~ p(t)
(3.32)
and
and by
(2.63) we o b t a i n
h+(x,t)
Therefore,
I
(3.33)
for the d r i f t t e r m s
= v(x,t)
the a s s o c i a t e d
± u(x,t)
stochastic
I = ~ p(t)
equation
± ~(x-q(t)
for the p r o c e s s
(3.34)
~t
is
g i v e n by
d~ t = ~ where
Wt
p(t)
- ~ ( ~ t - qCt))
is the W i e n e r p r o c e s s w i t h v a r i a n c e
dt + 1 .
OW t
(3.35)
66
To solve
this
equation,
let us
introduce
~to
the process
such that
O ~t = q(t) From
(3.29)
given
we deduce
+ ~t
t h a t the
(3.36)
" distribution
Po(X)
and f r o m
(3.35)
differential
we
=
-I/2 - T x
(~)
e
(3.37)
see t h a t t h e p r o c e s s
Therefore,
o
the p r o c e s s
(Section
II,6).
the h a r m o n i c
~t
that
This
in t h i s
for a n y time. strictly
of t h e d e n s i t y
process
__~M -I/4
at t i m e
described
the g r o u n d
state
of
=
[ me exp
the d e n s i t y case,
2
[-~-77 ~ x
e + i
p(x,t)
] t
2
(3.39)
J
is s t r i c t l y
the condition
Po(X)
zero does not assure the 2 T@(x,t) t for f u t u r t i m e
=
strict
positive 2 l~o(X) l
positivity
(t > O)
oscillator
. This
fact
by choosing
state
Go = - 1 / 4 =
process
with
in the c a s e of t h e h a r m o n i c
~o(X)
first
stochastic
(3.38)
is a s s o c i a t e d
(me)
example
p(x,t)
c a n be i l l u s t r a t e d
where
the
.
is just t h e o s c i l l a t o r
In t h e g e n e r a l
positive
as i n i t i a l
verifies
oscillator
~°(x,t) Notice
~
equation
o
Po(X)
and
2
~
d~ t = - e ~ t dt + m-- d W t
the
is i n v a r i a n t
by me
in
o ~t -
of
!
= ~
(Go(X)
e- x2/2
excited
l~o(X) 12
state.
+ i~I (x))
(3.40)
is the ground s t a t e and At time
is s t r i c t l y
t = O
positive,
~1 = (¼)-]/4xe-X2/2
the d i s t r i b u t i o n but
at t i m e
t
the density
2 p(x,t)
(where w e h a v e
chosen
- e
-x [x 2
- V~--x s i n t +
m = M = e = I) c a n v a n i s h
~llJ
(3.41)
at t i m e
t = ± ! 2
(mod 27). Therefore,
a node
appears
in
I I x = - - or x = - - -
respectively! Hence trivial
it is n a t u r a l
nodal
at e a c h h a l f
period
V~ to c o n s i d e r
set a n d v a n i s h e s
for
the
case where
some v a l u e s
of
x
p(x,t) and
t
has
a non-
. In t h i s
67
case,
log p
and,
consequently,
the oslaotic v e l o c i t y
u -
~z
Vp
2
P
are
only d e f i n e d on the c o m p l e m e n t of the nodes. In Chapter IV we will show that under rather large c o n d i t i o n s such a process w i t h singular drift exists and that the nodes are never attained.
In other words,
the nodes act as an impenetrable barrier.
III.4 C o n s e r v a t i v e N e w t o n i a n Processes w i t h S t a t i o n a r y D i s t r i b u t i o n In the p r e v i o u s example, we have c o n s i d e r e d the o s c i l l a t o r process w i t h s t a t i o n a r y distribution.
Let us h o w s t u d y the general feature
of p r o c e s s e s with stationary d i s t r i b u t i o n i.e.
Z--~P (x,t) 3t
Hence
p
= 0
(3.42)
does not depend on time
p(x,t)
As a consequence,
=
Oo(X)
(3.43)
the osmotic v e l o c i t y in the region where it exists is
time independent 02 u (x)
Using the Ansatz
-
Vp(x)
2
(3.44)
P (x)
(3.9) ~S(x,t) (x,t)
and
=
~
(3.45)
e~
(3.42) the continuity e q u a t i o n
(3.1)
takes the following form
V o (O q S) = 0
and the e q u a t i o n
(3.7)
9S
rewrites
4
~t
(3.46)
1
2 Alogp + ~
4 (VS) 2 - -~- (Vlogp) 2 + m ~ = O.
The system of coupled partial d i f f e r e n t i a l equations
(3.46),
(3.47)
(3.47)
with initial c o n d i t i o n
S(x,O)
= S
o
(x)
admits a solution of the following form
(3.48)
68
E = - { t + So(X)
S(x,t)
(3.49)
if and only if the e q u a t i o n 2 (~-- A-V) ~
=
E
(3.50)
admits solutions of the form i - - So(X) e O2
#(x) = ~ - ~
Indeed,
for
S(x,t)
4
4
m
for
x
such that
given by
Alogp - %
4
(3.49),
(3.51)
(3.50) rewrites
I
(Vlogp) 2 + [
(VSo)
2
! v
- m
(3.52)
p % 0 .
The c o n t i n u i t y equation becomes
AS
+ V_pp VS = O. p o
o
Hence
(3.53)
i2 So (x) (x) = ~
is solution of
eq
(3.50) and E - i ----~ t (3.54)
is a s t a t i o n a r y solution of Conversely,
given
(3.16).
(3.54), then
(3.51)
is solution of
splitting in real and imaginary parts, we get the eqs.
(3.50). By
(3.47) and
(3.48). Moreover,
the current v e l o c i t y does not depend on ti~e
v(x)
= VS o(x)
and the same occurs for the forward and b a c k w a r d drifts b_(x).
(3.55) h+ (x)
and
69
No~, the a s s o c i a t e d process
~t
v e r i f i e s the stochastic differ-
ential e q u a t i o n
3.56)
d~ t = b + ( ~ t ) d t + odW t .
It is a h o m o g e n e o u s M a r k o v i a n process with stationary distribution, hence
~t
is a s t a t i o n a r y process.
Let us remark that only the drift a s s o c i a t e d w i t h the strictly p o s i t i v e ground state w i l l be non-singular.
Each excited state since
o r t h o g o n a l to the ground state, will have a non-trivial nodal set. An interesting situation is the case where the nodal surfaces split the configuration
space in closed disjoint domains. The t r a j e c t o r i e s of
the process are trapped in one of the domain.
Such a d i f f u s i o n process
furnishes a model of c o n f i n e m e n t by i m p e n e t r a b l e barrier. We can understand this p r o p e r t y in a h e u r i s t i c way. The osmotic v e l o c i t y tisfies
(2.62),
surface
u(x)
02
= --2 71ogp(x)
p(x) = 0 . However,
hence has a singularity on the
the region w h e r e the function increases. T h e n
i n t e r p r e t a t i o n of
b+(x)
u(x)
, hence
b+(x)
If we remember that the h e u r i s t i c
is the m e a n v e l o c i t y of p a r t i c i e s w h i c h leave
x , then typical t r a j e c t o r i e s of
nodal surface.
sa-
the g r a d i e n t of a function points towards
points outside of the nodal surface.
the point
u(x)
In the next
~t
are r e p e l l e d by the
section, we will c o n s i d e r the stationary
case and prove that the nodal set is indeed never reached.
III.5 U n a t t a i n a b i l i t y of the Nodes for S t a t i o n a r y D i f f u s i o n P r o c e s s e s In this section we limit our c o n s i d e r a t i o n to the case of a s t a t i o n a r y M a r k o v d i f f u s i o n process
~t
solution of the stochastic
differential equation
d~ t = b+ where
Wt
(~t) dt + dW t
is a W i e n e r process
in
and w i t h a s t a t i o n a r y density that the
L loc I
c h a r a c t e r of
~
(3.57) with convariance matrix
p > 0 , p 6 C , p(x) > 0 p
a.e.
~ t
(Notice
does not exclude densities w h i c h are
not p r o b a b i l i t y densities but define s t a t i o n a r y m e a s u r e s ) . M o r e o v e r we assume in the f o l l o w i n g that the drift gradient,
h+
is a
namely
b+(x)
I
= ~ Vlogp
.
(3.58)
70
The
stationarity
b+(x)
implies
= -b_(x)=
that the c u r r e n t v e l o c i t y
u(x)
then
~t
is a s y m m e t r i c
vanishes, diffusion
v = O
and
process
(see A p p e n d i x ) . The d i f f u s i o n
p r o c e s s we c o n s i d e r I
is a s s o c i a t e d
w i t h the g e n e r a t o r
I
A = ~ A + -- ( V . I o g p ) - V 2 outside
the o p e n set
N£
(3.59)
defined by
N£ = {x 6 IRd Ip < e , 6 > O}
The d r i f t the p r o c e s s
b+
and the g e n e r a t o r
in the c o m p l e m e n t
of
a large c l a s s of d i s t r i b u t i o n s
lim e%O
TN
=
3.60)
A
are w e l l d e f i n e d
N£
. Our aim is to p r o v e
p , with probability
and we c o n s i d e r that,
for
one
~
(3.61)
where TN
= inf {t > O1~ t 6 N e}
is the first h i t t i n g time of
Ne
I n s t e a d of c o n s i d e r i n g let us i n t r o d u c e
p ~ I
(symmetric)
singularities
to
C3.53),
and
p > O
(~)
a.e.
of
A
defined
in
E
;f,g 6 C ~o (~d) r
(3.59)
(3.63)
P dx)
form • Vg p dx ; f g 6 C ~ ' o
formula
and d e g e n e r a c y
(~)
Vf
form
just the c l a s s i c a l
Let us n o w a s s u m e
This c o n d i t i o n
energy
] ~ = ~ d
we r e c o v e r
In c o n t r a s t
bilinear
= -(A f, g)L 2, ~d
F(f,g) For
(see A p p e n d i x ) .
the l i n e a r o p e r a t o r
the a s s o c i a t e d
E(f,g)
the so c a l l e d
(3.62)
£
Dirichlet
(3.64)
allows
(~d) .
(3.64)
integral. discontinuities,
p
that the f o l l o w i n g
condition
on
p
is v e r i f i e d
2
(~d~ the first w i t h [~I 6 H ioc ] S o b o l e v s p a c e (i.e. T~I and VI~I b e l o n g to L2 ( ~ d ) ) and 141 > O a.e. loc 0 =
141
is e a_u i v a l e n t
to
p 6 Llo c
(~d)
np
V__~p 6 L ~ o c ( ~ d
pdx)
71
Under the cond~tlon C ~ (~d) o
(~')
is closable
The symmetric E(fn-fm;
the syyor~etric for/~ (.3.64) defined
in
L 2 ( ~ d, p dx)
form
[
in the following
is called closable
gn-gm ) ~ O, n, m ~ ~,
in
on
sense
:
L 2 ( ~ d, p dx)
if
[(fn,f n) ~ O, n ~ ~ . To be closable
J d p~dx) O, n ~ ~ implies (~ , is equivalent with the following
property:
with respect
L 2 ( ~ d, p dx)
is complete
[1(f,g) Indeed condition a sequence [(fn,g)
~ O, n ~ ~
Moreover
= [(f,g)
(#9)
such that
implies
(f,g)
~d
associated
capacity
[8],[50,a,b],[94]. operator
that the domain
H
of H
(3.65)
{fn}n £ ~
,fn 6 C ~o (~d)
P dx)~ O, n ~ ~
(#)
More precisely
[
follows.
a unique diffusion
up to a set of zero
there
is a unique positive
associated to the closure (the domain
= (H f'g)L 2 ( m d, p dx)'¥ f 6 D(H)
is
then
there exists
to the form
D(H)c D ([)
[i
L 2 ( ~ d, p dx)
that if
under the condition
on
to the metric
for any g £ ~oo (]Rd)' &~d the closability
~t
[(f,g)
+
(fn'fn)L2(~d,
process
self adjoint
J (fn'fn)L2
of
[ [)
of
E
and such
and
, g £ D(E).
Given the positive self adjoint operator H we can construct a diffusion process with values in lRd, with transition function Pt symmetric in L 2 ( ~ d t p dx) such that - t H Pt f = e Proof of these
pdx)
,
f 6 L2(IR d, p dx)
facts can be found in [8],
The capacity measure
f
Cap(0)
is defined
[50,a],
(3.66)
[10~4.
for an open set
0
(w.r.t.
the
by
Cap(0)
= inf{[l(f,f):
f 6 L 0 = {f 6D([)
f > 1,a.e.on 0}} (3.67)
where [1(f'f) and
D(E)
is the domain
est closed extension taken to be is defined
= [(f'f)
+ ~
of
if
of the
+
(f'f)L 2(IR d, p dx)
(regular)
~ , denoted by
Dirichlet E
again).
L 0 = @.) For an arbitrary
set
(3.68) form
[
(the small-
(The infimum B
is
the capacity
as Cap(B)
= inf{Cap(0)
:
0 open
0 m B}.
(3.69)
72
The ~ini~
~n
which minimizes One has
~I
0 < e
The ~ n n c t i o n
(3~67) , this
< d e0
and
T0
t > 0
0
[50,a,b].
has zero c a p a c i t y
Remark
0
x £ ~d
from
it follows that
x
at time
O
that an o p e n set P(~t 6 B
for some
B.
Dirichlet
integral
141 6 H lI o c ( ~ d ) , 141> O
an o p e n set
to
starting
From this
is l o c a l l y a E - q u a s i c o n t i n u o u s
141
(e0,e0).
(0 = I)
the e q u i l i b r i u m
is h a r m o n i c
on
~d ~ 0
I on 0 .
Consider
exists
"equilibrium potential".
(3.70)
is g i v e n b y a f u n c t i o n w h i c h
and ~ i d e n t i c a l l y
e0 £ L0
probability
iff the p r o b a b i l i t y
for any
F o r the c l a s s i c a l of
function
= ~ x ( e -T0)
(see
I~ 0 = x) = 0
potential
= E [ e 0 , e 0] +
is the first time the p r o c e s s
h i t s the set B c ~d
f u n c t i o n b e i n g the Cap(0)
is e q u a l to the h i t t i n g
e0(x)
where
is as s u ~ e d b y a u n i q u e
~d~B
B
with
is c o n t i n u o u s )
just the D i r i c h l e t
f o r m for
function
Cap(B)
< g
with
a.e.
and such that
(namely for any
I~I
g > O
there
such that the r e s t r i c t i o n
finite Dirichlet
integral
of
(that is
p = I).
Let N = {x 6 ]Rd
and
~t
be the d i f f u s i o n
Theorem
3.1:
f r o m above,
l~(x)
= O}
process
associated
U n d e r the a b o v e c o n d i t i o n ,
TN
if
to
141
with
p =
141
is l o c a l l y b o u n d e d
is the first h i t t i n g
q.e.,
time of
x 6 ]Rd
(3.72)
N
(3.73)
T N = i n f { t > O l ~ t 6 N} and q.e. Proof:
(quasi e v e r y w h e r e
means
" e x c e p t e d on a set of c a p a c i t y
U s i n g the r e l a t i o n b e t w e e n
it s u f f i c e s
2
then
P[T N < + ~] = 0
where
(3.71)
to s h o w Cap(N
(see [50,a,b])
the c a p a c i t y
and h i t t i n g
time
that
n B r) = O , B r = {x 6 ]Rd !IxI < r} V r > O.
0"). (3.70)
73
Consider on
a function
B r • and
with
f £ C~
f = O
B r+l c
such that
0 < f < 1
(the complement
g(x)
= logl~(x) I . f(x)
ge(x)
= log[l~(x) Ive]
f v g = max
The assumption
on
(•~d)
of
• f(x)
with
f = I
Br+ I) and set
E > O ,
(f,g).
(%#)
on
I01
Since
1~I 6 Hloc(IRd) , I01
logl~l
6 L2(~d~, l~12dx)
implies belongs
It follows
that to
ge
is E-quasi
L ~ o c ( ~ d)
for
from this that
continuous.
p > 2
and
g 6 L 2 ( ~ d ,I~i2dx)
and IVgl21~12dx d
~ 2 [ IVl~112f2dx ~d
The same property
holds
for
e~olim ~Rd l~gel2
By regularization
+ 2 [ IVfI2(logi~l)21~2~dx j~d
gs . Moreover,
< + ~.
we have
l~12dx = I~d IVgl21@12dx
it is easy to see that
gc
"
is a E~-limit
of
C~-function and hence gE is a quasi-continuous function on the Diricho let space D(EP). We can therefore write (by using a Chebyshev's type inequality)
Cap(Ig£1 and letting
E + O
I
> I) < ~
(3.74)
E1(ge,gs)
we obtain
Cap(llog~l (x) II If(x) l > I) _< ~
_
and since
Cap(NflB r)
is smaller
than the left hand side of the above
equation,
the theorem
is proved.
Remark: cess
Xt
If the condition
(~)
IVgl
[~21dx +
is not verified,
can reach or cross the nodal
surface
I%1
O
(3.75)
the paths of the proof
I~I.
(See e.g.
[5O,5].) Consider that zero.
the case where
d = J , p 6 L]oc(~)
non negative
such
inf p(x) > O if O ~ a ~ b ~ + ~ then p(O) can take the value a<x~b The origin is unattainable from the right if and only if
74
b
i (see eg If
p
dx p(x)
- + ~
(3.76)
' b > O
o
[ 8]). satisfies
the c o n d i t i o n b
(~)
then we have
'2
I
I~
(3.77)
dx < +~
o
and using
Schwartz'
inequality
I I~T from which
shows
Diffusion
that
that
sider The
in p r e s e n c e the
FI
q
acceleration
is t h e
the c u r r e n t haviour a scalar Let
A
field
takes
c a s e of a d i f -
L e t us n o w c o n -
the
is p r e s e n t . form
(3.79)
in
(2.94).
We
assume
the
force
field.
(2.60).
which
This
b e the v e c t o r
E
and
The velocity last
Moreover,
accounts
(3.80)
+ v(x,t) xB(x,t) ]
of the p a r t i c l e .
time reversal.
potential
m
the
force.
electromagnetic
defined
= q[E(x,t)
charge
velocity
as
force
and magnetic
under
+ ~
Field
considered
of m a s s
to
= F I (x,fi) + F2(x)
F I (x,t)
the e l e c t r i c
we have
an e x t e r n a l
to be the L o r e n t z
where
goes
(3.78)
is u n a t t a i n a b l e .
Electromagnetic
l a w for a p a r t i c l e
a is t h e
I
side
the o r i g i n
sections,
ma(x,t)
where
(b) - loql~l (e) j
of a n e x t e r n a l c o n s e r v a t i v e
case where
dynamical
the r i g h t h a n d
indeed
that.
[logier
I~Pl 2
in an E x t e r n a l
In t h e p r e v i o u s fusion
conclude
--!--I dx > e
it f o l l o w s
e ~ 0 , which
III.6
dx
we
assumption F2(x)
for t h e
potential
B
and
assures
possible
the
respectively,
is c h o s e n
= -VU(x)
other ~
are,
v(x,t)
scalar
to b e
the usual
where
U
be-
is
forces. potential,
then
B
=
V
x: A
(3.81) E
~A ~t
75
If f u r t h e r m o r e a function
we m a k e
S(x,t)
the g a u g e
invariant
assumption
that there e x i s t s
such t h a t (3.82)
mv + qA = V S
we c a n r e w r i t e
the c o u p l e d aS ~t
m~ 2
s y s t e m of n o n - l i n e a r
4 (?logp
I/2 2 )
m~ 2
4
equations
(3.1),(3.2)
I/2 ~logp
2 + I 2m
-
the l a t t e r
(?S - qA)
+ U + q¢ = 0
I/2 i/2 ap + vp ovs at
I/2 +
-
equation
If f o l l o w i n g
p
(III.3)
solves
= p
= 0
(3.84)
equation.
i (x,t)
- qAoVp
w e m a k e the A n s a t z
1/2 (x,t)
i/2 As
is the c o n t i n u i t y Section
(3.83)
S (x ,t)
e m~z
(3.85)
the l i n e a r e q u a t i o n
a~ imo 2 ~-~(x,t) =2-~ 2 ~(-imo2V-qA) ( x , t ) ~
+
(U(x,t)
+ q~(x,t))~(x,t) (3.86)
V i c e versa,
given
the f o r m u l a
(3.75)
diffusion
defines
Diffusion
(~,g)
g
Let
in
~
a n d its c o n t r a v a r i a n t
type e q u a t i o n ,
and the d r i f t
b+
then
of a
A t , which
d-dimensional
, the c o m p o n e n t s g 13
Riemannian gij
manifold.
of the m e t r i c
verify
i ~k
=
(3.87) index.
, be a d i f f u s i o n
description
Manifold
components
the sum is t a k e n on r e p e a t e d X t , t 6 T c JR+
The a n a l y t i c tor
p
on R i e m a n n i a n
system
'' gZ3 g j k where
the d e n s i t y
be a s m o o t h o r i e n t e d
In a local c o o r d i n a t e tensor
of this S c h r ~ d i n g e r
process.
III.7 N e w t o n i a n Let
a solution
of
assumes, on
Xt C~(9~)
I A t = ~ ~g + b+
process with values
in IM
is g i v e n b y its i n f i n i t e s i m a l
• grad
functions
the
following
.
genera-
form
(3.88)
76
where
b÷. g r a d = ~ i ~---i = g.' h i
non random
C
-vec£or
ly on the time
field,
=
Agf
for any s c a l a r
The connection
=
between
Xt
to
Xt+At
Y~t
Xt+At and
b+
, with
Xt .
I~
In local c o o r d i n a t e s
length
b+(Xt,t)
we
(3.89)
and
~• i ~x z
(3.90)
and the p r o c e s s of
= lim At+O
means
Xt
at time
(At)-1
is that t
b+(Xt,t)
expectation to
Xt
(3.91)
with respect
tangent
to
to g e o d e s i c s
e q u a l to the g e o d e s i c s
is also the f o r w a r d
is
in the sense that
~ [yi At I Xt = x]
conditional
IYAtl
Xt
stochastic
from
distance
of
derivative
[29],[40],[85,f],[90,e].
dx = ~
dx I .... dx d
Due to the a s s u m p t i o n p(x,t)
f ,
on
is the v e c t o r a t t a c h e d
in the sense of Let
is a
depend explicit-
g z3 ~jf)
forward derivatives
~ [.IX t = x]
Xt = x .
which might
det[gij]
b+i (x,t)
where
operator
V~ ~ i ( ~
function
g
(mean)
the dg~ft,
he
t .
A is the L a p l a c e - B e l t r a m i g have
the
±n local c o o r d i n a t e s ,
of the law of
be the R i e m a n n i a n
we k n o w that t h e r e e x i s t s Xt
with respect
to
dx
volume
e l e m e n t on
~.
a smooth density , i.e.
d P ( X t E dx)
=
p(x,t)dx. Let
f 6 C~(M)
, then
~ [f(Xt)]
= SMf(x) o ( x , t ) d x
and
~ t ~ [f(Xt)]
=
SMf (x) ~~,P I ~ ..t ) d x . On the o t h e r h a n d , by the d e f i n i t i o n of L t , the left h a n d side is e q u a l to SM(Atf) ( x , t ) o ( x , t ) d x . By p a r t i a l i n t e g r a tions we a r r i v e
at the K o l m o g o r o v
forward
equation
(Fokker-Plack
equa-
tion) I Agp ~---{- p = ~-
where A t* f
d i v V = V~ ~i V~ Vl
d e f i n e d on
j o i n t of
C2
for any s m o o t h v e c t o r
f u n c t i o n by
A t* f
=
~I
field
Agf-div(b+f)
V
.
b e i n g the ad-
At .
Let us n o w d e n o t e by X t , i.e.
(3.92)
div (b+ p )
V
X_t
v X t , t E -T
has the same law as
,
the time r e v e r s e d p r o c e s s
X t . It is w e l l - k n o w n ,
see e.g.
to
77
[41] and s e c t i o n tesimal
v Xt
II.5 t h a t
b e i n g the
Y ~•A t
defined
the b a c k w a r d above,
infini-
. grad
"backward drift"
b~(x,t)
with
process with
generator v I - b A t ~ ~ Ag
bi
is a g a i n a M a r k o v
as
stochastic
one a r r i v e s
defined,
= lim At+O it YA
(3.93)
for
t £ T , by
CAt) -1 E [Y~At I X t = x]
with
derivative
-At of
replacing
At
(3.94)
" Then
b-
is
X t . By the same p r o c e d u r e
at the F o k k e r - P l a n c k
equation
for the r e v e r s e d
as pro-
cess t £ T: I - ~--~ p = ~ ~g p + d i v ( b _ p ) .
As in the E u c l i d e a n u
case, we c a n i n t r o d u c e
a n d the c u r r e n t v e l o c i t y
As in C h a p t e r
I u = ~ (b+ - b_)
(3.96)
1 v = ~ (b+ + b_)
(3.97)
II, we c a n d e r i v e
"osmotic
the
"continuity
the o s m o t i c v e l o c i t y
follows
manifold.
Let
(3.98)
Ag p = div(pu)
I u = ~ grad
This
equation"
equation" I
Moreover,
the o s m o t i c v e l o c i t y
v
~P - - d i v (pv) ~t a n d the
(3.95)
logp
(3.99)
u
is g i v e n by
.
f r o m the g e n e r a l i z a t i o n f,h £ C~ o
(3.100)
of f o r m u l a
(~ .52) to R i e m m a n i a n
(Tx~V~ t h e n
0 = IT~t ~ [f(Xt,t)h(Xt,t)]dt (3.101) = IT ~ [ ( D + f ( X t , t ) ) h ( X t , t ) ] d t
+ IT~[f(Xt,t~D-h(X~t)}dt
78
~-~ ÷ ~I Ag ~
where
D+ ~ •
functions)
D+
we a r r i v e + grad this
D_ ~ ~-~ - ~ Ag + b _ . g r a d to act on
We shall
equation
. To do this,
integrations
equation
(3.92),
field
on
~
acceleration
like
to h a v e
of v e c t o r s and Guerra , then
(v.u).
on
associated
the
concept
~ . The
[39],
forward
with
Let
the p r o c e s s
of m e a n
appropriate
[90,c].
the m e a n
(3.102)
forward
and
definition
has
F = Fi(x,t)
derivative
of
be a
F
is de-
by
D+F(x,t)
where
~ lim At+O
(At) -I E[m~ ~ at'At+At
my,y+AyF
y+Ay
obtained
parallel briefly
y(s)
that
is the v e c t o r
T
way
Yt(s)
a parallel
in the
of this
(so)~. and
Let
tangent
transport
hit)
~(0)
family
for
Ty+Ay~
at
terms
= F
such
of g e o d e s i c s
. Let
that
, getting Yt(So)
see
on
Let
~.
be a c u r v e
us t r a n s p o r t
h(t)
details
on
[39]). Fi
~
be
such
in a L e v i - C i v i t a
a vector
= h(t),
field
G(t).
it(s O ) = G(t),
of g e o d e s i c s
s = so
y(s),
my(se),y(s)F order
space
(for m o r e
, t 6 [0,1],
along
be the g e o d e s i c
the t r a n s p o r t
F(Xt,t) IX t = x] (3.103)
F 6 T ~ by D o h r n - G u e r r a ' s s t o c h a s t i c Y the g e o d e s i c s f r o m y to y + A y . We r e c a l l
+(s o) ~ G(t)
along
second
t+At)(Xt+~t'
, s o < s < s I , be a s e g m e n t in
s < s I . The
field
along
definition
h(O) = Y(So)
parallel
F
f r o m the v e c t o r
transport the
a vector
and
we w o u l d
by D o h r n
div v - g r a d
the m e a n
derivatives
given
vector
Let
partial
(3.98)
grad
now define
backward
Let
. Using
Fokker-Planck's
on
f r o m t h i s to the c o n c l u s i o n that -D_ = - ~--~ - b+ .g r a d I + ~ A . Using b = D_X t w e then get (3.100). F r o m g (3.100) we have, t a k i n g the t i m e d e r i v a t i v e and u s i n g
equation
fined
and u s i n g
derivative
easily
~u ~t
been
hp
of m e a n
(Logp).grad
the c o n t i n u i t y
Xt
(the o p e r a t o r
J
and
to b r i n g
b÷lgrad
with
. Let
B(s O) = F
differs in
Ty,y+AyF
{Yt(s), s o < s < s I, t 6 [O,1]} is d T B(s) = ~-~ Yt(s) It=O; this is a v e c t o r . By d e f i n i t i o n ,
f r o m the
s . This needed
Levi-Civita
then in
gives,
(2.11).
for One
TY(So),Y(s)F
displacement
of
~B(s) F
by
y = Y(So) , y + A y = y ( s ) computes
easily
D+ = 7 t + b + . g r a d
I + ~ ~DR
(3.104)
D_ = ~ t + b _ . g r a d
- ~I A D R
(3.105)
79
ADR ~: ~+R
being the Laplace-de
being the Ricci tensor,
Rham-Kodaira
Laplacian
on
~
, R
acting on vectors.
is given by A
where
Di
=
D.
Di
l
(3.106)
is the covariant derivative,
defined on vector
field
F
by
k ~ F j + ri DiFJ _ ~xi jk F
and
ri jk
age the Christoffel
(3.107)
symbols associated with the metric tensor
r~. kh 13 = g Fh,ij
(3.108)
I ~ Z rk,ij = 2 ( -~x - i gkj + ~X "" 3 gik R
is the Ricci tensor,
rk
~xJ
As in the case where a (Xt,t)
ki
-
rI
~x k
ji +
3
k
ki - Fkl
31
(3.110)
IM -- ]Rd , we define the mean acceleration
by
a(Xt,t) Using
(3.109)
defined by
=
Rij
~ gij) ~xk
I
=- ~(D+D_ + D_D+)X t .
D+X t = b+(Xt,t)
a(Xt,t)
=
and
D_X t = b_(Xt,t)
I = ~ D+b-(Xt,t)
l(~t+b+
we get
1 + ~ D-b+(Xt,t)
-+I -v-u I I v+u .grad ~ADR)--~-+y(~t+b_.grad-~ADR ) y I
= (~t v + v o g r a d v -
u.grad u - ~ A D R U ) ( X t , t )
~v ~-~ = a + u-grad u -
v-gradv+
(3.111)
hence I ~ADRU
Let us also remark that a purely p r o b a b i l i s t i c cess is qiven by the solution of the stochastic (in ItS's sense)
.
description
(3.112) of the pro-
differential
equation
80
(3.113)
dX t = b + ( X t , t ) d t + dB t ,
which
Bt
the standard B r o w n i a n motion on
standard B r o w n i a n m o t i o n
wt
in
~d
JM, w h i c h is related to the
by the following ItS~stochastic
differential e q u a t i o n
(3.114)
dBt = m i ( B t ) d t + o ~ ( B t ) d w k
m
i
i ok
and
being given in terms of the tensor m e t r i c by the e q u a t i o n i
I
m
ik Fi jk
=-~g
(3.115) ik o Of course,
i oj
Okj =
this is not an intrinsic description,
for such one see
[68],
f85,b]. Given
b+
, we can under suitable a s s u m p t i o n s construct
p , hence over, given
b_
and hence Xt
u
and
v
satisfying
and its d i s t r i b u t i o n
p
(3.95),
we can get
b+
mean forward resp. b a c k w a r d d e r i v a t i v e s and then get (3.95),
(3.99), a
Now,
being the m e a n a c c e l e r a t i o n of
in the
Same spirit as in section
servative N e w t o n i a n diffusion. mean,
If
Xt
Xt
V
on
~x:~
and
u,v
b_
as
satisfying
X h-
(3.2), we can define a con-
satisfies N e w t o n ' s law in the
in the sense that there exists a positive constant
valued function
and get
(3.99). More-
m
and a real-
such that
(3.116)
m2 (D+D_ + D_D+)X t = - grad V (Xt)
and such that in addition the c o r r e s p o n d i n g current v e l o c i t y is a g r a d i e n t field
V(x,t)
(3.117)
= grad S (x,t)
we will say, as in the E u c l i d e a n case w h e r e
~=
~d
, that
X
t
is
a
N e w t o n i a n diffusion. Introducing the complex function on the m a n i f o l d
(x,t) = ~p(x,t)
e
iS(x,t)
(3.118)
81
it c a n
easily
he
shown
that
~
verifies
the
partial
differential
equation
~
where Vice
A
(x,t)
versa,
if
~
and define
u
(3.95)
satisfy
and t h u s
tion
and
the
of w h i c h
t = O
Finally,
the
p
chapter,
we w i l l
+ V~(x,t)
operator
of
(3.106)
u = g r a d log p
(3.99).
stochastic
From
u
equation
for all t i m e s
on
and
and and
(3.119)
if we w r i t e
v = grad S , then v
we can get
for a p r o c e s s p(x,t)
~ = v~-e iS
=
Xt
u
and
in p a r t i c u l a r
, the d i s t r i b u -
l~(x,t) l 2 , if at time
l~(x,O) l 2. M o r e o v e r ,
the p r o c e s s
satisfies
in the mean.
of n o d e s
density
space
by
let us r e m a r k
same m e t h o d s
attainability
v and
distribution
equation
the p r o b l e m s
is s o l u t i o n
is t h e n
it h a s
Newton's
(x,t)
is the L a p l a c e - B e l t r a m i
g
v b+
= _ I
as
can b e
give
be r e a c h e d in the
of the n o d e s
in the
case
investigated
in S e c t i o n
can never
of the p r o c e s s
that
(3.5).
in the
which
stationary
In p a r t i c u l a r ,
by Newtonian
non-stationary works
is a R i e m a n n i a n
of R i e m a n n i a n
also
a proof
in the
manifold.
case
the n o d e s
diffusion.
case
manifold along
of the
In the n e x t of the un-
case w h e r e
the
state
IV. GLOBAL E X I S T E N C E FOR DIFFUSIONS W I T H S I N G U L A R DRIFTS
IV.I I n t r o d u c t i o n Within the context of d i f f u s i o n processes, mathematical
literature
(see e.g.
[55],
[68],
the main bulk of the
[104]) discusses the
question of existence and uniqueness of solutions of stochastic differential equations in ItS's sense under assumptions very r e m i n i s c e n t of those for d e t e r m i n i s t i c d i f f e r e n t i a l equations. of the stochastic d i f f e r e n t i a l e q u a t i o n characteristics condition
b+
and
Usually the coefficients
(the so-called i n f i n i t e s i m a l
b_ ) are r e q u i r e d to satisfy some r e g u l a r i t y
(such as a Lipschitz condition)
to ensure local existence and
uniqueness of a continuous solution and a growth condition on the drift b+
is imposed to avoid explosions,
off to i n f i n i t y within
finite time.
i.e. to avoid the process of m o v i n g Both from a m a t h e m a t i c a l p o i n t of view
and a look towards applications in other fields, such as physics or biology,
it is of interest to relax the standard conditions.
ical situations,
There are phys-
e.g. o c c u r r i n g in stochastic mechanics, where one de-
s i r e s to construct diffusions with extremely singular drifts but wellbehaved B r o w n i a n path. In Chapter III, we have i n v e s t i g a t e d the p o s s i b i l i t y of singular drift in the case of stationary processes, richlet forms.
using the properties of Di-
In this Section, we consider the case where
p
is not
n e c e s s a r i l y stationary and a pure p r o b a b i l i s t i c approach in terms of stopping time is used. The p r o b a b i l i s t i c a p p r o a c h has the advantage of e x h i b i t i n g e x p l i c i t l y that the diffusions Let us consider a wave function dinger equation
~
avoid nodes of the density. in
~d
satisfying the Schr~-
~2 g ~
= i 2mm A~ - iV~
w i t h initial condition
~(x,O)
(4.1)
= Co(X).
We write formally ~(x,t) with
R
and
S
real.
= e R(x't)+iS(x't)
(4.2)
In this Section, we choose
w i t h o u t dimension,
then the current v e l o c i t y takes the form Then we define two vector fields b+(x,t)
= ~
v = ~ VS m b+ and b_
S
[Re V~ + Im ?~--~] = ~
by
(VR + VS)
(4.3)
83
b - (x,t) and a n o n - n e g a t i v e
= ~
[-Re
function
+Im
~
p(x,t)
=
z_~]
: ~
(4.4)
(-?R + ?S)
2 l~(x,t) I
b+
and
b_
are only
w e l l - d e f i n e d on the c o m p l e m e n t of the nodal set N = {(x,t) of
(4.1)
6 ~d × ~
l~(x,t ) = O}. Using the complex conjugate equation
it is possible,
fusion process to
p.
at least formally,
Indeed,
an easy c a l c u l a t i o n shows that
satisfy a F o k k e r - P l a n e k equation,
3t Thus,
=
2m
to a s s o c i a t e a M a r k o v dif-
Ap - div(pb+)
=
l~o(X) 12
must
(4.5)
if we c o n s t r u c t a d i f f u s i o n process having
p(x,O)
p
namely
b+(x,t)
as initial p r o b a b i l i t y density,
as drift and
i.e.
dX t = b + ( X t , t ) d t + dW t (4.6) P[X o £ A] = L for any Borei set
A c ~d,
p(x,O)dx
where
Wt
is a d - d i m e n s i o n a l B r o w n i a n motion
w i t h c o v a r i a n c e ~t i, then since the F o k k e r - P l a n c k equation (4.5) has a m unique solution for b+ and p sufficiently smooth, we know that the probability density of the d i f f u s i o n process
Xt
is just
p(x,t)
=
l~(x,t) 12.
F r o m this point of view, the p r o b a b i l i s t i c i n t e r p r e t a t i o n of SchrSd i n g e r ' s e q u a t i o n is very natural. guments are p u r e l y formal, (4.5)
exists:
indeed,
But u n f o r t u n a t e l y
all the above ar-
since we don't know w h e t h e r the solution of
b+(x,t)
has no m e a n i n g if
~(x,t)
= O. All the
classical theorems about the e x i s t e n c e of solutions for stochastic differential e q u a t i o n s could not be used d i r e c t l y in this case. On the other hand, p h y s i c a l intuition leads to the conjecture that the sample paths of the d i f f u s i o n process do not get trapped by the nodal surface of the density
(see Section III.4 for a h e u r i s t i c argu-
m e n t and Section III.5). This chapter is devoted to the p r o b l e m of c o n s t r u c t i n g the diffusion process w i t h nice B r o w n i a n part and singular drifts w h i c h are needed for s t o c h a s t i c mechanics.
Using a purely p r o b a b i l i s t i c a p p r o a c h
in terms of suitably d e f i n e d stopping geometrical
times as well as some physical and
ideas, the c o n s t r u c t i o n of d i f f u s i o n w i t h singular drifts
can be carried out w i t h r e l a t i v e simplicity. [15] and i n s p i r e d by
[17a] and
[90e].
This m e t h o d was used by
84
In the previou s
last section
of this
Let us first describe are given in
chapter,
we will
discuss
briefly
some
work in this field.
at each time
the p r o b l e m we have
t 6 ~+
= [0,~)
to solve.
a probability
Suppose we
density
p (',t)
IRd . Define
U = {(x,t)
and the nodal
6 IRd xIR+
d
b+ : U ÷ ~
the forward
> 0 }
(4.7)
set
N = U c = { (x ,t)
Let
I p(x,t)
6
IR d xIR+
be given and suppose
Fokker-Planck
equation
St@ = -div(Pb+)
I p(x,t)
that
on
p
= O } .
and
(4.8)
b + are related
by
U
+ yAP
(4.9)
2 =~
being
the d i f f u s i o n
and b+satisfyregularity drift near
b+ w h i c h
is defined
the b o u n d a r y
sion process
coefficient.
conditions.
Xt
of
on
U
with drift
we have
utions
of the stochastic
b+
to i n v e s t i g a t e
where
Wt
process over,
is a standard
exists
is m e a n i n g f u l
be very
density
and u n i q u e n e s s
p.
In other
of global
sol-
equation
(4.10)
process
with covariance
derivative
measurable
a
a diffu-
+ dW t
Wiener
if p
singular
in c o n s t r u c t i n g
and p r o b a b i l i t y
differential
its m e a n f o r w a r d
for all b o u n d e d
consists
existence
dX t = b + ( X t , t ) d t
equation
we are only c o n s i d e r i n g
and can therefore
U. The p r o b l e m
words,
This
However,
functions
is equal f
to
2vt ~.
If such a
b + and, m o r e -
and all Borel
sets
A
we have
P{f(X t) 6 A} = I dx
p(x,t)f(x).
(4.11)
A
IV.2
Existence
IV.2a
Heuristics Before
of N e l s o n ' s
going
through
to get some p h y s i c a l be d e f i n e d
globally
Diffusion
Processes
the m a t h e m a t i c a l
intuition
discussion,
about w h a t m i g h t p r e v e n t
and why this
is not
so.
it is w o r t h w h i l e a diffusion
to
85
Let us assume for the m o m e n t that the sample paths of the process
Xt
(4.6)
has a local solution,
i.e.
are d e f i n e d at least for some finite
time interval. What may p r e v e n t a t r a j e c t o r y from being d e f i n e d for all times? This can occur only if the t r a j e c t o r y approaches the nodal set w i t h i n finite time
and then the drift
b+
is u n d e f i n e d
if the p a t h escapes to infinity w i t h i n finite time
~d
(Fig.
N
1.a) or
(Fig. 1.b).
md
~
N
t
Fig. 1.b F r o m a p h y s i c a l p o i n t of v i e w the u n a t t a i n a b i l i t y of the nodal set seems rather plausible. it is not the drift the current
p b+
Indeed,
b+
N
in the c l a s s i c a l theory of diffusions
itself w h i c h has a p h y s i c a l i n t e r p r e t a t i o n but
and even if
b+
is singular on
N
the p r o d u c t
pb+
may stay finite. P h y s i c a l l y speaking this means that almost no diffusing particles run into the nodes. Moreover,
there is another m e c h a n i s m
w h i c h can be i n t e r p r e t e d as an i n d i c a t i o n that the p a r t i c l e s never reach the nodes:
since the singular drift field points a w a y from the node,
it
will produce a r e p u l s i o n strong enough to prevent the p a r t i c l e from reaching the nodal surface To p r e v e n t explosions additional c o n d i t i o n
N
.
(i.e. the escape at infinity)
one needs an
(cf. T h e o r e m 4.3).
IV.2b U n a t t a i n a b i l i t y of the Nodes and Global E x i s t e n c e In the sequel, we assume that
where
C a'b
of order Moreover,
(A.I)
p 6
C2'I(IR d x IR+)
(A.2)
b+ 6 cI'O(u)
denotes the space of functions with continuous d e r i v a t i v e
a
in the space variables and of order p
and
b+
b
in time variable.
satisfy the F o k k e r - P l a n c k e q u a t i o n
(4.9).
86
Locally
the d r i f t
grant a unique
field
local
b+
= inf{t In fact, the
existence
is r a t h e r
solution
regular
(Xt)0~t~ T of
~ 0 IX t ~
and u n i q u e n e s s
theorems
up to the s t o p p i n g time
U}
(4.12)
are in the s t r o n g
sense.
We w i l l n e e d
f o l l o w i n g lemma [l16d
Lemma
4.1:
Let
f(x,t)
random variable
be a n o n - n e g a t i v e
satisfying
function
O S T ~ ~ ^ k
It is c o n v e n i e n t
~ ~iRd dx I o
to i n t r o d u c e
Moreover, cess
we set
(Xt)o~ts T
Theorem
4.2:
Yt =
Then
Suppose
for any c o m p a c t
Proof:
For
(Xt,t). We are n o w r e a d y to p r o v e
set
A
b+
tinuity
6(Yt,K
distance
we d e f i n e
be a s s o c i a t e d
such that
n N)
(A.I)
to and
p
via
(A.2)
in
~d
× ~+
.
set (4.16)
6 ]Rd x JR+ J Ixl < £ , t _< K}
space
has p r o b a b i l i t y
of the s a m p l e p a t h s
q u e n c e of this c o n t i n u i t y
(4.15)
> O} = 1
the c o m p a c t
of the p r o b a b i l i t y
{p(Yo ) > O}
that the pro-
f i n i t e time.
K c ]Rd x 19+
A = {p(Yo ) > O, Yt 6 K Vt < T h k, The set
within
Xt , O ~ t < T
and a d r i f t
J inf t
K = {(x,t)
and a s u b s e t
N
p diw v 6 L 1 19d loc ( × ~+) "
the E u c l i d e a n k,Z 6 ~
set
also that
P{~ 6 ~
denoting
v
on N
equation
(A.3)
(4.13)
(4.14) O
Let the p r o c e s s
are s atisfied.
be a
on U
does n o t h i t the n o d a l
the F o k k e r - P l a n c k
dt p ( x , t ) f ( x , t ) .
the c u r r e n t v e l o c i t y
= I b+ - ~ V l o g p
[
T
k > O. T h e n
k
[ I dt f ( X t ' t ) ] o
v
and let
for some
T ~
6
and the s t a n d a r d
(4.10)
t ~ Yt
one.
~
by
inf t
P(Yt
Throughout
= O}.
(4.17)
the p r o o f the con-
w i l l be e s s e n t i a l .
A first conse-
is the fact t h a t in o r d e r to p r o v e
the the-
87
o r e m it s u f f i c e s introduce
to s h o w t h a t
for e a c h
{t < 7 ^ k J Yt { K
Tn
7 ^ k
k
and o n l y A
if
Since
stopping
JYo j ~ Z
and
in t e r m s of s t o p p i n g
arbitrarily
we
by
n
or
× ~ • Next,
p(Yt)
is n o n - v o i d
the s e q u e n c e
1 _< ~ }
(4.18)
and we set
(Tn)n6IN
is n o n - d e c r e a s -
T = lim T n . C l e a r l y , T > 0 if n > 0 . Thus, we m a y r e w r i t e the set
p (Yo)
times
We s h all n o w look at t h o s e
f r o m below.
T
6 ~
time
1 < ~
A = {T > 0 , p ( Y T ) n
becomes
time
1 ~ ~ }
p (Yt)
or
another
V(k,i)
< • A k IY t ~ K
otherwise.
ing, we d e f i n e
= 0
n 6 IN a s t o p p i n g
T n = inf{t
If
P(A)
Vn
trajectories
small or,
An application
(4.19)
Q
for w h i c h
in other
of ItS's
E IN}
words,
the d e n s i t y
l o g P (Yt)
p (Yt)
i s unbounded
lemma yields T
Z{T>O } l o g P(YTn) = I{T>O } l o g P(Yo ) + Tn + I
ndt
---¢-- + b+" ---~+,~AlogP
(Yt)
o dWt'VP -~
(Yt)
n 6 IN
(4.20)
O
I{...}
being
the c h a r a c t e r i s t i c
t e r m in the a b o v e If on for
K
Tn > O
formula and
side of
(4.20)
Moreover,
Therefore,
then from
Yt £ K Tn > O
the s t o c h a s t i c
is a m a r t i n g a l e
{T > O} c {jyo j ~ i}
Now,
the e x p e c t a t i o n
of e a c h
is to be a n a l y z e d .
t E [O,T n]
by c o n t i n u i t y . t 6 [O,Tn].
function.
? p ( Y t ) is boundedl
follows
integral
(indexed by
it f o l l o w s
and
that
p(Yt ) >
on the r i g h t h a n d
n ) of m e a n
zero.
Since
that
]R [ J I { T > o } l o g p (Y o) j] -
J
Jlog p(Yo ) 13
o
(4.21
= ~ dx p(x,O) I{jxj~i } Jlogp(x,O) J
The
last t e r m in
and h e n c e
(4.11)
is f i n i t e
IE[I{T>o}logp(Yo) ]
L e t us n o w c o n s i d e r Y T n 6 K, n 6 IN , and
since
exists
the left h a n d
p
is l o c a l l y b o u n d e d
by
(A.I)
and is finite. side of
(4.21).
I { T > O } log p (YTn ) is b o u n d e d
If
T > O
then
from above uniform-
88 ly in ~ E ~ and n 6 IN. Hence bounded from above, Assume now that
P(A) > O. Our aim is to deduce a contradiction
from this assumption.
We will denote by
of a real-valued function Since
A c {T > O}
~[I{T>O } log P(YT )] is uniformly n
and
f
f = f+ - f-
the decomposition
into its positive and negative parts.
log P ( Y T ) n
~ -log n
on
]E[I{T>O } (logP) ( Y T ) ] -> [IA(l°gp) n
A
it follows
(YT)] n (4.22)
> P (A) log n As
~[I{T>o}log
(YT )] n
is uniformly bounded from above, this implies
lim ~ [I{T>O } log P(YT )] = - ~ n~ +~ n
'
(4.23)
and therefore T [ I n dt ~ ~tp + b+ • V__Q_ + vAlog p } (Yt) ] = - ~ .
lim ~ n~ + ~
L
o
P
On the other hand, by reformulating ker-Planck equation 8tP 'p + b+
the integrand by means of the Fok-
(4.5) we obtain on • Vp ~ =
U
v Ap _ div b+
which implies
8tPp+ If
Tn > O
and
--
b+ • V_~p + v Alog P
= v
t E [O,T n]
Yt 6 K
then
- div v
Q
and hence
T ]E [ I n o
dt [ S~t p + b+ • Vp -~ + ~ Alogp ]- (Yt) J,] P P T =~
[ I n d t I{Tn>O}
[~AP~-divl-
(Yt) ]
O
T
-<~ [ I ndt I{YtEK} [ ~A-~-P- div v]-(Yt) ] o <
dx d
(4.24)
P
dt T K (x,t) +
[~ A @ - p
div v]
(x,t)
89
% v II
dx dt
l~p (x,t) l + II
K
dx dt pldiv(x,t) I K
w h e r e the last but one step follows from Lemma i. Since the r e g u l a r i t y of
p
implies that
#~ds
K
is compact,
dtlAP(x,t) I < + ~ ,
whereas
the finiteness of the second term in the last formula follows from
(A.3).
Thus we have o b t a i n e d a u n i f o r m upper bound for the e x p e c t a t i o n I[Tndt b~
[ ~t - 7p + b+
• Vp ~- +
~ Alog p] (Yt) i and this contradic%s
(4.24).
O
As a result we conclude that ity, i.e.
P[A] = 0
A
cannot have a n o n - v a n i s h i n g probabil-
is established.
R e m a r k I: For a symmetric d i f f u s i o n process v ~ 0
and
(A.3) is satisfied.
Remark 2:
b+
are not be chosen
The r e g u l a r i t y c o n d i t i o n for
p and
o p t i m a l l y and as a reward for this we get t r a n s p a r e n t proofs. In the next theorem, time cannot occur;
•
it will be p r o v e n that explosions in finite
in other words,
the process
(Xt)o~t< ~
cannot dis-
appear at infinity w i t h i n finite time. This follows e s s e n t i a l l y from the following a s s u m p t i o n on the p r o b a b i l i t y density (A.4)
There is no continuous path some finite time interval
sup t<s
IXtl = + ~
and
t ~ Xt [0,s)
inf t<s
p :
defined on
such that
P(Xt,t ) > 0 .
In addition, we have to s t r e n g t h e n the a s s u m p t i o n on
p
and its deri-
vatives from a spatially local to a global one. Then this yields a unique global c o n s t r u c t i o n of the d i f f u s i o n processes. tence and u n i q u e n c e s s are meant T h e o r e m 4.3:
Let the process
time defined in
that for all
Then Proof:
X t , 0 ~ t < ~,
~
(4.12), have a p r o b a b i l i t y density
such that A s s u m p t i o n s
(A.I),
Here both exis-
in the strong sense.
(A.2)
and
(A.4)
being the stopping p
and a drift
are satisfied.
b+
Suppose also
k £ I~ (A.5)
Ap(.,k)
(A.6)
S dxlApl
(A.7) P[~ = +~] = 1.
6 Cb(~d) 1
£ Lloc(
~+) 1
S dx pldiv vl 6 Lloc(
~+)
.
As in the p r e v i o u s t h e o r e m we will use in the following proof
suitably d e f i n e d stopping times. For
~,i,n 6 ~
we introduce the stop-
90
ping
times :
= I
0
if
IXol >
S (I)
(4.25)
[
A k
if
IXol -<
i S(2)n = inf { t < ~ ^ k J P(Yt ) < ~ } 1 {t < ~ A k I p ( Y t ) ~ ~ } is n o n - v o i d
if
otherwise.
We also define
(4.26)
and we set
the n o n - i n c r e a s i n g
(2) = Sn
~^k = k
sequence
(2) S n = S (I) ^ S n which
gives
rise
(4.27)
to the l i m i t i n g
stopping
time
S = lim S = S "I'( ~ ^ ~ ^ k . n n Clearly
S > O
if and only if
The t h e o r e m will be p r o v e d
JXoJ N ~
if we could
P[~ > k] = 1
Vk
(4.28)
and
p (Yo)
> O .
show that
6 ~.
(4.29)
But we can r e w r i t e {C > k} = {inf P (Yt) t<~Ak Moreover,
it is s u f f i c i e n t
sets a r o u n d that
P(B)
the origin. = O
to c o n s i d e r
Since
for all
> O }
P[P (Yo)
k,Z 6 ~
(4.30) paths
> O] = 1
where
B = {S > O, inf P (Yt) = O } t<~Ak
starting
B
within
compact
it suffices
to show
is d e f i n e d
= {S > O, P ( Y s ) n
by -< in V n 6 ] N } . (4.31)
Using
ItS's
lemma we o b t a i n S
I{S>O } I°gP(Ys
)= n
I { s > o } l o g p ( Y o ) + I n d t [+~ P[P o
b " V-~ +p
~£1°gp](Yt)
S +
dWt.
(¥t)
n 6 IN.
4.32)
o If
S
n
> 0
and
t 6 [ O , S n]
then
l p ( Y t ) _> ~
and
it
follows
from
S
,[fon d t J ~
(Yt) j2] < +co
and t h e r e f o r e
the s t o c h a s t i c
integral
(A.5)
91
i•n
dWt V o P
Since
(Yt)
is a martingale
{S > O} c {IXoJ ~ Z}
~[II{s>o}IOgD(Y°)
we can write =
'] ~E[I{Ix°'s£}JI°gP(Y°)J]
By (A.I) the density exists and is finite. use the following
of mean zero.
!xlSi
dxp(x,O) JlogP(x,O)J. (4.33)
p is locally bounded and hence ~[I{s>o } log (Yo)] To discuss the left hand side of (4.32), we will
decomposition
1 = I{Sn=O}
+ I{O<Sn<~Ak}
+ l{Sn=~Ak }
= I{Sn=O}
+ I{O<Sn<~Ak}
+ I{Sn=k}
•
Then [I{s>o}A{Sn=O}(I°gp )+ (YSn )] N ~[I{l x o l
= I
)+(Yo ) ]
dx P(x,O) (logP)+(x,O),
(4.34)
Ixl~ ]E[i{s>o}N{O<Sn<~Ak}(logp)
+ (YSn )] = ~ [ I { O<Sn< ~^k (logp )+(Ys n )] = O (4.35) +
IE[I{S>O},A{Sn=k}(I°gp )+ (Ys n ) ] =IE[I{Sn=k}(l°gp) < sup
(Yk) ]
(logp) + (x,k) •
(4.36)
x6 IRd Since
p (.,k) 6 C 1 n L 1
and
Vp(-,k)
p(.,k)
6 C b . Thus the decomposition
6 Cb
(by (A.5)) it follows that ~[I{s>o}:Iogp)+(YSn ] has led
of
to three finite uniform bounds and hence
~[i{s>o}logp(YS
formly bounded from above. Let us now assume that P(B)
B c {S > O}
logp(y S ) ~ - log n n [I{s>o}(l°g@) and this implies
on
-
(Ys n
B
> O. Since it follows
n
)]
and
that
)] ~ E [ I B ( I O g P ) - ( Y S )] Z P(B) n
log n
is uni-
92
liml~[I{s>o}log p(YS ) ] = - ~ ' n~ ~ n
lim~,[ f~dt
(4.38)
r~tp + b+ • ~ + vAlogp](Yt ) ] = -=
n-~::
L P
P
(4 39)
"
"
0
Using the Fokker-Planck
~
f o
+b+ - ~+~Alogp
dt
~
equation we obtain then
f
dx
dt
~d
IAp(x,t) l +
o
and the two terms are finite by that
dx
dtp(x,t)
div v]
div v (x,t)
o
(A.6) and
(4.40)
(A.7). This yields a contra-
to Stochastic Mechanics
In stochastic mechanics
p(x,t)
the probability
=
[~(x,t)l
the diffusion
density
p(X,t)
2
(4.41)
constant
~
is equal to
~.
We have therefore
# O}
(4.42)
and b on U is given by (4.2a). Our aim is now to express + ditions (A.3) and (A.7) in terms of the wave function. 4.4:
Let the wave function
Then conditions Proof:
(A.9)
2 d IR+) V~ E Llo c0R x .
S
(A.2), (A.I)
(A.3)
and
the con-
be such that
~ 6 C 2'I 0Rd x]R+)
(A.8) implies
valued function
~
(A.8)
(A.I),
is related
equation by
U = { (x,t) 6 IRa × IR+ I ~(x,t)
Theorem
(x,t)
P(B) = O .
to the solution of the Schr~dinger
Moreover,
f
~d
diction and we conclude
IV.3 A p p l i c a t i o n
~ I d dx dt[~Ap ~ o
(Yt
of Theorem 4.2 are satisfied.
(A.2). On
U
let us introduce
(the phase of the wave function
= pi/2 eiS Vlog~ = ~I Vlogp + i V S
a real-
~ ) by (4.43) (4.44)
93
from w h i c h it follows that on
U
the current v e l o c i t y takes the form
v = 29VS. Moreover, we have 1V~, 1 , 2 = p ~[(Vlog~) 2 + .VS. I ~ I 2.] ~ - Vp-VS = - 12v v-Vp
D e n o t i n g by
~
the complex conjugate of
~
.
(4.45)
we have
PV = Im(~ V~).
Condition
(A.3)
(4.46)
follows then by integrating and using Green's identity.
Indeed, the b o u n d a r y t e r m remains finite by Remark:
(A.8) and
The c o n d i t i o n s we impose are, of course,
(4.46).
stronger than n e c e s s a r y
because the proof of T h e o r e m 4.3 depends only on the n e g a t i v e part of 2 I + vAp-pV-v. N o t e that (A.9) also implies llV~ lJ2 £ L l o e ~ R ) w h i c h is just the finite action
(A.12) defined by Carlen and Zheng.
A l t h o u g h the a s s u m p t i o n of T h e o r e m 4.4 will be true in many q u a n t u m mechanical
situations,
the p o t e n t i a l
V
it w o u l d be nice to have conditions
and the initial wave function
~o(X)
us now state some t h e o r e m s w h i c h give s u f f i c i e n t conditons and r e f e r e n c e s therein). .Et e -z~-- ~(x)
For the stationary case, w h e r e
Let
~
=
= E~
is a m e a s u r a b l e function and
V £ Cm0Rd), part of
(see [15]
~(x,t)
be a w e a k solution of
~2 (- ~ A + v)~
V
Let
we have the following t h e o r e m
T h e o r e m 4.5:
where
in terms of
= ~(x,O).
then
E
~ £ cm-[d/2]+10Rd);" here
the eigenvalue. [d/2]
If
denotes the integer
d/2.
So, in the t h r e e - d i m e n s i o n a l case arity as the potential.
In p a r t i c u l a r
~
has at least the same regul-
V 6 C 2 ( ~ 3), then
fied. In the case of the t i m e - d e p e n d e n t s i t u a t i o n we have T h e o r e m 4.6:
Let
m £ ~
and suppose that
(A.10)
~o £ H2m(]Rd)
(A. II)
V
and its d e r i v a t i v e s up to order are continuous
and b o u n d e d in IRd
2m-2
(A.8)
is satis-
94
then
(a version of) the solution
~t = e
-it(-~ A+V) -~. 4 ° of the
m
C Z(]R+, H 2(m-Z) (jRd)).
n
~t 6
SchrSdinger e q u a t i o n satisfies
Z=0 As a consequence T h e o r e m 4.7:
If conditions
2m ~ [5] + 3
then
In particular, two;
(A.6)
Theorems
2
and
are satisfied for
(A.7) hold.
~o 6 H40R 3)
m
and that
has to be greater than V
and its d e r i v a t i v e s
are continuous and bounded. 6 and 7 give conditions under w h i c h the stochastic mechan-
ical d i f f u s i o n exists,
although the b o u n d e d n e s s condition on the poten-
tial is p h y s i c a l l y unsatisfactory. (A. IO),
(A.11)
in the t h r e e - d i m e n s i o n a l case
i.e. we need that
up to order
(A. IO) and
(A.11) may not hold but
In a given situation, A s s u m p t i o n s
(A.6)
and
(A.7) may be true nevertheless.
IV.4 A l t e r n a t i v e Methods to C o n s t r u c t S i n @ u l a r D i f f u s i o n s The results exposed in this chapter show that a wide class of diffusion p r o c e s s e s with singular drifts can be c o n s t r u c t e d and that both global e x i s t e n c e and uniqueness are in the strong sense. Let us first discuss the results obtained.
Conditions
similar to
(A.3)
and
(A.7)
are
t a n t a m o u n t to all c o n s t r u c t i o n s of d i f f u s i o n s w i t h singular drifts and from a p h y s i c a l point of v i e w they are not unreasonable. hand,
On the other
it does not seem t o lie within the framework of the m e t h o d to
relax the smoothness conditions
(A.I)
and
(A.2)
considerably.
As a slight
g e n e r a l i z a t i o n we can replace the r e q u i r e m e n t for a d e r i v a t i v e to exist by Lipschitz
condition, w h i c h will give u n i f o r m bounds,
too. As a last
remark it should be pointed out that the m e t h o d carries over to the case where the state space
~d
is r e p l a c e d by a R i e m a n n i a n manifold.
There has been some previous w o r k in this field. The stationary case was first c o n s i d e r e d by A l b e v e r i o and H ~ e g h - K r o h n mona
[23], N a g a s a w a
[89], Albeverio,
The analysis in [6] and [23]
[7a] and then by Car-
Fukushima, Karwoski
and Streit
[6].
is based on the theory of D i r i c h l e t forms
and works under mild r e g u l a r i t y p r o p e r t i e s of the p r o b a b i l i t y density, w h i c h can be d i s c o n t i n u o u s
(see also Chapter III).
shown, using p r o b a b i l i s t i c methods
In [89] N a g a s a w a has
(Dynkin formula),
that the d i f f u s i o n
process does not cross the nodal surface of the e q u i l i b r i u m d i s t r i b u t i o n p(x) =
l~(x) l2, where
~ is
a solution of a S c h r ~ d i n g e r - l i k e equation.
In a paper by B l a n c h a r d and Zheng
[17a] the stationary case was dealt
with by using a p a t h w i s e c o n s e r v a t i o n law.
95
The n o n - s t a t i o n a r y
situation
was
solved only recently.
case w h e r e t h e c o n f i g u r a t i o n
space
complished
But the c o m p a c t n e s s
ped.
by N e l s o n
[90e],
For an E u c l i d e a n
tablished
analytic
singular
point
bolic partial ution
of view.
compactification remains
of
IRd,
slight
The
a honest
generalization
a n e w class strategy
Guerra
interval;
to this
positive.
way
to discuss
idea to c o n s t r u c t the p r o b l e m
Both Carlen
set in finite
condition
Let us c o n c l u d e Remarks:
(A.12)
to intro-
singular
drifts.
on this
class that
even though,
steps:
on a b o u n d e d and takes
time
the
com-
in the limit we
in the
completion
pro-
singular. [115]
with
used t i g h t n e s s
results
on a R i e m a n n i a n
a s i n g u la r
drift
the d e n s i t y
that
Meyer
for semi-
manifold.
consists
p
The main
in r e d u c i n g
is e v e r y w h e r e
a kind of global
In fact,
guarantees
Section
(A.5)
of C a r l e n
finite
and Zheng
the d i f f u s i o n s
strictly
action showed
con[116]
do not r e a c h
and
[22b]
by some remarks. (A.7)
are to be c o m p a r e d
and Zheng
to the finite
[115b]
f dx p(u 2 + v 2) E L~oc0R+).
speaking,
though
Carlen's
shown that by a
time.
this
i) A s s u m p t i o n s
condition
Strictly
Zheng
some w o r k
of the f o l l o w i n g
diffusions
He proves
and Zheng need
a local
be seen
processes,
the diffusion.
that also
tions,
a metric
has
it
is indeed a dif-
it is p o s s i b l e
consists
the d i f f u s i o n s
the no d a l
action
class
con-
this measure,
Nonetheless,
h a v i n g very
sol-
the one p o i n t
step requires
[60c]
treatment
processes
situation where
to c o n s t r u c t
Guerra
p , Carlen
denoting
process
last
equa-
from a rather
is to solve a p a r a -
and
solution.
of regular
diffusions
to the
This
to this metric.
some d i f f u s i o n
In a d i f f e r e n t
dition
of Carlen's
drifts may b e c o m e very
martingales
differential
Having
stochastic
diffusion.
then he introduces
still get
p~
es-
the f u n d a m e n t a l
(~d)IR+ , ~ d
fundamental
from the class
pletion with respect
cedure,
it the
of d i f f u s i o n
leading
starts
Using
~ =
coefficients.
one has only a w e a k
duce
on
can be drop-
[22a,b,d]
the p r o b l e m
and to obtain
in a s t a n d a r d way.
to check that under
method produces
of s t o c h a s t i c
equation.
measure
this was ac-
condition
Carlen
approaches
equation
of this
fusion with the right since
Carlen
~d
manifold,
The hard part of his m e t h o d
differential
a probability
space
of solutions
drifts.
p~(y,t;x,s)
structs
configuration
weak existence
tions w i t h
is a compact
In the
there
in a loose
as follows.
is no i n c l u s i o n sense
(A.12)
relation
implies
between
(A.6)
and
these
(A.7).
condiThis
can
96
Provided that no surface term turns up we have
(4.47) U
U
Thus the conditions
Idx pu 2 6 L iocgR+) I
fudX pIV-(Vp--~) . 6 LlocgR+),
does not quite imply
which were sufficient
for
(A. 6) , since on
Ap = p[(.~.)2 + ~7.(-~-)] iApj = p[ (V~.p)2 + IV" Similarly,
2 luvl -< u2 + V 2
f Therefore
dx pluvl
(4.48)
(-~) I].
(4.49)
and thus
if
g ~
dx p(u 2 + v2).
(4.50)
(provided there is no surface contribution), dx pdiv v = ~
L~oc0R +)
is an
U
by virtue of
rather than merely
dx pu
(4.51)
(A.12). However,
V -u. In conclusion,
(A.7) require a [V.vl-term
(A.6) and
(A.7) constitute
a
different kind of finite action condition than that of Carlen and Zheng, but often they will be a consequence ii)
of
The basic s t r a t e g y in our proofs
on
logp,
method
(A.12).
is to find appropriate
and this is done by means of ItS's lemma.
is similar to the one employer by Nelson
not require the finite action condition condition to
S~dt Sdx plu°vl
< ~
which,
estimates
In this respect,
our
[90c]. His proof does
(A.12) but works with the weaker according
to
(4.51), is related
(A.7) .
iii)
In [6, Th.
the assumption density
4.2] the unattainability
falls off to zero sufficiently
linearly).
In a way this corresponds
to
has the drawback of not d i s t i n g u i s h i n g tial properties
of nodal set is proved under
that, in perpendicular direction to the nodes, the probability
of the density.
fast
(essentially,
faster than
(A.6), although our assumption between perpendicular
and tangen-
V. S T O C H A S T I C
V.O
VARIATIONAL
PRINCIPLES
Introduction Before p r o c e e d i n g
ational ation
principles,
in c l a s s i c a l
to a d e t a i l e d
it is p r o b a b l y mechanics
The c a l c u l u s of v a r i a t i o n mechanics
and, more
laws of physics. ments
generally,
of c l a s s i c a l
mechanics
of some v a r i a t i o n a l
equations
but only new ways of m o t i o n
of looking
In a given
functional ctions
of v a r i a t i o n
for w h i c h
involving
is also
variational
an energy
functions
with
In a concrete
problem,
the d e t e r m i n a t i o n
it states
integral.
The
new theories, equations
law.
the f o l l o w i n g
find a function
is an e x t r e m u m w i t h
of new
are the sol-
and the r e s u l t i n g
by N e w t o n ' s
in
state-
principle;
systems
do not r e p r e s e n t
is c o n c e r n e d
the latter
of the domain.
to discuss
dynamical
at dynamics,
set of a d m i s s i b l e
remarks.
principle
for the d e t e r m i n a t i o n
applicable
and H a m i l t o n
are the same as those d e r i v e d
The calculus lem:
problem
of E u l e r - L a g r a n g e
a few general
as H a m i l t o n ' s
classical
vari-
r e v i e w the situ-
as a u n i f y i n g
as a guide
is known
of many
of s t o c h a s t i c
thatwe
and that we make
has been useful
One of the most w i d e l y
that the t r a j e c t o r i e s utions
examination
desirable
respect
prob-
of a given to all fun-
the first p r o b l e m we have
of the class
of a d m i s s i b l e
func-
tions. Let us return
to stochastic
dXt=b+(Xt,t)dt
If
Xt(1)
and
X (2)
diffusion
equations
+ dW t •
are d i f f u s i o n
processes
t
x~i)" :x(i)o ÷ rj b+(i)(x (i)~,~)d~ +W t
i : 1,2
o let us remark cess.
that
Therefore,
the sum
we must
view of f o r m u l a t i n g evant
V.1
stochastic
for s t o c h a s t i c
some classes
Xt(1) + Xt(2)
enlarge
is no longer
the class
variational
mechanics.
This
leads
a diffusion
of p r o c e s s e s principles
pro-
we consider
which
in
are rel-
us to the c o n s i d e r a t i o n
of
of s e m i m a r t i n g a l e s .
The Classes
S(F)
On a b o u n d e d
and
interval
S(P,F) I =
[O,T] c IR+
let us c o n s i d e r
a family
9B
(Pt)t61 and,
of g - a l g e b r a s
moreover,
third
condition
ensures
We r e c a l l tingale
if
which
such that
is i n c r e a s i n g ,
Po that
contains every
that a process
Xt
admits
the
Pt
Xt
from the
is P - c o m p l e t e
is a
canonical
continuous
all t h e P - n e g l i g i b l e
right
sets.
The
(see A p p e n d i x ) .
(Pt)t6i ~ c o n t i n u o u s
semimar-
decomposition
Xt = Xo + M t + A t
(5.1)
such that (i) X (ii) sequence a
is a P - m e a s u r a b l e r a n d o m v a r i a b l e . o is a (Pt)t61 - l o c a l m a r t i n g a l e , i.e.
o Mt
(Tn)n6 ~
of s t o p p i n g
(Pt)t6i - m a r t i n g a l e (iii)
A t 6 Pt
At
is an a d a p t e d
f o r all
a function
Let
T n,
fixed
such that
n, M ° = 0 a.s.
process
of b o u n d e d
a n d for a l m o s t
variation
Tn+ ~
exists
every
on e v e r y
variation,
fixed
bounded
a
M T At n i.e.
~ £ ~ , At(e)
interval
is
is
of t i m e
= 0 a.s.
o
We must such a way
t £ I
of b o u n d e d
[O,t] ~ I, A
times
for e v e r y
there
now restrict
that S(P)
decomposition
this
the class
class
be the
includes
collection
Xt = X° + M t + At
i) ii)
X°
diffusion
of s e m i m a r t i n g a l e s such
we
consider
in
processes. admitting
the
that
6 L2(Po )
M t is a c o n t i n u o u s [O,a]
iii)
of s e m i m a r t i n g a l e s some
with t
At =
jl O
M
square-integrable
o
H s ds , w h e r e
such that
martingale
on
= O
~E
IHsl
Hs ds
is a
(Ps)-adapted
< +=
(finite
process
energy
condition ) . L e t us
first
remark
that the
diffusion
processes.
tive"
Xt
son, On
of
we write s(P)
we
in the
sense
often
H t = D + X t-
define
lIXlls(p)
We
last condition
Moreover,
are n o w p r e p a r e d
of N e l s o n
a norm
m ~
the p r o c e s s
I
to s t a t e
(see Sect.
ll.lls(p)
+
our
is n o t Ht
II.2
by all
"forward
). F o r t h i s
derivarea-
by
iHsl2ds
first
satisfied
is t h e
result.
(5.2)
99 Lemma 5.1: Proof :
(S(P),
il~lls(p)) is a Hilbert space.
By Schwarz's
inequality we have
]E [( I t IHsldS) 2] -< t :E [ I t IHs.2ds ] o o T Since
XT -
Xt
= It
HsdS
+ MT - Mt
it
(5.3)
follows
T
(5.4)
Xt + (MT - Mt) = XT - I H ds . t s Using now the fact that the random variables orthogonal,
Xt
and
(MT - Mt)
are
we obtain T 2 XT 12] + 2(T-t) ]E[It IHsl ds ] .
~ [ I X t 12] +]E[ ( M T - M t ) 2 ] < 2 ~
(5.5) Thus, let
{x(n)}n6 ~
be a Cauchy sequence in
S(P), i.e.
" - x(n+m) HS(p) ~ 0 for n ÷ + ~ . This implies in particular sup fIX In)" m that sup ~ [ IM~n) --TM(n+m) I2 ] ~ O for n ~ l m X (n) - X (n+m) I2 ] ~ O for n ~ + ~ and sup
~[I
o
o
m
T SUPm ~[ ~ IH(n)s -H(n+m) 12dS]s plete, there exist
~ 0
M T(~) , X o(~)
for and
Since
n ~ + "nt(~)
L2
is com-
such that
lim ~[IM~ n) - M T(~) i2] = l i m ~ [ I X (n)o -X(~) 12]o n~ ~ n~
= O
(5.6)
~T lim ]E [ IH (n) -H(~) 12ds] = O n÷~ Jo s s
and
Denoting now by
X (~)
X~ ~) = where
the element of
S(P)
defined by
ItH~ (~) + X (m) ~)ds + M t o o
M t(~) = ~ [ M ~ ) }Pt] lira
(5.7)
(5.8)
it is easy to verify that
IIx(n) - X (~) II S(P) = O .
(5.9)
n~ []
Now,
given
conditions
a second
as
Pt
filtration
F
such
that
FT_ t
we say that a continuous process
satisfies
the
X 6 S(P,F)
same
if
100
X t £ S(P t)
and
XT_ t 6 S(FT_t).
Let
fo
XT_ t = X T +
KsdS
it
+
. We
denote
D X t = -KT_ t • On
S(P,F)
c S(P)
we
I, we
Lemma
(S(P,F),
5.2:
Remark:
On
is e q u i v a l e n t
V.2
Strongly Let
set.
We
exists
can p r o v e
the
il ° II
+
(X, ll'll)
be a n o r m e d
a functional C > O
a,b
a subclass We
£ K
of the
are i n t e r e s t e d
sure
that
IOT(I'D+ X s j2 +
ID X s l 2 ) d s ]
(5.12)
and
space
-f[(1-1)a+Ib]
of c o n v e x
in the
let
K c X
is s t r o n g l y
be
a convex
convex
sub-
if t h e r e
that we h a v e
I 6 [O,1].
class
and
The
>CI(1-1)
strongly
llb-all
convex
2 (5.13)
functionals
form
functionals.
following
problem:
Find
conditions
which
as-
a function
f:
attains
+
f: K c IR
such
(1-1)f(a) + I f ( b )
for all
norm
S(P,F)
Functionals
a constant
(5.11)
) is c o m p l e t e .
IXTI2
Convex
say that
If" l~(p,F)
that
following
= E[IXo}2
to
the n o r m
JlollS(P,F)
S(P,F)
IIXII2
introduce
T = {[IXo 12 + Io(lD+Xsl2 + I D _ X s l 2 ) d s ] } I/2
IIXIIs(P,F)
As in L e m m a
(5.10)
X~
IR
its m i n i m u m .
semicontinuous We n o w p r o v e
if
We
lim n
a theorem Let
recall
that
xn = x giving
Theorem
5.3:
tinuous
functional
defined
f: K ~ IR
bounded
from below
there
a function
implies such
lim n
be a s t r o n g l y
exists
f(x n)
sufficient
on a c l o s e d
x
is c a l l e d
lower-
= f(x).
conditions:
convex
convex
a unique
f
set 6 K
and
lower-semicon-
K c X. such
If
that
f
is
101
f(x ) = inf f(x) x6K Proof:
Let
{ Xn} nE]N
lira n ÷~
f(Xn)
(5.14)
be a sequence
= inf x6K
f(x)
in
K
such that
> -
T h e n w e h a v e also I
lim f(~(Xn+Xm))~ n,m ~ ~
= inf f(x) x6K
Using now the strong convexity
C~I LIxn _ Xm]12 we c o n c l u d e
that
there exists
X
semicontinuity
f(x
of
f
) = inf xEK
f
is a C a u c h y
such that
lim n~
implies
f
unless
s equence.
Since
K
Moreover,
is closed, the l o w e r -
f(x)
the u n i q u e n e s s , n o t e t h a t if I on K, t h e n ~ ( y + x ) E K and 1 " f(~(y+x
_ f ( ½ ( X n + Xm ) )
Xn = X
To p r o v e of
I = I
for
~ ~I f(x n) + ~I f(Xm)
{Xn}n6 ~ 6 K
of
)) < inf x6K
y
and
x
r e a l i z e the m i n i m u m
f(x)
y = x
•
V . 3 The Y a s u e A c t i o n Let
V(x,t)
one c o n s i d e r s jT= c In s t o c h a s t i c
be a p o t e n t i a l
the a c t i o n I~
function.
m "2 ,t)}dt {2 Xt - V ( X t
mechanics,
In c l a s s i c a l
mechanics,
functional
we define
"
(5.151
following
Yasue
[112] a s i m i l a r
ac-
tion
jT =~ At t h i s tential.
point,
it
is
[~(kD+Xsl
+ iD_Xsl
- V(Xs'S)]ds
convenient
t o make some h y p o t h e s e s
(5.16) about the po-
102
We suppose straight
that the second derivatives
Of V(.,t)
along all
lines are bounded:
d2
d V(x+le,t)
< k
Vx
6 IR
and for all unit vectors
e.
(5.17)
d12 Moreover,
we suppose
V(x,t)
that there exists
< C(I + Ixl) 2
The first basic
¥x £ IR -d
a constant
C
such that
Vt £ I
fact about the functional
(5.18)
jT
is presented
in
the next theorem. Theorem convex
5.4~
If
~ £ S(P,F)
is such that
jT(6)
< +~
and if the
set (5.19)
K( = {X 6 S(F,F) I X ° = ~o ' XT = (T } is closed,
then the functional
T 2 < m/k . As a consequence, which minimizes Proof:
Remark
jT
K(
first that the class of admissible
and denoted by T J1(X)
J2(X)
convex on
a unique element
T (I-I)JI(X)
Z = X,Y.
mechanics. Then
Let
functions X,Y
K~
we consider
be two elements
Z o = Z T = O. Define
functionals (5.20) (5.21)
that T
T
T
+ IJ1 (Y) - J1 [ (I-I)X+ IY] = l(1-1)J I (Z) m ~(1-t) = ~
(5.22)
Ilzll 2 S(P,F)"
(5.17) we obtain
(1-1)V(Xs,S) + I V (Ys,S) - V [ ( I - I ) X s + iYs,S] Skl(1-1)iZsl From
if
in
m IT 2 =~E [ ~ (ID+Xsl + ID-- X S 12)ds] "o T = ~ [ I V(XstS)ds] o
It is easy to verify
Using
K~
jT
is much the same as for classical of
is strongly
there exists
(5.5) we see that I~ ]E[J Ztl2 ]at ~ ~1 T 2 IE [ I~ED+Zsl2ds]
.
(5.23)
103
Thus, using the similar inequality I
[hZsl2]ds
and therefore
for
D_Z s , we obtain
~ ~I T 2 lIZIl2S(P,F)
from (.5.22)
T (I_I)J2(X) + ~j] (y) _ j][ (~_~)X+ ~y] _ < k4 T 2 I(I-I) iiZii2(p,F) Hence combining (I-I)jT(x)
(5.2])
+ IJT(Y)
and therefore
jT
and
(5.24)
it follows
- JT[ (I-I)X+IY] is strongly
(5.24)
>- ~ ( m - k T 2) IIZIL (F,F)
convex on the closed convex set
K~
K~ c S(P,F). Hence by Theorem 5.3 there exists a unique element in which minimizes jT To discuss the connections the solution of the stochastic
between the minimizing
element and
Newton law, we prove first
Lemma 5.5: Let X,Y be in S(P,F) are also in S(P,F). Then, denoting
and suppose that D+X t
a(Xt) = yI (D+D_ + D D+)X t
and
D.X t
(5.25)
we have
~[
a(Xs)' Ys ds] = Yt" }(D++D-)Xt
iT o
(5.26) - E [ IoI(D+X~D+Ys
Proof:
+ D_X s.D_Ys)ds]
Using the formula of integration
T ~ [IoD_D+ Xs • YsdS]=Yt'D+Xt
[
So
I
D+D Xs°YsdS] = X t'D~x t
which implies the result.
T o
-
by parts
(see 2.52) we obtain
fT ]E []oD+Xs • D+YsdS ]
hZ
-W[]O
D_X s o D,YsdS]
(5.27)
(5.28)
104
Assuming,
moreover,
IV(x+ly,t)
that the p o t e n t i a l
- V(x,t)
V
is such that
- I V x V ( x , t )- Yl ~ C~2( 1+Ixj2 + lYl 2) (5.29)
for all
x,y 6 ~ n
and all
1
with
O < ~ ~ 1
then it is e a s y to
c h e c k the f o l l o w i n g p r o p e r t y : Let
X,Y £ S(P,F)
such that
d jT(x+Xy) dl
I=O
Yo = YT = O
=
m E[ ~
t h e n we h a v e
D+Xt.D+Yt +
D Xt • D Y t ) d t ] -
T ~[
We are n o w p r e p a r e d Theorem and
5.6
(5.28),
the a c t i o n
for all
to s t a t e o u r n e x t theorem.
(Yasue) :
Suppose
then for
T
functional
x I ~ ~
Next,
we w o u l d
5.6 i m p l i e s
(5.17),
is c h a r a c t e r i z e d
such t h a t
= ~[
point
s2
7 x V ( X t 't) • Ytdt]
Newton's
of
by (5.31)
Yo = YT = O .
like to d i s c u s s
(5.18)
m law in the mean.
Theorem
the f o l l o w i n g
5.7:
X 6 S(P,F)
satisfies
small e n o u g h the u n i q u e m i n i m i z i n g jT
(D+Xt-D~Yt+D - Yff D_Yt)dt]
Y 6 S(P,F)
Corollary
V: ~ n
12
N2 ~[
(5.30)
Yt " ? x V ( X t ' t ) d t ]
Let
satisfies
V
and
T
Newton's
be as in T h e o r e m
5.3.
Suppose
that
law in the m e a n
m a(X s) = - V x V(Xs,S)
(5.32)
and that it is s u c h t h a t D ~ X t , D_X t 6 S(P,F), the u n i q u e m i n i m i z i n g
p o i n t of the Y a s u e
But u n f o r t u n a t e l y , extremal
point,
a solution over,
of the s t o c h a s t i c
nothing
a diffusion we o b t a i n
the c o n v e r s e
the e x i s t e n c e
ensures
process.
of w h i c h Newton
law.
X
is
jT.
is p r o v e d ,
The m i n i m i z i n g
is n o t n e c e s s a r i l y
Let us also r e m a r k that, p o i n t of
two r e s t r i c t i o n s
are n o t v e r y s a t i s f a c t o r y
tic m e c h a n i c s .
action
is not valid.
t h a t the m i n i m i z i n g These
t h e n the p r o c e s s
jT
in
S(P,F)
moreis
m a k e c l e a r that the r e s u l t s
f r o m the p o i n t of v i e w of s t o c h a s -
105
V.4 Construction Variational
of Diffusion
Processes by a F o r w a r d Stochastic
Principle
In this section, we discuss cesses with constant diffusion nal principle.
We emphasize
as for classical mechanics tion is equivalent (~,F,P) val tion
coefficient
(stochastic)
semimartingales
Now,
let
let
(Pt)
and recall that
space
space and on a bounded time interfiltra-
from the right and such that (S(P), :ll.Li S)
S(P)
F°
of continuous
with
IHs 12 ds]
(5.33)
(Lemma 5.1).
(Wt)t£ I
be a Brownian motion with initial value
be a filtration
Sw(P ) c S(P)
Newton law.
sets. We consider the class
IIXl~(p)=~[ LXT 12 + is a Hilbert
of the ac-
as in Sect. V. I, an increasing
w h i c h is continuous
all P-null
using a stochastic variatio-
in the sense that the m i n i m a l i t y
we consider,
(Pt)t6i
diffusion pro-
that the result we obtain is much the same
is a given probability
I = [O,T]
contains
to the
a m e t h o d to construct
containing
the subset of
S(P)
W°
and
~ (WslS ~ t). We denote by
such that:
(i) X ° = W °
(5.34)
X 6 Sw(P) (ii) the m a r t i n g a l e On
Sw(P)
we can define a new distance
X t = ]2HsdS + W t
and
part of
as follows.
(Xt)
given by
N Jls
Sw(P )
Let
this new distance
(5.35) is equivalent
<-
JIX-YJi 2S(p)~(T + I)IIX-Yll 2
The next lemma describes
elementary
properties
i)
S (F) w
is an affine subset of
ii)
Sw(P)
is complete
iii)
to the distance
we remark that
LiX-Yil 2
Lemma 5.8:
(Wt).
Yt = ~2 KsdS + Wt ' then we set
lix-YLI 2 = ]E[ ~oT IH s - Ksl 2 ds] To show that on
is
For
X,Y
6 S
w
(P)
]E[ I~ IX t - g t l 2 d t ]
(5.36) of
Sw(P).
S(~)
for the distance
H. Ji
we have
< ~I T 2 ]IX-yIL2
(5.37)
106
Proof:
i) and ii)
N [
are a c o r o l l a r y
IX t - Y t l 2 d t ]
and f r o m S c h w a r z ' s
I
of L e m m a
= ]E [
1
5.1. Let
X,Y 6 Sw(P),
then
(H s - g s ) 2 d s l d t ]
inequality
(Hs-Ks)dS
12
rt
< t jolHS-KsI2ds
.
T h e n we get
[
Let
V(x,t),
consider
IXt-Ytl2dt]
_<~ [
x 6 ]Rn , t 6 I
the a c t i o n
t dt
IHs-Ksl2ds]
be a p o t e n t i a l
functional
J
defined
function.
~
On
S
I]X-YII .
w
(P)
we
in the f o l l o w i n g way:
Let
X t = fot H s d S + W t , t h e n
J(X) To o b t a i n exactly
]El IoT m(g IHt i2 - V ( X t ' t ) ) d t ]
=
the s t r o n g c o n v e x i t y
the same c o n d i t i o n s
of the a c t i o n
as in V.I.
•
(5.38)
J, we i m p o s e on
Using Theorem
V
2.1, w e o b t a i n
the f o l l o w i n g T h e o r e m 5.9: Then,
for
Let m T2 < E
there exists
V
be such that J
is s t r o n g l y
a unique element
~
of
-~
< J(~)
= inf X6Sw(P)
Proof:
The p r o o f of T h e o r e m
Theorem
5.4,
J(X)
Sw(P)
(5.18) are s a t i s f i e d . convex
~
of
J
the same c o n d i t i o n s
in
Sw(P).
Sw(P )
and
J, i.e.
< +~.
5.9 is e n t i r e l y
N e x t we t u r n to a d i s c u s s i o n
on
which minimizes
analogous
and so it w i l l be left as an e x e r c i s e
tremal point
(see
of the p r o p e r t i e s Let us s u p p o s e
to the p r o o f of [17b]).
of the u n i q u e ex-
that
V
satisfies
as in V.3.
We are n o w in p o s i t i o n this
(5.17) and
the a c t i o n
to f o r m u l a t e
the first b a s i c t h e o r e m of
section:
Theorem
5.10:
(5.17),
(5.18)
extremal
Suppose that and
p o i n t of
V: ]Rn x I ~ I R s a t i s f i e s m T2 < k " An e l e m e n t
(5.29).Let J
if and o n l y if
the c o n d i t i o n s X 6 Sw(P)
is the
107
H s
is7
= ~ ]E [
IoS
=
Remark:
-
V x V(Xu,u) dUlP s]
I'T1
~IV x V(Xu,u) d u + ]E[ jo~
Using the notations of Chap.
vx
V(Xu,U)dulP s] .
(5.39)
III, by the remark in Section V.I,
(5.39) is just equivalent to
= -vx V(Xs,S)
m D+D+X s
(5.40) D+X T Proof:
= O
a.s.
We recall first a classical result.
two integrable measurable
functions,
\ ]o f(s)d
X t = StHsdS + W t
Yt = StKsdS + Wt
in
Sw(P)
f(t)
and
g(t)
be
O < T < +
joJo I{s~t } (s,t)f(s)g(t)ds
s2(s s• g(t) dt )
=
Now, let
g(t)dt
Let
then for every
dt
f (s)ds.
(5,41)
be the extremal point of
J . For any
we have
T J(Y) - J(X) = 7E[ Io m H t (Kt-Ht) - V x V ( X t , t ) (Yt-Xt)dt] + o (NY-XII) (5.42)
Since
X
is the extremal point of
J
in
Sw(P), we observe that
T ]E[ Io{m Ht(K t -Ht) - V x V(Xt,t ) (Yt- Xt) }at] = 0
(5.43)
which can be rewritten as T t ]E[ Io {m Ht(K t - H t) - V x V(Xt,t)-Io (K s -Hs)dS}dt] Using
= O . (5.44)
(5.41) we have also
2[
{m H t ( K t - H t) - ( t x V(Xs'S)ds) (Kt -Ht)}dt]
= O.
(5.45)
Inserting now Kt we o b t a i n
- Ht = ~ [ m
Ht
IT - jtVx
V(Xs,S)dslP
t]
(5.46)
108
rT m H t = ~ [ jtVx V(Xs,S)dsJ Pt ] Conversely,
if (5.47) holds we have also
mal point of
((5.43) and
X
is the extre-
J.
m
In (5.40) the relation
m D+D+X s = - V x V(Xs, s) can be interpreted
as a stochastic Newton equation. is too restrictive, Let
(5.47)
Since the condition
DX T = 0
a.s.
we modify a little bit the action functional
f 6 C2(IR d)
such that
Vx
J .
6 IR
I If(x) l < C(I + Ixl 2) d2 - - f(x + le) < k dl 2
for all unit vectors
I f ( x + l y ) -f(x) -Vf(x)
yJ -< C12(I + lxl2+ lyJ 2)
vx,y £ IRd
Theorem5~.11:
Let
are satisfied and tional on
Sw(F)
V: IRd
f 6 C2(IR d) verifies
(5.48)
£ [O,I].
(5.17),
(5.1). Let
(5.18) and Jf
(5.29)
be the func-
defined by
T 2 < ~m,
an element
V~
x I ~ IR such that
T Jf(X) = J(X) + ~]E[f(XT)] Then, if
e 6 ]Rd
the functional
X 6 Sw(P)
(5.49)
• Jf
is strongly convex. Moreover,
is the unique extremal point of
Jf
if and
only if
Hs
=
-
~
Vx V(Xu,U)du + ~[
(~ V x V(Xu,U)du
~ Vx f(XT))IP s] . (5.50)
Remark:
(5.50~ can be put in the form mD+D+Xs= -VXT V(Xs'S)
} (5.5~)
D+X T Proof:
= ~ Vx f(X T)
The proof being almost the same as that of Theorem -(5.10), we
leave the details to the reader. Now, let us consider the solution tic differential
equation
Xt
of the following stochas-
109
where
Wt
and let
dXt = b+(Xt,t)dt ~ dW t has the quadratic variation
(5.52)
d<Wi,wJ> t = ~6 ij dt
(5.53)
f
and T
V
be two functions
such that
Vx f(x) = b + ( x , T )
-
V
x V(x,t)
If the hypotheses
(5.54)
= m
(x,t)÷(b+(x,t) .Vx)b+(x,t)+ ~ Axb+(x,t)
of Theorem
5.11 are satisfied,
extremal point of the action
~
in
X
is the unique
Jf : Sw(P) ~ IR.
We have proved that the functional element
then
. (5.55)
Jf
has a unique m i n i m i z i n g
Sw(P)
t ~t = ~ H s
ds + W t .
If we can establish,
moreover,
(5.56) that
is a Markov process,
we will
be able to conclude that
H s =b+({s,S) In other words,
(5.57)
the process
~t
will be a diffusion process.
We formu-
late the second basic theorem of this section: Theorem 5 . 1 2 :
Under the same hypotheses
extremal point of the action functional
as in Theorem Jf : Sw(P)
5.11 the unique
~ ]R is a M a r k o v
process which is the solution of the following stochastic
differential
equation
d ~ t = b+(~t't)dt + dWt ' where
b
~o = Wo
(5.58)
is given by T b+(~t't)
Proof:
= ~[
m V x V ( ~ u , U ) d u + ~ Vxf(~T) l~t ]
For a proof of this more technical Let us emphasize
mechanics
result,
see
[17b].
that Theorem 5.12 gives a m e t h o d of stochastic
to prove the existence of the solution of the stochastic
differential
equation
(5.59)
110
d6 t : b +
(~t,t)dt + dW t
Now, We discuss briefly the relation between the above result and the results obtained in Chap. son, we have considered
III.
In Chap.
another stochastic
III,
following E. Nel-
acceleration
defined as
a(Xt ) = 21 (D+D_ + D_D+)X t . Let
9(x,t)
= e R(x't)+i
S(x,t)
~t = - 2-m A~(x,t) Setting
H(x,t)
= ~
{V x R(x,t)
(5.60)
be a solution of Schr6dinger's
+ V(x,t)~(x,t). + V x S(x,t)}
we know that under some regularity
(5.61)
and
conditions
equation
p(x,t)
= I~(x,t) 12
(see Chapter IV), there
exists a diffusion process
dX t = H(Xt,t)dt
+ dW t
(5.62)
d < w i , w J > t = ~ 6 ij dt m whose probability
density at time
(5.63
t
is just
p(x,t)
and, moreover,
that the following stochastic Newton equation holds
m a(Xt,t) Now, expressing
= -V x V(Xt,t)
a'(Xt,t)
(5.64)
=D+D+X t as a function of
easily shown that the above equation
m a'(Xt,t)
= -Vx[V(Xt,t)
X t , it can be
is formally equivalent
to (5.65)
- V'(Xt,t)]
with
v,(x,t)
~2
= -{-
Ae R
(5.66)
R
e
V.5 Other Approaches
to Stochastic Calculus of Variations
In this section, we present the main ideas of the various to introduce
a calculus of variations
allows to characterize
the dynamics
of probabilistic
sed in Chapter III by extremal properties of the process.
attempts
for stochastic processes which systems
of some non-linear
as discusfunctionals
111
The first attempt to introduce v a r i a t i o n a l methods w o r k of N e l s o n ' s also
in the frame-
stochastic m e c h a n i c s is due to K. Yasue
[114a] for a review). This point of v i e w exploits
[112]
(see
in an e s s e n t i a l
way the s o - c a l l e d integration by parts formula w h i c h is used to d e r i v e a g e n e r a l i z a t i o n of the classical E u l e r - L a g r a n g e equation.
As m e n t i o n e d
in V.3, this a p p r o a c h leads to conceptual difficulties. A second approach,
the so-called " F l u i d o d y n a m i c a l picture",
is
based on a g e n e r a l i z a t i o n of the classical H a m i l t o n - J a c o b i equation and inspired by the methods of stochastic control theory also
(see [62b] and
[9oe]). Let us briefly p r e s e n t the b a s i c facts of this m e t h o d
(more details can be found in the existing literature).
To derive in
c l a s s i c a l m e c h a n i c s the H a m i l t o n - J a c o b i e q u a t i o n from a v a r i a t i o n a l principle,
a v e l o c i t y field
q(t) = V(q(t),t)
v(.,.)
m u s t be introduced such that
and the equations of m o t i o n are
v(x,t)
= ~ ?S(x,t) m
~t s + 2 ~
(5.67)
(?S)2 + V = O.
(5.68)
As shown in Section 1.3, the structure of the equations g o v e r n i n g the M a d e l u n g fluid is not very different. We consider stochastic d i f f u s i o n p r o c e s s e s in
]Rd. We assume that the process starts at time
density
p (.,0)
and
{Xt}t6 ~R
with values
t = 0 +with a given
evolves in time according to the stochastic dif-
ferential e q u a t i o n
dX t = b + ( X t , t ) d t Like
v
+ dWt.
in the classical framework,
d y n a m i c a l constants.
(5.69) b+
has to be now d e t e r m i n e d by
The critical diffusions m a k e the stochastic action
stationary w i t h respect to the v a r i a t i o n of the drift v e l o c i t y field. As e x p l a i n e d in Chapter II we can also consider the b a c k w a r d velocity field
b
(Xt,t) b-
and we recall that
Vp = b+ ÷ 02 p
For the time interval
[O,T] we can introduce the following action func-
tional T
o where
V
(5.70)
is some scalar potential.
112 The v a r i a t i o n a l p r i n c i p l e gives conditions on the drifts b_
b+
and
based on the stationarity of the action functional, namely, DA = O
under a p p r o p r i a t e b o u n d a r y conditions.
One can show that critical dif-
fusions have the following structure in terms of the density of the process and some auxiliary scalar function
p (.,t)
S
I I v(x,t) = ~ (b+ + b_) = ~ VS
(5.72)
~t(PV) = -V(pv)
(5.73)
£1/2 ~t s + 2 ~
(VS)2 + V - 2v2m 2
I/2 = O .
(5.74)
P This last e q u a t i o n is a stochastic g e n e r a l i z a t i o n of the H a m i l t o n - J a c o b i equation,
in which Bohm's q u a n t u m p o t e n t i a l appears.
In other words,
one gets the M a d e l u n g fluid equations w h i c h , as d i s c u s s e d in Chapter I, are e q u i v a l e n t to the S c h r 6 d i n g e r equation. The connection to q u a n t u m m e c h a n i c s is e x p r e s s e d through the following formulae i
2~
=
2~m
~
,
0I/2
S
e~
(5.75)
£ + V .
(5.76)
~2 igt~ = H ~ , H = - ~
As shown by E. N e l s o n
[90el one can also see that the m e a n action func-
tional d i s c u s s e d above
(and p r o p o s e d in [62b])
is e q u i v a l e n t to a re-
n o r m a l i z e d mean classical action integral. An other a p p r o a c h to stochastic calculus of variations, called "path-wise picture", L a g r a n g e equation.
the so-
is based on a g e n e r a l i z a t i o n of the Euler-
In this strategy we look at c o n f i g u r a t i o n s m a k i n g
the m e a n classical action stationary with respect to the v a r i a t i o n of the sample paths. To do this, a d i s c r e t i z e d v e r s i o n of the mean classical action is considered.
Using then the e l e m e n t a r y p r o p e r t i e s of the
product of finite differences,
one is able to calculate the v a r i a t i o n
of the mean classical action. For a complete account of this a p p r o a c h we refer to [60],
[87a,b]
(see also
[81] for an application).
Let us emphasize an important d i f f e r e n c e between the two last app r o a c h e s we briefly described. w a r d v e l o c i t y field
b+
is
like the v e l o c i t y field
v
In the p a t h - w i s e approach,
In the f l u i d o d y n a m i c a l picture the for-
c o n s t r a i n e d by construction to be a gradient, in the c o r r e s p o n d i n g classical situation. the v e l o c i t y fields are in general no longer
the g r a d i e n t of some scalar functions.
113
In c l a s s i c a l m e c h a n i c s the path is often o b t a i n e d by using a variational method
(Hamilton's principle)
and final positions.
J.C.
structive c o u n t e r p a r t of these v a r i a t i o n a l chanics
[ll4b,c],
for a given pair of initial
Zambrini has given in a series of papers the cona p p r o a c h in Stochastic Me-
r e a l i z i n g a p r o g r a m i n i t i a t e d in 1931 by E. Schr~dinger
[98b~]
by c o n s t r u c t i n g a class of d i f f u s i o n p r o c e s s e s X t indexed by T T I = [- ~, ~ ] w i t h two given p r o b a b i l i t y d e n s i t i e s P T (x) and pT(y).
The class of a d m i s s i b l e p r o c e s s e s
in Zambrini's
p r i n c i p l e is not limited to M a r k o v i a n processes, being r e p l a c e d by the B e r n s t e i n proper~y,
stochasticvariatlona! the Markov p r o p e r t y
w h i c h is a one d i m e n s i o n a l
v e r s i o n of the local M a r k o v p r o p e r t y used for example in c o n s t r u c t i v e field theory.
Zambrini gives a new p r o b a b i l i s t i c
the heat e q u a t i o n w h i c h mechanics.
i n t e r p r e t a t i o n of
is the closest classical c o u n t e r p a r t of q u a n t u m
VI, TWO V I E W P O I N T S
VI.I
General
CONCERNING
a) Stochastic leading
astic m e c h a n i c s
to look at stochastic
mechanics
events
equation
viewpoint,
classical
the S c h r 6 d i n g e r
ation w h i c h enables stochastic particle.
fluctuations field
equation
background
of freedom
b) The o r t h o d o x
complete
description stochastic
assigns
a Markov
operator stochastic
cannot exclude
one arrives
and J.T.
c) S t o c h a s t i c
"orthodox
mechanics
system.
for
interpreta-
function
as a
F r o m this point
q u a n t u m mechanics,
4-
space
to each
Even if one does
physical
mechanics
problems
at new questions
which
theory,
one
may p r o v i d e
and that using are
a quantity which
(see e . g . R .
Werner
the
meaningless
of q u a n t u m mechanics.
at a counter,
in the framework
Lewis
and
field.
can be taken
the wave
state
interpretation
the b a c k g r o u n d
system of fi-
as fully isolated
as a r e a s o n a b l e
in q u a n t u m m e c h a n i c s
unarmbigously defined Truman
By
that stochastic
time of a p a r t i c l e
with
in c o n f i g u r a t i o n
quantum mechanical
for the s t a n d a r d
fully u n d e r s t o o d
e.g.A.
and each q u a n t u m mechanics
the p o s s i b i l i t y
methods
first hitting
comes
H
of the
to give a more d e t a i l e d
regards
process
equ-
that q u a n t u m
a background
is a way of p r o d u c i n g
diffusion
for studying
or unnatural
of microphysics.
for
of the d i f f u s i n g
no p h y s i c a l
description
can be used
which
of m o t i o n
asserts
of the state of a p h y s i c a l
not
stochastic
mechanics
mechanics
S chr6di n g e r
new methods
with
to this
differential
The solution
can be c o n s i d e r e d
any i n t e r p r e t a t i o n
of view,
According
interaction
Indeed,
),deriv-
of q u a n t u m physics
the t r a j e c t o r i e s
field hypothesis
in i n t e r a c t i o n
of some aspects
accept
the drift.
quantum mechanical
and stochastic
tion" we mean
w hich
concepts.
Stoch-
description
(E. N e l s o n [90]
is not an e q u a t i o n
gives
electromagnetic).
degrees
description
equation
one to d e t e r m i n e
such a system is always
granted
terms"
and all other elements
are the result of classical
(presumably
nitely many
in classical
objective
system but a linear partial
differential Nelson's
of microphysics.
a realistic
and p r o b a b i l i s t i c
the state of a p h y s i c a l
as a new q u a n t i z a t i o n
description
to provide
ing the S c h r 6 d i n g e r
mechanics.
can be c o n s i d e r e d
to an a l t e r n a t i v e
"attempts
of q u a n t u m p h y s i c a l
from purely
MECHANICS
Remarks
There are three ways
procedure
Q U A N T U M AND S T O C H A S T I C
[109])
The
is not be-
of stochastic
mechanics
(see
as a general
description
of
[105]). can be v i e w e d
115
a class
of d y n a m i c a l
invariant
systems
associated
obtained
remark
semigroups.
Guerra
and P. R u g g i e r o
relevant
In this
tails
process
or as a a u x i l i a r y
physically
theory
[63].
from the point
tool w h i c h
and to prove
we will
discuss
can be used to express
theorems.
briefly
of view of s t o c h a s t i c
mechan-
of a p h y s i c a l
some topics
mechanics.
in q u a n t u m
For greater
de-
mechanics
we will
which
consider
other p h y s i c a l
applications
are not c o n n e c t e d w i t h q u a n t u m
theory.
Interference If we
send a b e a m of electrons
in it, more
generally
Its c h a r a c t e r Let
~(x,t)
of mass
through
a screen w i t h
one observes
a diffraction
two slits pattern.
by q u a n t u m mechanics.
the solution
for a p a r t i c l e
initial
a crystal,
can be p r e d i c t e d
i ~ with
to look at stochastic
as a r e p r e s e n t a t i o n
mathematical
In the next chapter,
]R 3
equi-
first noted by F.
see [90a,b].
of s t o c h a s t i c
VI.2
appears
quantities
chapter,
was
The two proces-
by u n i t a r i l y
See YI.9.
on the a t t i t u d e we choose
the s t o c h a s t i c
mechanics.
observation
the
time and the process
since they are g e n e r a t e d
This intriguing
between
of q u a n t u m m e c h a n i c s
to imaginary
state by s t o c h a s t i c
the same
valent
reality
translational
a connection
formulation
continuation
to the g r o u n d
ses are e s s e n t i a l l y
Depending
that there exists
to the E u c l i d e a n
by a n a l y t i c a l
associated
ics,
by some isotropic,
noice.
L~t us finally process
disturbed
Of the free S c h r ~ d i n g e r
equation
m
= - 2--m A ~ condition
on
(6.1)
given by x2
~(x,O)
This
=
solution
fraction To this
(2~o2) -3/4--
corresponds
by G a u s s i a n solution
fusion process Introduce
~(x,t)
slits
e
202
to a G a u s s i a n wave of half w i d t h
the s t o c h a s t i c
which
(6.2)
mechanics
o
associates
can easily be c o m p u t e d
now the w a v e
= N[~(x-a,t)
packet
describing
centered
the dif-
at the origin. a Gaussian
dif-
explicitly.
function
+ ~(x+a,t)]
(6.3)
116
N
being a n o r m a l i z a t i o n factor. In a frame of reference m o v i n g with the beam, the function
can be viewed as the wave function a s s o c i a t e d to the t w o - G a u s s i a n slit experiment since the p a r t i c l e is at time ±a.
Let
I = oma2/M
sociated to
~
and
~ = ma/~,
located at the part
then the drift of the process as-
assesses the complicated form
t - 212 sinh 412~'x 414+t 2 414+t 2
t -212 b+ =
t = O
t + 212 sin 2 t ~ . x 414+t 2 414+t 2
x +
a
414+t 2
(6.4)
412~.x 2t~.x cosh ~ + cos 414+t 2 414+t 2
and the c o r r e s p o n d i n g process can no longer be computed explicitly. However, we can easily see that only the coordinate along the direction joining the two slits is relevant and that the drift is bounded, but the process comes very close to having nodes.
Indee d , the following
fact can be proved 2 a) If
t << ~ ~
the process is p r a t i c a l l y i n d i s t i n g u i s h a b l e
from the equally w e i g h t e d mixture from
+a
and
-a . 2 ma t >> ~ - ~ -
b) if
of the one slit processes starting
x = (2n+I) ~
the drift becomes very big near the lines
t , n £ Z
and points away from them.
It follows that
the particle is confined in one of the regions between these lines and this trapping p h e n o m e n o n produces the d i f f u s i o n pattern.
VI.3
Observables - Measurement
VI.3a
Observables In q u a n t u m mechanics,
joint operator in a state
~
to each o b s e r v a b l e corresponds a self-ad-
A. The q u a n t u m e x p e c t a t i o n value of the observable is given by
<~,A~>
In stochastic mechanics, vation of the diffusion process
observables c o r r e s p o n d to the obserX t . The basic o b s e r v a t i o n s are given
by a cylinder function of the form Valued, bounded and measurable.
f(Xt1'''''Xtn )
f
being real
In stochastic mechanics,
is a random variable. For p o s i t i o n observables,
an o b s e r v a b l e
at a fixed time, there
is a one-to-one c o r r e s p o n d e n c e between the two descriptions. fusion
Xt
Xt
<~,X(t)~>, where
is
A
a s s o c i a t e d with the wave function X(t)
~,
For the dif-
the e x p e c t a t i o n of
is the p o s i t i o n operator in the Heisen-
117
berg r e p r e s e n t a t i o n . matically mentary
verified
In other words, in s t o c h a s t i c
hypothesis,
process,
one has
For more
e.g.
general
correspondence.
if
p(x,t)dx
p =
the Born
mechanics denotes
the p r o b a b i l i t y
l~(x,t) 12dx
observables
interpretation
there exists
of this
random variable
depend
w a y on
~
Moreover,
the f o r m a l i s m
of s t o c h a s t i c states
mechanical
observables
general
VI.3b
and s t o c h a s t i e
situation
Momentum
by R. W e r n e r
[62a]
complementarity
the co h e r e n t symmetry struct
states
diffusion
coordinates
have
attempted
in s t o c h a s t i c
and m o m e n t u m
associated
Let us c o n s i d e r
stance,
in this more
processes
V(x).
variables
on the same to a p p r o a c h
mechanics
to
and m a n a g e d
Their
F. Guermo-
to deal w i t h
By e x p l o i t i n g
the f u L
they could con-
strategy,
of mass
In s t o c h a s t i c
m
however,
momentum
and b a c k w a r d
moving
mechanics,
turn out to have mean values w h i c h
the forward
footing.
where
does
other potentials.
of the q u a n t u m m e c h a n i c a l
we can take
in s t o c h a s t i c
the p o s i t i o n
of the Hamiltonian,
a point p a r t i c l e
of a p o t e n t i a l
expectation
to
quantum
of q u a n t u m m e c h a n i c s
oscillator.
to momentum.
not seem to be able to g e n e r a l i z e
chastic
studied
[9Oe].
[1093.
of the h a r m o n i c
in p o s i t i o n
influence
has been
oper-
see
extended
between
than
does not
details
has been
and the r e l a t i o n
to the L 2 - f o r m a l i s m
space and m o m e n t u m
ra and L. M o r a t o mentum
mechanics
on more
no s e l f - a d j o i n t
For more
role of c o n f i g u r a t i o n a l
is in c o n t r a s t
one treats
of the
Process
The p r i v i l e g e d mechanics
depends
in general
and t h e r e f o r e
to such a r a n d o m variable.
the case of m i x e d q u a n t u m
density
no such o n e - t o - o n e
which
one time the e x p e c t a t i o n on a s e s q u i l i n e a r
a supple-
.
Indeed, for a r a n d o m v a r i a b l e
ator c o r r e s p o n d s
is auto-
and is no longer
coincide
operator
drifts
under
several
the sto-
w i t h the
P. For in-
on the current
velocity
~ [b+(Xt,t)]
= ~ [b_(X£,t)]
= ~ ~ (Xt,t)] = < ~ ( - , t ) , P ~ ( - , t ) >
But none of those r a n d o m v a r i a b l e s operator
P, already
To i n t r o d u c e [3Ob],
a momentum
D. De Falco,
of the a s y m p t o t i c
their v a r i a n c e
differs
in s t o c h a s t i c
S. De M a r t i n o
behavior
has the same d i s t r i b u t i o n from those of mechanics,
and S. De Siena
of the t r a j e c t o r i e s
(6.4)
as the
P
M. D a v i d s o n
[35b]
make use
for a free particle.
118
Let
Xt
be the p o s i t i o n process in a situation where a p o t e n t i a l is
present.
C o n s i d e r now the solution
tion with initial c o n d i t i o n at time wave function
~
at time
~°'t(x,t)
~o,t t
of the free S c h r 6 d i n g e r equabeing given by the interaction
t :
= ~(x,t)
(6.5)
This leads to the free p o s i t i o n process
~T'°'t given by
dXT,t = b+o,t (XT o,t ,T)dT + dWT,t where
W °'t
is a W i e n e r process with variance
In particular, we can impose
xOlt Xt t = The process cess
X
(6.6)
_o,t ' WT = WT
X O't
can
at time
m
.o,t. (independent of x t ).
"per fiat" Davidson's
construction [30b]
(6.7)
"
be thought of as being
t. F o l l o w i n g
~t = lira T++~
--
[3Ob] and
"tangent" to the pro-
[35b] we define
xO,t T T
(6.8)
A c c o r d i n g to the result d i s c u s s e d in Section
VI.6
,this limit exists
a.s. and it has a p r o b a b i l i t y density equal to the m o m e n t u m distribution of the q u a n t u m state
~. Thus in the case of arbitrary p o t e n t i a l
a random variable has been c o n s t r u c t e d whose d i s t r i b u t i o n coincides with the m o m e n t u m d i s t r i b u t i o n in q u a n t u m mechanics. In a recent paper, S. Golin
[58b]
has carefully analyzed this
i m p l e m e n t a t i o n of m o m e n t u m in stochastic m e c h a n i c s by d i s c u s s i n g the ground state of the one d i m e n s i o n a l harmonic o s c i l l a t o r I + x ~(x,t) = (2~o2)-I/4 exp{~(i~t
2 )}
(6.9)
202
where
o 2 =-~-2m~ " In this case, the m o m e n t u m process
~t
can be d e t e r m i n e d
explicitly -
~t = m~e
t
-
2[~t +
eY(T-t)dWT] o
where
(6.10)
119
(t) ~ arc tan ~t - ~log
and
~t
is the position process
(I + ~2t2)
solution of the stochastic
differential
equation
d~t = - ~ t or, in integral
+ dWt
(6.11)
form,
~t = e-~t[~o + Ite~T dWT]
(6.12)
o where
Wt
is the Wiener process with
Golin's
variance
~m
analysis pointed out some manifestly
of the m o m e n t u m process.
a) The m o m e n t u m process in the free case.
~t
a non-vanishing
way of implementing
was required in the definition b) Using ItS's formula
I o [~
of we
potential
except
is present,
the definition of
because you cannot simply turn off the potential
~t = m e -~/2
features
has no operational meaning,
Indeed, w h e n e v e r
there is no experimental
unphysical
The most important ones are:
at time
~t '
t. But this
~t " can obtain a new representation
- ~(T-t)]eY(T-t)
~T dT
of ~t (6.13)
t from which we can deduce that fortunately,
the derivative
terpretatedas force. Indeed, variance
of the harmonic
~t ~t
has two continuous
of the m o m e n t u m process
the variance of
relation using
From these u n s a t i s f a c t o r y concludes [58b] tum
(classical)
mechanics
by measure preserving ance properties VI.3c Repeated
from the
way of giving the position-mo-
unitary
of m o m e n t u m
of the process is unacceptable
(canonical)
transformations
~t
Golin
(see in quan-
have to be replaced in stochastic mechanics
transformation.
A general discussion
in the stochastic mechanical Measurements:
As discussed
cannot be in-
is different
~t (we shall discuss this point in Sec. VI.4). shortcomings
that such a definition
). More generaliy,
~t
Un-
force.
c) There is no straight-forward m e n t u m incertainty
derivatives.
A Case Against
in Section
(VI.3a),
framework Stochastic
of covari-
is still missing. Mechanics?
for position measurements
formed at a fixed time stochastic mechanics
per-
and quantum mechanics make
120 exactly the same predictions. and P. Talkner
It was argued by H. Grabert,
[58] and E. Nelson
repeated measurements
[90e,f]
predictions.
this Section is to prove that a careful consideration reduction in stochastic mechanics
The aim of
of the wave packet
shows that in fact the quantum mechan-
can also be derived in the stochastic mechanical
work. We refer to measurement
[16]
and
[60b]
01
in stochas-
in relation with the problem of repeated measurements.
a) Example
HI
of
in stochastic mechanics.
tic mechanics
ians
frame-
for a more detailed discussion
Let us first sketch some apparent paradoxa appearing
lators
for
obtained in the framework of stochastic mechanics
were in conflict with the quantum mechanical
ical correlations
P. H~nggi
that the correlation
I. Consider two dynamically
and and
02 H2
uncoupled harmonic oscil-
with circular frequency in the Hilbert
spaces
~ . We have two Hamilton-
H I = H 2 = L2(]R).
The Hamil-
ton operator of the total system is of the form
H = H I @ ~2 + 11 @ H2 and acts on
(6.14)
H = HI ® H2 .
For any observable
of
AI
HI
its time evolution
in the Heisen-
berg picture is given by
e
itH
(A1 @ ~2
and is completely
)e-itH
itH I = e
independent
-itH I AI e
®
of the choice of
are dynamically
uncoupled.
at time
and a position m e a s u r e m e n t
t = 0
Since the corresponding
H2
(6.15)
since the systems
Let us perform a position measurement
Heisenberg
on
02
at time
position operators
[X 1 (0), X 2(t) ] = O, the quantum mechanical can be associated with this experiment. tions,
J2
periodic
in
t
b) Example
t > O.
correlation
<X I (0) X 2(t)>
To carry out e x p l i c i t computa]E [X I~ X t] correlation
Then
is proportional <XI(0)X2(t)>
to is
[16]. 2.
harmonic oscillator relation
01
commute,
let us suppose that the state of the system is Gaussian.
the stochastic mechanical correlation e -£0 t whereas the quantum mechanical
on
A similar apparent paradox appears in the ground state.
for a single
Its stochastic mechanical
cor-
can easily be calculated [Xo Xt] = °2 e-~Itl
' °2 = 2mM
(6.16)
121
-, n 6 ~ the H e i s e n b e r g p o s i t i o n t = -nx m so t h a t w e m a y c o n s i d e r <X(O) X ( t ) >
For
operators
commute
correlation.
not agree
X(t)>
with
as in E x a m p l e
of the
Pin
energies about
Consider the
the
As we w i l l
scattering
correlation
and
which
shows,
in a s c a t t e r i n g
state
final
The elasticity
momentum.
and de-
are d e f i n e d
~out
mX t t
mX t t '
VI.6
a resolution
we consider
only
fashion.
limits
Wt
is a W i e n e r
in this
performed,
One
exist
to i n t r o d u c e
of the p r o c e s s
in fact,
with
probability
changes
of the
first
features Markov
To be
are r e s o l v e d
of s t o c h a s t i c
processes
with
after
measure times
t h e drift.
variance
mech-
on the d r i f t
xO x x = b+°(Xt°,t)dt dX t
between
any a t t e m p t
after
the
a new process.
x + dWt °
it seems
system
for the d e s c r i p t i o n
because
X~ °
. Thus,
on the
the c o r r e l a t i o n
measurement
For t > O, the n e w p r o c e s s Xo tic d i f f e r e n t i a l e q u a t i o n
~t
a measurement
Therefore,
automatically
in
(6.19)
a new process
position
is to be p r o p o s e d .
the o t h e r s
+ dW t
at d i f f e r e n t
t = 0 we h a v e
paradoxa
2 since
of the b a s i c
process
framework,
W e cannot,
of t h e s e Example
of the d i f f u s i o n
dX t = b + ( X t , t ) d t
particle
(6.18)
quantum mechanical distributions. It t u r n s 2 2 ~ [~in ~ out ] is d i f f e r e n t f r o m the q u a n t u m
that
similar
natural
the
by
correct
is the d e p e n d e n c e
where
= lim t~+
u' t_
correlation. Now,
a fully
in the c o r r e l a t i o n
variables
in S e c t i o n
the
however,
explicit,
sult
a particle initial
is c o n t a i n e d
random
lim t+- ~
see
and have
mechanical
time
mechanical
fall-off.
is r e f l e c t e d in the fact t h a t the c o r r e s p o n d i n g k i n 2 2 commute [Pin ' P o u t ] = O and q u a n t u m m e c h a n i c a l i n f o r m a -
~in =
tem.
(6.17)
stochastic
Pout
The corresponding
anics
X(t)
scattering
tion
out,
(-1)no 2
the
3.
and
etic
one
=
I, an e x p o n e n t i a l
c) E x a m p l e n o t e by
and
But
<X(O)
does
X(O)
as the c o r r e s p o n d i n g
at time
of the
first measurement
t =O
sys-
the v a l u e s
to l o c a l i z e
Suppose
is s o l u t i o n
t > 0
has b e e n
that
yields
of the n e w
the at
the rethe v a l u e stochas-
(6.20a)
122
X
lim t+O where
W~ °
(6.20b)
is a Wiener process with the same variance
increments
independent
a functional function we have
a.s.
Xt° = x °
of those of
of the quantum
state.
after the measurement
by
x
W t , t ~ O. Denoting ~o
as
Wt
The drift
and with b x°
the quantum mechanical
with
lim t+O
#~o = d(X_Xo),
The probabilistic t > 0
information
is entirely
reduction
has naturally
According
to this analysis
function
~ [ X ° X t]
In this way,
it
Xt
is in this
the wave packet
into stochastic
is obviously
at
mechanics.
not the auto-correlation
x
(6.22)
for the correlation
of the Schr~dinger
experiment.
Indeed,
we
correlation.
equation
(3.24)
for the
takes the form I
=
measurement
x ° ~ [Xt°]
~o
oscillator
where the kernel
X~°,_ whereas
with the quantum mechanical
The solution
x ~t°(x)
in
whatsoever.
been incorporated
that gives the prediction
harmonic
position
but the quantity
dX ° P(Xo,O)
now get agreement
(6.21)
about repeated
contained
context of no significance
I
wave then
x
b+ ° =--~m (Re + Im)Vlog#t°
time
is
dx' Kt(x,x')
Kt(X,X')
x ~o°(X ') = Kt(x,x O)
(6.23)
is given by the following
explicit
formula
i/2 Kt(x,x,)
=
(
me i2~M sin~t
)
{ me exp - ~
(x2-x '2) -
~
(e-i~tx-x')2} -2i~t 1-e
" -j
(6. ~ 4) X
Hence the drift
b+ °
x b+O (x, t) =
Consequently,
takes the form x x tan~t
the stochastic
o sinet
(6.25)
"
differential
(6.20)
is linear and can be
solved x Xt°(s) = ~ o s ~ t s S t,
sin~t cotg~s)
sin~s ~ 0 .
xo sin~t Xo [ dWT ~xo + sines X s + sin~t J sin~T s (6.26)
123
For
t
being
c onsta n t
a multiple
(-1)nxo
a.s.
of
~ , the r a n d o m v a r i a b l e
Thus
the c o r r e l a t i o n
(6.22)
x is just the Xt° is simply
(-1)n ~ dXo ~(Xo'O)X2o = (-1)n ~ (X~) and it c o i n c i d e s
VI.4
with
Indeterminacy
A
and
[A,B]
where
version
of an i n d e t e r m i n a c y
is due to W. H e i s e n b e r g
Let
correlation.
Relations
The e a r l i e s t mechanics
the q u a n t u m m e c h a n i c a l
(6,27)
B
[64]
relation
in q u a n t u m
in 1927.
be two H e r m i t i a n
operators
such that
= c
(6.28)
c £ ~ , then H e i s e n b e r g
proved
that
I 2 Var A • Var B ~ ~ c
(6.29)
Var A ~ > 2
(6.30)
where
and
<.>
denotes
In 1930, (6.29).
as usual
the q u a n t u m m e c h a n i c a l
E. S c h r ~ d i n g e r
Defining
[98a]
the c o v a r i a n c e
then S c h r ~ d i n g e r ' s
version
is c l e a r l y Several
framework. istic
of i n d e t e r m i n a c y
feature
[36],
have o b t a i n e d recently
by S. G o l i n
relation
by
takes
the form
(6.32)
can be d e r i v e d
In 1930,
relation
I. Fenyes
[45],
L. de
in the stochastic is a character-
R. F ~ r t h
for the heat
S. de M a r t i n o
mechanical
[58a,c]
B
(6.31)
I, their e x i s t e n c e
processes.
D. de Falco, stochastic
and
I + ~l<[A,B]>l 2
relations
uncertainty
process.
A
form of
(6.30).
in C h a p t e r
of d i f f u s i o n
velocity
for the W i e n e r
than
indeterminacy
As d i s c u s s e d
a position
M. Cet t o
stronger
a stronger
-
Var A • Var B ~ Cov2(A,B)
which
established
of the o p e r a t o r s
I ~ ~
Cov(A,B)
expectation.
[51b]
derived
equation,
la Pena A u e r b a c h
and S. de Siena
indeterminacy
the i n d e t e r m i n a c y
relations. relations
i.e. and
[35] As p r o v e d which
can
124
be derived
in stochastic
ger's v e r s i o n
We c o n s i d e r stant by
~ . Let
u
mechanics
are fully e q u i v a l e n t
of q u a n t u m m e c h a n i c a l the case w h e r e
f
the d i f f u s i o n
be a function~of
the osmotic
velocity,
indeterminacy
space
coefficient
and time.
the f o l l o w i n g
to S c h r 6 d i n -
relations.
Denoting
formula
is a conas usual
is o b t a i n e d
by inte-
gration by part
[f u] = - 9 ~ [?f]
Using now the Schwarz velocity
u
(6.33)
inequality
has zero mean,
Var f
.
and the fact that the osmotic
we obtain
Var u = ~ [ ( f - E ( f ) ) 2 ] E [ u 2]
~2[(f-E(f))u]
= E2[f u]
then Var
If we set
f Var u ~ 2
f(x)
= x, we obtain
Vat x Var u
and by means
~2[Vf]
This now yields
(6.35)
inequality
(v
tum m e c h a n i c a l ator
P
op
velocity)
position
- momentum
indeterminacy
rela-
mechanics
of the d i f f u s i o n
operator
X
op
process
coincides.
+ 2
Xt
(6.37)
and of the q u a n t u m quan-
Moreover,
the m o m e n t u m
oper-
satisfies
Var P
op
= m2(Var
CoV(Xop,Pop)
Therefore,
the current
(6.36)
Var x(Var u + Vat v) Z Cov2(x,v)
The d i s t r i b u t i o n
being
a Cov2(x,v)
the f o l l o w i n g
tion in stochastic
now
2
~
of the Schwarz
Var x V a r v
(6.34)
the above
(6.38)
u + Var v)
= m Cov(x,v)
stochastic
mechanical
(6.39)
indeterminacy
relation
is
125
e q u i v a l e n t to S c h r ~ d i n g e r ' s
stronger version of the p o s i t i o n - m o m e n t u m
u n c e r t a i n t y r e l a t i o n in q u a n t u m m e c h a n i c s
Var Xop Var Pop ~ C°V(Xop'Pop) by setting the d i f f u s i o n constant In [68a]
and
[58c]
+ ~4
(6.40)
v = j~ 2m "
f o r c e - m o m e n t u m u n c e r t a i n t y relation,
angle v a r i a b l e s - orbital angular m o m e n t u m i n d e t e r m i n a c y relations, t i m e - e n e r g y i n d e t e r m i n a c y relations are d i s c u s s e d in the framework of stochastic mechanics.
It is w o r t h w h i l e m e n t i o n i n g that all these un-
c e r t a i n t y relations are a general feature of stochastic systems fusions) stant.
and that the d i f f u s i o n c o n s t a n t
~
(dif-
could be any p o s i t i v e con-
In fact, the i n d e t e r m i n a c y relations depend on a purely kinema-
tical feature of diffusions,
namely the n o n - d i f f e r e n t i a b i l i t y of their
sample path. Remark I.
R e v e r s i n g the point of view,
following question:
it is natural to ask the
Given the q u a n t u m m e c h a n i c a l u n c e r t a i n t y relation,
what can we infer about the notion of the q u a n t u m p a r t i c l e ? As discussed by L.F. Abott and M.R. Wise
[I]
the H e i s e n b e r g p o s i t i o n - m o m e n t u m
u n c e r t a i n t y p r i n c i p l e is r e f l e c t e d in the fractal nature of the q u a n t u m m e c h a n i c a l paths, viz.
the paths have
dorff d i m e n s i o n of a closed set lowing way. Let
A
H a u s d o r f f d i m e n s i o n 2. The Haus-
in
~d
cave or convex. The H a u s d o r f f m e a s u r e of strictly p o s i t i v e tive m e a s u r e
~ O
(may be infinite)
~ # 0
Z(B) ~ h(IBl).
can be defined in the fol-
h(t) be an increasing function of A
such that
such that for all balls B
h(t)
= t~
either con-
w i t h respect to
if and only if
The H a u s d o r f f d i m e n s i o n of
t > O
A
A
h
is
carries a posi-
of d i a m e t e r
IBI,
is the s u p r e m u m of the
has this property.
But this is exactly
the r e g u l a r i t y p r o p e r t y of the sample paths of d i f f u s i o n processes. The W i e n e r process has H~ider continuous paths of any order
e < I/2
(see Chapter II). Remark 2. lation Using
We can ask for the p o s i t i o n - m o m e n t u m u n c e r t a i n t y re-
~t
as defined in Section
(VI.3b). Using Schwarz inequal-
ity, we get
Var
x Varz ~ Cov2(x,~)
(6.41)
For the ground state of the harmonic o s c i l l a t o r Cov 2(x,~)
= f U
e -~
(6.42)
126
w h i c h does not coincide with
VI.5
CoV(Xop,Pop)
= 0
in this case.
Locality The causality p r i n c i p l e asserts that any p h y s i c a l l y real phenom-
enon cannot be affected by a disturbance w h i c h occurs later in time. If we accept r e l a t i v i t y theory, this implies the locality principle: Any p h y s i c a l l y real property cannot be influenced by something that occurs outside its b a c k w a r d light cone. The experimental results of q u a n t u m mechanics are subject to randomness and there are correlations in the results of m e a s u r e m e n t s on widely separated particles w h i c h have interacted in the past e.g.
(see
[43]). Bell's inequality
[1i]
is the most dramatical illustration of
the relation between probability theory and q u a n t u m mechanics.
This
inequality is a constraint which has to be satisfied by any purely p r o b a b i l i s t i c model of discrete spin. This inequality is v i o l a t e d in q u a n t u m mechanics, w h i c h implies that q u a n t u m m e c h a n i c s has no underlying p r o b a b i l i t y model The locality principle and the experimental confirmation of the p r e d i c t i o n of q u a n t u m mechanics
(and also of stochastic mechanics)
forces us to conclude that d e t e r m i n i s m is ruled out and that there is an intrinsic randomness in nature w h i c h has nothing to do with our ignorance of the initial data. I n d e p e n d e n t l y of the nature of space time,
locality can be dis-
cussed in terms of separability of correlated but d y n a m i c a l l y u n c o u p l e d systems.
In q u a n t u m mechanics,
if there is no q u a n t u m m e c h a n i c a l inter-
action between two systems and if we are only interested in observables of system I, we may ignore system 2 completely as explained in Section (VI.3c). This very convenient feature of q u a n t u m m e c h a n i c s satisfied in stochastic mechanics.
Nelson
a system for w h i c h the a u t o c o r r e l a t i o n choice of the H a m i l t o n i a n
H2
is no more
[90c] gives an example of I depends on the ~ [X t X~]
of the second system. This is due to
the fact that the diffusion takes place on c o n f i g u r a t i o n space M I × M2
and that both components of the drift depend,
the total configuration.
The stochastic mechanics
if the particles are arbitrarily far separated, ed by the second.
M =
in general,
is non-local:
on
even
the first one is affect-
127
VI.6
S c a t t e r i n g Theory In s c a t t e r i n g experiments,
ured directly,
the a s y m p t o t i c m o m e n t u m is not meas-
one is only able to m e a s u r e p o s i t i o n s and times. To de-
termine the final momentum,
one can use the follwoing method.
p a r t i c l e was close to the s c a t t e r i n g center at time d e t e c t e d in a
counter
at the point
the distance b e t w e e n the scattering
x £ ~3
0
If the
and if it is
at time
T > O
and if
center and the place of d e t e c t i o n
is m u c h greater than the range of interaction,
it is reasonable to as-
sume that during most of its flight the s c a t t e r e d p a r t i c l e m o v e d nearly freely w i t h a m o m e n t u m close to a p a r t i c l e of mass
Pf . This implies that
i. Therefore,
to study the time e v o l u t i o n of we have to consider that
Pf ~ ~
, for
in stochastic m e c h a n i c s it is natural I z t = ~ X t . Given a p o t e n t i a l V(x)
diffusion
w h i c h can leave the region w h e r e
the p o t e n t i a l is strong and to show for such d i f f u s i o n that for process wt
the following limit
lim wt(~) t++ ~
I ~ Xt(~) = Pf(e)
= lim
exists p a t h w i s e w i t h p r o b a b i l i t y one. D.S. such a result in the free case w h e r e
V ~ O
has proved for a large class of p o t e n t i a l s type)
(6.43)
t++
that the r a n d o m v a r i a b l e
Pf
Schuker
[99]
and
E. Carlen
has proved [22c,e]
(potentials of K a t o - R e l l i c h
exists almost surely,
is square
integrable and has the same d i s t r i b u t i o n as the q u a n t u m m e c h a n i c a l m o m e n t u m for the c o r r e s p o n d i n g solution For large
t, wt
~
l@(pt,t) 12t 3
t31~(Pt, t) 12 =
is the Fourier t r a n s f o r m of
lira t++
I ~ Xt(~)
(6.44)
4 •
~ v+(~)
exists almost surely. Let a s s o c i a t e d to this process.
B+ ? F
~3
such that
(6.45)
B+ =
n ~ {XulU > t} be the tail • field t>l Clearly, v+ is B+-measurable. It is natu-
ral to ask the following question: field
the p r o b a b i l i t y
I~(P)12
Let us now consider a d i f f u s i o n process in
variable
l~(x,t) 12
wt " A simple c a l c u l a t i o n shows that
lim t++ where
X t , so is
final
of the S c h r ~ d i n g e r equation.
is a measure of the m o m e n t u m and since
is the p r o b a b i l i t y density of density of
~
Can
v+
g e n e r a t e the whole tail
If this is the case then any b o u n d e d B % - m e a s u r a b l e random admits the r e p r e s e n t a t i o n
128
F(~) = f(v+(~))
a.s.
for some b o u n d e d Borel function f o n ~ 3 . This q u e s t i o n is p h y s i c a l l y very important.
Indeed,
if
v+
does not generate the tail field
B+ ,
L
this implies that there exists extra scattering i n f o r m a t i o n besides the final m o m e n t u m which can be gained by observing only the large time b e h a v i o u r of the sample paths of the d i f f u s i o n process. esting paper E. Carlen
[22c]
In a very inter-
- using coparabolic Martin representa-
tion m e t h o d s - proved that for a large class of p o t e n t i a l s field
B+
V
the tail
a s s o c i a t e d to the diffusion process of stochastic m e c h a n i c s
is indeed
generated
by
v+ . In stochastic m e c h a n i c s the scattering
o b s e r v a b l e s c o r r e s p o n d to B + - m e a s u r a b l e functions and in q u a n t u m m e c h anics the only s c a t t e r i n g o b s e r v a b l e s are functions of the q u a n t u m mechanical m o m e n t u m operator ~V which is the operator theoretic analogon i of the statement that the tail field is generated by v+ . By Carlen's result
B+
does not contain any extra information, w h i c h agrees w i t h
the answer given by q u a n t u m mechanics. Using other methods,
let us m e n t i o n the results of P. Biler
in the one d i m e n s i o n a l case and M. Serva
[96]
[12]
for central p o t e n t i a l s
w h i c h both discuss potential scattering in stochastic mechanics. Nelson
[90e] considered a G a u s s i a n wave packet under the free
evolution and computed the c o r r e l a t i o n m a t r i x of the initial m o m e n t u m and the final m o m e n t u m and found it to be - e - ~ , width of the Gaussian.
Therefore,
i n d e p e n d e n t of the
the two m o m e n t a differ a l t h o u g h their
density coincides and this result shows the d i f f i c u l t y of d e f i n i n g a pathwise analogon of the s c a t t e r i n g m a t r i x in stochastic mechanics. Similarly, is equal to
the c o r r e l a t i o n coefficient of the square of the m o m e n t a -2~ -e . Hence, there is no p a t h w i s e energy conservation,
i.e. the trajectories of the c o n f i g u r a t i o n process do not exhibit elastic scattering. Let us m e n t i o n finally that there is a case in w h i c h a direct relation between s c a t t e r i n g quantities and p r o b a b i l i s t i c q u a n t i t i e s come out naturally, namely the limit of low energies in w h i c h the quantum m e c h a n i c a l cross section is given by a geometrical quantity, scattering length. haviour for
In
Ixl + + ~
[5]
of the drift a s s o c i a t e d by the D i r i c h l e t form
approach to q u a n t u m mechanics w i t h the S c h r ~ d i n g e r operator through
b+ = V l o g ~
the
the relation between the asymptotic be-
, H~ o = Eo~ o , ~o
and the spectral p r o p e r t i e s of
H
at
H = -A + V
being the ground state wave E°
is discussed.
It is also
shown that the leading term in the a s y m p t o t i c b e h a v i o u r of
b+
for
129
Jxl ~ + ~
determines
the s c a t t e r i n g
tion about the e f f e c t i v e details,
see
tering,
[5].
range p a r a m e t e r
To o b t a i n
the p r o c e s s e s
length
complete
associated
a
but gives
no informa-
of the p o t e n t i a l information
to the excited
V. For more
about the scat-
states
must be con-
sidered.
Vl.7
Spinning
Particle
One attempt
to d e s c r i b e
ics is b a s e d on the B o p p - H a g g particles
as q u a n t u m
particles with
spin in stochastic
model
which
rigid bodies.
tion space of a point p a r t i c l e M = ~3×
S0(3),
being Therefore wave turns
which
a double
[29b] In this
the f o l l o w i n g
of
covering
~
on
out that
~
must be either an integral
a half-integral
which
spin wave
of the two classes
give
Dankel
function.
In absence
and D. Dohrn,
what
F. Guerra,
of e l e c t r o m a g n e t i c
on
M ?
It
function
of wave
to a d i f f u s i o n mechanics
diffusion
are the smooth
spin wave
A superposition
does not c o r r e s p o n d
[29b]
~ = ~ 3 × SU(2),
rise to a d i f f u s i o n
The theory of spin in s t o c h a s t i c by T.G.
the configura-
space
must be answered:
functions
mechan-
spinning
is the m a n i f o l d
M. Our goal is to c o n s t r u c t
question M
framework,
with o r i e n t a t i o n
has the u n i v e r s a l
covering
interprets
on
M
or
functions .
has been e l a b o r a t e d
P. Ruggiero [40].
field the S c h r 6 d i n g e r
equation
will
be
i ~ ~
where
A
related
=
-~
A~
(6.46)
is the L a p l a c e - B e l t r a m i to the mass
we assume
m
spherical
and the m o m e n t
symmetry
theory,
To take this
fact into account,
mechanics
the t r a n s l a t i o n a l couple,
which
deterministic according ical
verifies
I + 0
of inertia
of
the limit
I ~ O
s
involving I
of spin.
the Pauli e q u a t i o n
a point particle
Dankel
shows
of the e x i s t e n c e
ics is open.
For a h e u r i s t i c
of
that if
SU(2),
s
argument,
ones
see
[90e].
since
fully de-
is no classical ~
transforms
wave
with multiplicity I ÷ O
O .
In
the q u a n t u m m e c h a n -
and the c o r r e s p o n d i n g
of the limit
is
must be taken.
fact that there
for spin
The q u e s t i o n
In a non-
but it is u n i n t e r e s t i n g
representation
also exists
constants
of the particle,
of freedom and the r o t a t i o n a l
the w e l l - k n o w n
analogon
M
of inertia
limit exists,
degrees
express
to a spin
limit
the m o m e n t
this
on
for the sake of simplicity.
relativistic
classical
operator
function 2s + I .
in stochastic
mechan-
Let us also
130
mention
that W. Faris
[44]
has shown that stochastic mechanics gives
a p e r f e c t l y consistent p r o b a b i l i s t i c d e s c r i p t i o n of the E i n s t e i n - R o s e n Podolsky-Bohm
experiment,
a more practical version of E.P.R. ex-
periment that involves spin. Let u s n o w briefly m e n t i o n a more pragmatic point of view to introduce spin in a stochastic framework. For a spin I/2-particle, this approach starts from q u a n t u m m e c h a n i c s and tries to interpret the continuity equation for equation.
l~t(x,o) 12, o = ±I, as a forward K o l m o g o r o v
In this procedure,
to each smooth solution w i t h o u t nodes is
associated a M a r k o v process
Yt = {(Xt'ot)
6 IR3 x { - 1 , 1 } }
which reproduces q u a n t u m averages for coordinates and a selected component of the spin
w h i c h is treated as a discrete random variable.
For more details, we refer to [31],
[32],
[33].
This general heuristic p r i n c i p l e which is based on the identification of the q u a n t u m m e c h a n i c a l continuity e q u a t i o n for
Pt =
I~t 12
with forward K o l m o g o r o v equations for suitably chosen random processes, is also useful in other p h y s i c a l l y interesting cases [26],
[32]).
VI.8
Pauli P r i n c i p l @ Let us consider in
~3 , N
(see, e.g.
particles which cannot be distin-
guished. The c o n f i g u r a t i o n space of this system is the M a n i f o l d consisting of all u n o r d e r e d
[14b,c])
N-uples
{XI,...X N}
in
M
~ 3 N , where the
X. are distinct points of ~ 3 . This is a d i f f e r e n t i a b l e m a n i f o l d l which is not s i m p l y - c o n n e c t e d if N > I. Indeed, the universal covering space of
M
is
~ = ~ 3 N /D,
D
being the set of all ordered N-
uples such that two or more points coincide. The fundamental group of M
is the symmetric group
when a smooth wave function
S N . To construct d i f f u s i o n on ~
on
M
This is the case if the wave function
M
we ask
generates a d i f f u s i o n on ~
M .
is either symmetric or anti-
symmetric but not a s u p e r p o s i t i o n of the two. It follows from this result that the e x c l u s i o n principle,
e.g. the d i s t i n c t i o n b e t w e e n Bose-
E i n s t e i n and F e r m i - D i r a c statistics is a consequence of the basic principle of stochastic m e c h a n i c s and is not an additional hypothesis. For more details, we refer to [9Oe],
[66]
131
VI.9
The C o n n e c t i o n Between S t o c h a s t i c M e c h a n i c s and E u c l i d e a n Quantum Mechanics The E u c l i d e a n f o r m u l a t i o n of q u a n t u m m e c h a n i c s is obtained by
a n a l y t i c a l c o n t i n u a t i o n to imaginary time. The existence of such analytical c o n t i n u a t i o n follows from the p o s i t i v i t y of the Hamiltonian. -tH The time evolution is now given by the semi-group e In this framework,
the S c h r ~ d i n g e r e q u a t i o n is r e p l a c e d by a d i f f u s i o n equation and
therefore a stochastic i n t e r p r e t a t i o n is very natural and suggestive. However,
it should be e m p h a s i z e d that in this approach the diffusion
processes play a purely auxiliary role since they do not take place in "real time" and are only used as a m a t h e m a t i c a l tool to prove theorems about operators on Hilbert space Let
~o
(see e.g.
[57],
[100a].
be the ground state of the H a m i l t o n i a n
H =
~2 -
2---~ A +
H~o =
(6.47)
V
(6.48)
0 .
Since
90
v = 0
and the a s s o c i a t e d process which is solution of the stochastic
is strictly positive it follows that the current v e l o c i t y
differential equation
[6Oa],
dXt = b+(Xt)dt +
[95]
2~ dWt
is stationary. Let us consider as simplest example the ground state of the onedimensional harmonic oscillator x2 ~o =
(2~a)-I/4 e
4a
a = ~
(6.49)
which leads to the drift vector
b+= - ~x
(6.50)
and to the F o k k e r - P l a n c k equation = ~
3t
2m
~ 2p
+ wP + wx ~p
3x 2
This e q u a t i o n can e x p l i c i t l y be solved. Namely, we have
(6.51)
132 f p (x,t) = i
p (y,O)
p(y,0,x,t)dy
(6.52)
with p(y,0,x,t)
= [ 2 ~ ( t ) ]-I/2exp{
I (x _ e-~t y)2} 2~$ (t)
(6.53)
An easy calculation shows that [X ° X t] = $(t) = de -~Itl
(6.54)
which looks very "Euclidean". More generally,
for the ground state process the Fokker-Planck
equation can be written
?P0~p] and has the stationary solution
(6.55)
P = P0
2
= ~o "
It turns out that in this case
-Iti ~ [Xo Xt] = <~o' X e
X ~O>L2(~d)
(6.56)
This relation is very remarkable since it links a "real time" q u a n t i t y on the left hand side to an "imaginary time" quantity on the right hand side. This connection was first noted by F. Guerra and P. Ruggiero [63]:
"Euclidean quantum mechanics
(or field theory)
is the ground
state process of stochastic mechanics". To show the mathematical origin of this relation, that the Fokker-Planck operator
HFp
let us remark
given by
g2 HFp .... ~
5 + ~V. (b+.)
(6.57)
with b+ = 2~m
V~o 9o
is not only symmetric in
(6.58) L 2 ( ~ d ,Podx)
but actually unitarily equi-
valent to the self-adjoint Hamiltonian operator L 2 ( ~ d ,dx)) HFp
=
U -I ~o H U~o
H
(defined in
(6.59)
133
where
U~e
is the unitary operator
from
L2(]R d ,dx)
given by multiplication
by
L2(IR d ,P0dx)
onto
90
From this it follows that the Fokker-Planck
equation has the
formal solution p(x,t)
t = e- ~ HFP p(x,O)
t > 0
tH
-I = U~0
M
e
U~0
O (x,O)
tH =
Moreover,
~0
e
this relation
a relaxation
process,
the spectrum of
- ~g ~ 1
p(x,O)
shows that the Fokker-Planck
the relaxation
equation describes
times being directly related to
H. The unitary equivalence
that the study of the unitary operator
of
H
e-item
and
HFp
tell us
in quantum mechanics
can naturally be connected with the study of the Markov semigroup e-tHFp/M(homomorphically
extended to
e-itH/i).
process
The associated
e -itHFP/M,
unitarily equivalent
is called a distorted
to
Brownian motion
(see [8]). Remark:
The relation
linking the real time quantity
the imaginary time object since the real time
VI.iO
<~0' x e - t ~ M X ~ o > L 2 ( ~ d ) c a n
quantity
The Semiclassical
is not accessible
with
to measurement.
Limit
An approach based on stochastic mechanics to study certain aspects of the semicl~ssical ics, i.e. the limit
~ [X ° X t] be true only
has been very useful
limit of quantum mechan-
~ ÷ 0 . In this limit the stochastic
differential
equation dX t =bi(Xt,t)dt
+
,~,lF dW t ~;
6.61)
can be analyzed using the theory of large deviations processes
[49],[103,~073.
Jona-Lasinio,
Adapting
F. Martinelli
very interesting
and E. Scoppola
multiwell potential
instability
exhibiting
ground state is degenerate).
G.
[79] have discovered new,
features of the semiclassical
case like the tunneling
for stochastic
the Freidlin and Ventzel method.
limit in the stationary
due to localized deformation
several equal minima
of a
(i.e. the classical
134
The m e t h o d consists in studying the process associated to the ground state
~o
of the q u a n t u m system.
In this case,
b+
is a gradient
2 b+ = 2~m Vlog ~o and from the equation
(6.62)
(2.52) we conclude that
V.b+ + b~ = 2 (V-E) m
(6.63)
and can separate the p r o b l e m in two steps. The first one c o n s i s ~ in studying the solution of e q u a t i o n
(6.63) when
~ ~ O. Indeed, the logarithmic
d e r i v a t i v e of the ground state wave function contains the essential information on the tunneling. The second step consists
in computing by
p r o b a b i l i s t i c m e t h o d s the spectrum of the g e n e r a t o r
A of the process
which is a s s o c i a t e d to the H a m i l t o n i a n by by
(6.59), in the limit ~ 0 ,
-~A + E o = ~ I
H ~o = HFp
to estimate the splitting of the ground
state level. The results o b t a i n e d show quite generally that both the localization state
of the
wave
function
and the splitting of the
ground state are very sensitive to small local d e f o r m a t i o n s of the potential.
In particular,
this is t h e position of the d e f o r m a t i o n rather
than its absolute value w h i c h is the relevant factor. This work was extended recently in various directions More generally,
[72],
[I00b3.
in the framework of stochastic m e c h a n i c s the
semiclassical limit consists in comparing a classical smooth path w i t h a diffusing one in its neighbourhood. In the weak noise limit large deviations the form
[103
(for example as
M ÷ 0)
the theory of
[107] leads in the simplest case a b e h a v i o u r of
e- S/M w h i c h can not be handled by usual p e r t u r b a t i o n theory,
indicating how it is natural to use the methods of stochastic mechanics to study n o n - p e r t u r b a t i v e effects.
135
VI.11
Bose Quantum Guerra
using
field
the ground
a large b u t orthonormal
each of w h i c h state wave
Removing
investigated
finite box B c basis
the cut-off
found that
Enclosing
~s
it
free field
of i n d e p e n d e n t
the d i f f u s i o n
fields
the free
and e x p a n d i n g
the study of the
performs
quantum
harmonic
associated
with
function.
i.e.
the g r o u n d
is the free E u c l i d e a n S(x-y)
[63] have
to the study of an a s s e m b l y
oscillators,
Ruggiero
[6o d]
of v i e w of s t o c h a s t i c m e c h a n i c s .
into
a complete
is r e d u c e d
Theory
and R u g g i e r o
from the p o i n t scalar
Field
Markov
in the limit B ÷ ~ 3 G u e r r a
state process
field w i t h m e a n
for a scalar
and
free
field
zero and c o v a r i a n c e
given by S(x-y)
= E
[(~(x,t)
~(y,t) ]
i _
ik. (x-y)
I
dk
e
(2z) 3
]R 3
2~ (k)
i/2 with ~(k)
=
(k z + m z)
The u n d e r l y i n g of mass
stochastic
differential
equation
for the f i e l d
m can be w r i t t e n
I/2 d~0(x,t) where W(x,t)
gives
For m o r e
[90 f] N e l s o n
j(x)
+ m 2)
dW(y,t)
~(x,t)
of field
] = ~(x-y)
dt + dW(x,t)
he suggests
is p o s s i b l e
theory.
Markov
field
[63].
~j for scalar
Moreover
dt
of the free E u c l i d e a n see
the G u e r r a - R u g g i e r o
fields
~(x).
of q u a n t u m m e c h a n i c s framework
details
extends
a family of random coupling
[ dW(x,t)
a new interpretation
in real time.
In
(- A x
is such that
E
This
= -
without
procedure
currents
j with
to c o n s t r u c t the
linear
that no real u n d e r s t a n d i n g considering
the larger
136
In a recent paper of the q u a n t u m mechanics
[22 f] E. Carlen uses
dynamics
for free
of the free scalar
sample path p r o p e r t i e s
to single p a r t i c l e
In this
framework
the K l e i n - G o r d o n
which
for a free p a r t i c l e
e quatio n
for a particle
equati o n
is used to define
Ph.
states
equation
sense have
filtered
as their
fluctuations
functions
w i t h one
position.
A detailed
can be found
in
stands
of the field.
in the same r e l a t i o n
to
field as does the N e w t o n
the c l a s s i c a l
of the K l e i n - G o r d o n
out the v a c u u m
the
corres-
in IR d m x = o to the free S c h r ~ d i n g e r d of mass m in ~ In both cases the k ~ n e m a t i c a l
Blanchard,
shown that one p a r t i c l e
diffusions
equation
for the q u a n t i z e d
equation
Very r e c e n t l y
mechanical
A d + m 2) ~ = 0
equation
equation
representation
the stochastic
field of mass m and to i n v e s t i g a t e
configurations
22 (-~t 2 is a k i n e m a t i c a l
to c o n s t r u c t
of the stochastic
ponding
Schr~dinger
the S c h r ~ d i n g e r
fields
localized
phase
E. Carlen in w h i c h
space of the system.
and G.F.
the c o r r e s p o n d i n g
are s t r o n g l y
localized
field c o n f i g u r a t i o n s in a p h y s i c a l l y bump
localized
have
solutions
in the N e w t o n - W i g n e r
(obtained by filtering
meaningful
covariant
way)
near the N e w t o n - W i g n e r
account of the c o n s i d e r a t i o n s
[13 bis].
Dell'Anton~o
sketched
here
VII.
VII.I
General This
theory,
chapter
one,
physics in terms
w i t h many
degrees
there
of freedoms,
of degrees
for the other.
one.
These
equations.
of degree
servative understood given
from the to In
which
are well a system
it is p o s s i b l e
now all the
of m o t i o n
adapted
disturbed,
Let us consider
the v a r i a t i o n s
In a p h e n o m e n o l o g i c a l
are the result
framework randomly
of problems,
only to c o n s i d e r
equations
to select
of w h i c h
fast degrees
the e q u a t i o n s become
then
description
of the i n t e r a c t i o n
with
are
a
slower
of freedom
of m o t i o n
stochastic of this
of
diffe-
type the
the e n o r m o u s
number
of freedom of the environment.
Let us now be more our model.
Nelson's
systems
such that
Incorporating
the
of e x t e n d i n g
processes.
of freedoms
source we have
fluctuations
MECHANICS
of q u a n t u m m e c h a n i c s
mathematical
is a large class
of stochastic
in a noisy
rential
a general
of a class of dynamical
classical
slower
to the p o s s i b i l i t y
is a d e r i v a t i o n
to p r o v i d e
modelized
than
is d e v o t e d
originally
the d e s c r i p t i o n
small number
LOOK AT S T O C H A S T I C
Remarks
which
classical
A NON-QUANTAL
We c o n s i d e r free
field.
precise
about
a large number The m o t i o n
and the c l a s s i c a l
the general of p a r t i c l e s
of an individual
deterministic
physical
basis
travelling particle
equations
of
in a con-
is quite well
of motions
are
by '
1
xi = m P i (7.1) Pi
F
describing
if on large
is not the case, collisions
+ F(xI,...,XN)
the i n t e r a c t i o n
ber of p a r t i c l e s more
= -V
involved scale
the
with
justifies
the other particles. a statistical
system is stable,
local i r r e g u l a r i t i e s of the
and n e a r b y
dom way the c l a s s i c a l for a r a n d o m b e h a v i o u r
encounters picture
on m u c h force
of p a r t i c l e s
described
of the particles.
The
treatment. smaller
They are the
num-
further-
scale
field as well
tend to m o d i f y
before.
large But
this
as
in a ransource
138
Both these reasons jectories chastic
of the p a r t i c l e s In many
process.
seems natural
to a s s u m e
to consider
are m o d e l e d
situations
of r a n d o m
changes
that the p a r t i c l e
its present
state. These rather
that the r a n d o m process words
the process
where Wt
Xt
is its p o s i t i o n
the standard W i e n e r
taking
into account
ly speaking
model we have
in mind
ral situations.
for w h i c h
and collithrough
imply m a t h e m a t i c a l l y process.
In other
differential
equation
dt + ~ dW t
process.
(7.2)
b+
~
that we made
stage
field and
coefficient
of the environment.
the additional
diffusion
can be g e n e r a l i z e d at this
a velocity
is a d i f f u s i o n
properties
and c o n s t a n t
However
and r e a s o n a b l e
of the past
assumptions
is a d i f f u s i o n
it
zero w e i g h t
irregularities
only the m e m o r y
at time t,
implies
isotropic
of a sto-
those p r o c e s s e s
of the stochastic
the diffusive
(7.2)
homogeneous,
gives
convenient
i.e.
due t o local
innocent
= b+(Xt,t)
xt(~)
the tra-
point of view
process
it seems
we consider
is solution
d~
stochastic
keeps
in w h i c h
by the paths
random processes
sions makes
model
from a p h y s i c a l
Furthermore
only M a r k o v
frequence
a s t a t i s ti c a l
that the
paths.
to d i s c o n t i n u o u s
the
justify
assumption
(see section
to take
II.3).
into account
is is not n e c e s s a r y
Strictof
The
more gene-
to consider
such a refinement. Up to now the drift short range
forces
efficient
~
considered.
a fact w h i c h
reflects
scale.
However, larger
is unspecified. into account
The paths
of the process
if one reminds scale,
the
smooth
one can define
(see ChapterII),
which
allows
force
co-?V
are not differentiable,
character
of the
on a small
force
acceleration
of derivative
to write
if the
field
of the e n v i r o n m e n t
a stochastic
of the notion
Indeed,
by the d i f f u s i o n
of the d e t e r m i n i s t i c
the r a n d o m c h a r a c t e r
tained by a g e n e r a l i z a t i o n process
b+
have been taken
, the influence
has not been
on a much
field
field ob-
for a d i f f u s i o n
a Newton's
law in the
mean: ma = -VV (x)
According
(7.3)
to this procedure
the u n d e r l y i n g
stochastic
equations
are c o n s t r u c t e d a) by g e n e r a l i z i n g probabilistic
the classical
b) by a g e n e r a l i z a t i o n for d i f f u s i o n
kinematics
in order
to allow
for
description of the c l a s s i c a l
motion,
which
gives
dynamical
to the drift
law a p p r o p r i a t e b+
a dyna-
139
mical meaning.
In other words
bridge
the d i s o r d e r
between
and the overall The next bability
force
step consists
the
stochastic
existing
field
acting
at large
in i n v e s t i g a t i n g
d e n s i t y of the p r o c e s s e s
i.e.
the
Newton
law is a
at the m i c r o s c o p i c
scale
scale.
the p r o p e r t i e s
functions
of the pro-
p(x,t),
x
6 ~d
such that
[f(xt)] ]E
where
denotes
f = Jl~d p(t,x)
the e x p e c t a t i o n
f(x)dx
with
(7.4)
respect
to the random process
The d e n s i t y p satisfies the F o k k e r - P l a n c k e q u a t i o n and an Xt a d d i t i o n a l c o n s t r a i n t coming from the N e w t o n ' s law in the mean. To solve explicitly
this
couple
it is c o n v e n i e n t reversing
of n o n - l i n e a r
to suppose
its sign under
that the current
time
III).
In a sense
this
noisy
turbulence
is on a much
to describe.
tical
further
The current
be e m p h a s i z e d
reversal)
smaller
physical
about the p h y s i c a l the m e c h a n i s m
meaning.
nature
responsible
velocity
corresponds
scale
than the
is e x p e c t e d
of N e l s o n ' s In N e l s o n ' s
of the noise,
and
b+
(the part of
b+
field
(see Chapter
to situations features
we obtain,
stochastic approach
neither
for the d i f f u s i o n
p
arises
It must
although
mechanics,
no statement
is needed.
where
we want
to be observable.
that the e q u a t i o n s
in form to the e q u a t i o n s
a different
involving
is a g r a d i e n t
assumption
velocity
at this point
equations
iden-
have is made
In our model
from a real p h y s i c a l
process. Let us also
remark
tems by d i f f u s i o n s sal.
Indeed,
that the c o n v e n t i o n a l
in v e l o c i t y
appealing
to the
mechanics
one can realize
the rSle
of the Planck
space formal
that
description
is not c o n t r a d i c t o r y analogy
of our model with
if the d i f f u s i o n
constant
~
of such
constant,
, is very
small
sys-
to our propoquantum
which
the
plays
stochastic
process Xt d e p r e s s e s in a e x p o n e n t i a l way by a factor of the form s e'~ the w e i g h t of those paths w h i c h are far from the "classical" ones, i.e.
those
Indeed,
corresponding
random
Also
from the p o i n t
using n u m e r i c a l extremely
methods
although
~ = 0, w h i c h become
are
of view of g a i n i n g
of Monte
solutions
deterministic
Carlo
type
in the
of
information stochastic
(7.1).
limit
~ + 0.
from m o d e l s
methods
are
powerful.
In Chapters process
to
trajectories
III and IV we have
is still w e l l - d e f i n e d the drifts
rity of the drifts
when
are not defined on the nodal
shown that a N e w t o n i a n
the d e n s i t y on the nodes.
surface
p =
[~I 2
Indeed,
Np = {(x,t)
diffusion has
the
zeros,
singula-
~dx~+Ip(x,t)=0}
140
produc e s
a repulsion
which
is strong e n o u g h
from ever r e a c h i n g
the nodal
the nodes
separate
space)
of
p
into d i s c o n n e c t e d
~d
If the process will
never
all time.
= N
Xt
say that the
family surface
group
manifold
If
]M =
the d i f f u s i o n diffusion
VII.2
Impenetrable
cussed
barriers
III.
been given dynamical
to the
formation
to b i o l o g i c a l systems
A Model
These
density
barriers
[2'] B r b , ~
[84
All
for d y n a m i c a l
as well
sys-
have been dis-
are d e s c r i b e d
patterns
the
are again valid.
mechanics
of the u n d e r l y i n g
of spatial
systems
of
Patterns
and hence
as of stochastic
and IV.
of the p r o b a b i l i t y
Applications
VII.2a
for diffusions
as well
groups
from one
in terms
III.).
is constant
of Spatial
we can
and a N e w t o n i a n
(see Chapter
and F o r m a t i o n
yi for P barrier
the R i e m a n n i a n
is given
(~t~)z1j
~
, it
several
consider g
gii~ =
94
into
and no p a r t i c l e
we must
in the case w h e r e
theory
in Chapters
surfaces
by
in
i
In c o n c l u s i o n
is split
the m e t r i c
a
with values
Phenomena
tems of q u a n t u m
particles
where
for some
F lp.
of the d e n s i t y
g),
Fi p
in
is not c o n s t a n t
obtained
Trapping
XO E
P is c o n f i n e d
N
(~d,
the c o n f i g u r a t i o n among them:
N and will stay in P acts also as i m p e n e t r a b l e
N
coefficients
process
c onclus i o n s
in
surface
of typical
~
generally
case that
(7.5)
started
P can pass to another.
Remark:
(or more
stationary
P
Kt
by the nodal
to keep the c o n f i g u r a t i o n
in the
Fi
surface
and
~d
n U P i=l
reach the nodal
for the process
Suppose
pieces w i t h no c o m m u n i c a t i o n
was
The nodal
set.
by nodal
diffusion
are m a n i f o l d as several
process. and have
physical
[4] ~3].
of the F o r m a t i o n
of J e t - S t r e a m s
in the Protosolar
Nebula It is an old h y p o t h e s i s prOtoSolar (dust).
nebula
In one
by Descartes steadily
Consisting
(1644),
Kant
solar
explain
the origin
planets
from the
the T i t i u s - B o d e
all
law
(1755)
was d i s c u s s e d (1796)
and has been
given
in the distances
this R
n
from a
originally
in the d i s c u s s i o n
There have been m a n y earlier
Classically,
formed
of a gas of small p a r t i c l e s
and Laplace
of the r e g u l a r i t y
(1766),
system was
this h y p o t h e s i s
later d e v e l o p m e n t s
system.
sun.
solar
es~entially
form or another
accompanying
origin of the
that the
regularity in the
of the
attempts Rn
of the
was d e s c r i b e d
form
to
by
141
R
n
= a + bc n
for suitable constants
(7.6)
a,b,c.
One idea w h i c h has been i n t e n s i v e l y
d i s c u s s e d r e c e n t l y is a sort of m o d e r n v e r s i o n of the K a n t - L a p l a c e ring formation:
namely that, before the a g g r e g a t i o n into planets,
Centric roughly planear rings were ice, p a r t i c l e s and dust,
formed.
con-
The rings consist of gas,
c i r c u l a t i n g inside the rings but w i t h no
c o m m u n i c a t i o n with n e i g h b o u r i n g rings. The formation of the planets should then have h a p p e n e d in a later state by a g g r e g a t i o n from the jet-streams
from N e w t o n i a n diffusions.
The
same kind of ideas can be applied also to the formation of jet-streams around planets
(Jupiter,
Saturn...).
Our stochastic model provides a general m e c h a n i s m able of exp l a i n i n g the formation of the jet-streams around a m a i n body planet):
mutual chaotic c o l l i s i o n between dust grains m o v i n g in the
gravitational of toroidal mass
(Sun or
M
field of the central body tend to focus into jet-streams
shapes c e n t e r e d on the central body.
The central body of
acts by some spherical symmetric p o t e n t i a l
V(Ix I)
and
is immersed in some d i s o r d e r e d gas of small p a r t i c l e s acted upon by V
and i n t e r a c t i n g by collisions or p s e u d o - c o l l i s i o n s ,
e.g.
the p r o t o s o l a r nebula of the most common c o s m o l o g i c a l models.
like in The
basic idea consists in thinking of a typical particle in the nebula as performing,
under the steady influence of the a t t r a c t i o n of the
central body and innumerous chaotic c o l l i s i o n s with other particles, a stochastic d i f f u s i o n process.
In other words we assume that a typical
p a r t i c l e moves along the trajectories of a N e w t o n i a n d i f f u s i o n process Xt
with a p o t e n t i a l
V
given a p p r o x i m a t e l y by the g r a v i t a t i o n a l
a t t r a c t i o n and that there exists an i n v a r i a n t d i s t r i b u t i o n as the p o t e n t i a l is a t t r a c t i v e and the time scale involved is large. Of course the i n v a r i a n t d i s t r i b u t i o n is thought to hold as long as the d i f f u s i o n a p p r o x i m a t i o n is valid. From the results of Chapter III distribution
p = l~I2
eigenvalue problem barriers
we then know that the invariant
is given by the solution of a S c h r ~ d i n g e r type
H~ = E~
and that the nodes of
for the N e w t o n i a n d i f f u s i o n process
explanation
Xt,
~
hence y i e l d i n g an
for the n o n - c o m m u n i c a t i n g rings in the nebula.
being central the e i g e n f u n c t i o n s 1 = 0,1,...n-l,
m = -I,...,+i
~n,l,m(X)
in
act as
L 2 ( ~ 3)
The potential with
are of the form
~ n , l , m (x) = Rn,l(IXl)
m(@, ~° ) ~i
(7.7)
142
with
Rn, 1
solution of an ordinary second order d i f f e r e n t i a l e q u a t i o n m and 41 (@,~) the usual spherical harmonics.
(the radial equation) The
Ixl
d e p e n d e n c e of the zeros of
zeros of the radial function
~n,l,m
Rn, I.
Setting
is d e t e r m i n e d by the Pn,l,m =
we can calculate the a s s o c i a t e d current v e l o c i t y
l~n,l,m I~
Vn,l, m.
The angular
m o m e n t u m in the Z-direction is given by
L z = I]R3dX e Z "(X x V n , l , m ) = c m with
c
constant.
(7.8)
This is the classical angular m o m e n t u m of the nebu-
la. Using the c o n s e r v a t i o n of the total classical m o m e n t u m and choosing Oz
along this d i r e c t i o n we conclude that the invariant m e a s u r e s to be
c o n s i d e r e d are of the form
Pn,l,l(X) Recalling now that that
Pn,l, !
=
~(@,~)
is, for
1
l~n,l,l(X)I 2
(7.9)
is p r o p o r t i o n a l to
eil~(sine) 1
we see
large, c o n c e n t r a t e d to a small angular
region about the e q u a t o r i a l plane. This c o r r e s p o n d s to the fact that the p l a n e t a r y system is essentially two-dimensional.
The t r a p p i n g regions
("jet-strean%s") are regions c o n f i n e d between concentric spheres c e n t e r e d at the center of the m a i n body and two cones. [13]and [4 bis] for n u m e r i c a l results. the m o r p h o l o g y of galaxies.
For more details see
~,b],
See also [3c,43 for an a p p l i c a t i o n to
VII.2b Cloud Covering of the Planets The available picture of planets w i t h a substantial atmosphere exhibits on a large scale regular structures,
namely zonal bands.
To mode-
lize such p h e n o m e n a statistical methods are very attractive a l t h o u g h it is very hard to justify them from the p r i n c i p l e s of fluid dynamics~ Indeed think of clouds as being c o m p o s e d of "particles" either droplets of icy flakes. Apart from the g r a v i t a t i o n a l
forces,
these
"particles"
feel very c o m p l i c a t e d forces from the surrounding turbulent
atmosphere.
We do not intend to take into account the details of these
influences but assume that it can be r e p l a c e d by a d i f f u s i o n mechanism. F u r t h e r m o r e we shall make no precise statement about the overall force only assuming it is spherical symmetric and derives V(r).
As in the section
sidered are of the form
VII.2.a. (7.9).
from a p o t e n t i a l
the invariant m e a s u r e s to be con-
Nodal surfaces are either spheres
around the origin c o r r e s p o n d i n g to the zeros of the radial part
Rnl
of the a s s o c i a t e d wave function or cones defined by the zero of Legendre functions
Pl'm
Hence possible
zones of c o n f i n e m e n t are anuli
143
This m o d e l planetary scale
does
structures
in mind
are
range
The Van Allen
involved,
namely
of the o b s e r v e d
field
that means
integer rela-
atmosphere
radiation
results
and
that one cannot
hope
varies
of rockets
and satellites
It was
must be charged
field.
(i.e.
elec-
Assuming
tend
moment
~
formula
that
vu
(7.10)
(the c o m p o n e n t
of
of i n c r e a s i n g
to keep
~
constant
the total
velocity,
particles
are r e f l e c t e d
since
the p a r t i c l e s field.
protons
westward.
parallel
back
time
drift
vll
belts
by the i r r e g u l a r i t i e s The Lorentz
owing
force
acting
increase
are called
Thus
to the
field.
This
a magnetic
of t o r o i d a l
shape
to the i n h o m o g e n e o u s
that the c h a r g e d
on a p a r t i c l e
will
Suppose
it is equal
to zero.
field e l e c t r o n s
of the e l e c t r o m a g n e t i c
to B
the p a r t i c l e must
of lower m a g n e t i c
are a c t u a l l y
longitudinally
suggests
v~
fallen
particle
fields.
only until
are r e f l e c t e d
In a dipole m a g n e t i c This
Then
has
into regions
radiation
charged
to B) takes
field.
it can increase
the p a r t i c l e s
The Van A l l e n
magnetic
but
v
perpendicular
of higher m a g n e t i c
magnetic
at w h i c h
v
that the g y r a t i n g
from regions
into a r e g i o n
mirror.
of the v e l o c i t y
it is clear
kind of region w h e r e
for
is given by
is the c o m p o n e n t
to be r e f l e c t e d
that
then the m a g n e t i c
This m a g n e t i c
vl
From this
of
that the
particles
magnetic
slowly with p o s i t i o n constant.
of zones
soon e s t a b l i s h e d
= m 2-B v&
to
in this model.
in 1958 by Van A l l e n
in the Earth's
is n e a r l y m
vary on a
temperature
Belts
the Earth.
B
of mass
in p l a n e t a r y
the d i s c o v e r y
trapped
of a p a r t i c l e
where
large
if one
a fit with
fit to the o b s e r v a t i o n s
entertaining
surround
and protons)
parameters
precise
has been
which
a particle
good
as far as the composition,
Radiation
One of the most investigations
the m a g n e t i c
is very
that one can make
are concerned,
get a more
moment
of the
of these
low numbers.
pressure
trons
observations
few free p a r a m e t e r s
the p h y s i c a l large
source
with
and it is nice
tively
radiation
parameters
feature
3c3.
of the model
n,l,m
VII.2c
on d i f f e r e n t
for the general
that
i) there
ii)
too much
but accounts
[2 ,
The a g r e e m e n t keeps
not d e p e n d
atmospheres
drift
particles
eastward
and
are d i f f u s e d
field. of charge
q
is given
144
classically
by
F = qv
v
being
as the
the v e l o c i t y .
force
sion w i t h should a
acting v
the m a s s
m.
satisfies
A
being
some
cylindrical
if we
finity
function
the e f f e c t i v e the
B
of
relative
r =
that
~
of
=
stochastic
in C h a p t e r
the
force
accelaration
divided
III.6.
then
force
that
by the
~ = p112 e is
- q A)2~I
1
(7.12)
~A -y~-~
and
,
plane.
~ =
as
(7.13)
(x2+y2) I12 reduces
potential
Introducing
now
to a t w o - d i m e n s i o n a l
potential
A
symmetry
The m o s t g e n e r a l
given
by
(7.14)
-+ q pZA(p,z)] 2
+ z2) i/2 g o i n g _eff U±lll have the
goes
to i n f i n i t y following
to zero
in a g i v e n
shape
1
\
t
axial
= 2A + p ~A ~
Bz
an e f f e c t i v e
[m ~ 2 J l !
with
can be w r i t t e n
p, ~) (7.12)
and
A
in the m e r i d i a n
the v e c t o r
q
potential
requirement
z
with
(x 2 + y 2
potential sign
y
(z,
equation
assume
with
For
the L o r e n t z
(-i m qzV
B this
~A ~--~,
Deff +llIL~l= Then
i 2m
field
coordinates
SchrBdinger-like
because
diffu-
constant.
satisfying
Bx =-x
do this the
(7.11)
a Newtonian
equation
an e l e c t r o m a g n e t i c
a magnetic field
we c o n s i d e r
undergoing
is a g r a d i e n t
type
~@ _ ~t
diffusion
We c o n s i d e r
reversal.
as e x p l a i n e d
SchrSdinger
We
substitute
m v + qA
framework
particle
velocity.
we
Assuming
the
stochastic
time
Xt
i m ~2
producing
In our
under
momentum
the
being
(7.11)
the c u r r e n t
be i n v a r i a n t
generalized
B
on a c h a r g e d
of the p r o c e s s
magnetic
x
k
at indirection
depending
on
145
This
implies
state,
that
whereas
for ql ~ 0 the
if ql < 0 bound
some c i r c u m s t a n c e s bound
states)
q2p2A(p,
z)
= B0P-1
The model
accounts
for the
i) There
exists
following discrete surface
zone,
westward
or e a s t w a r d
electrons 3) T r a p p i n g lines
cribed
the pure point
s p e c t r u m of
detailed
zones
discussion
show a general
according Protons
to'the
drift
density. drift
either
charge,
since
to the west,
are roughly
shaped
according
to the
field.
contain more
energetic
of the model
see
particles
than
[23.
in a Plasma
idea consists consisting
by the paths
of a p r o b a b i l i t y
related
belts.
for T r a n s p o r t
structure
facts:
zone of c o n f i n e m e n t s
of the m a g n e t i c
outer
The basic
like
to the east.
4) Inner belts
magnetic
admits
of the m a g n e t i c
behaves
observational
particles
ql must be positive.
A Model
(i.e.
is not empty.
2) In each
VII.3
fall-off
potential
It can be shown that
has no bound
To show that u n d e r
is c o n f i n i n g
on the
The e f f e c t i v e
to the nodal
For a more
like e q u a t i o n
can appear.
potential
must be made
distances.
(- i m o2V-qA) 2
stakes
the e f f e c t i v e
assumptions
field at large
Schr~dinger
of t h i n k i n g
the m o t i o n
of n u m e r o u s
small m a g n e t i c
of a stochastic
non-isotropic
of p a r t i c l e s island
differential
in
as desequation
is dX t = bt(Xt,
t)dt + DodW t
(7.15)
where D k
being
the B o l t z m a n n
the e l e c t r o n
mass
ly p r o p o r t i o n a l [533,
kTT m
and
(7.16) constant, T
to the viscosity.
the m a g n e t i c
field
form
~x ~* o =
0* ay 0
In the
B 0 acting
the z~axies and the d i f f u s i o n of the
T
0
the a b s o l u t e
a characterictic
matrix
temperature,
term of d i f f u s i o n
slab g e o m e t r y
on the plasma
considered
is e s s e n t i a l l y
~ is a symmetric
constant
m
inversein
along
matrix
0 (7.17)
146
To determine the drift for m a g n e t i c
b+
ma = ev a
we use Newton's
law in the mean, w h i c h
force takes the form x B0
being the stochastic acceleration and
of the process
Xt"
osmotic v e l o c i t y
v
the current v e l o c i t y
From this dynamical assumption follows that the
u
and the current v e l o c i t y
v
non-linear coupled partial differential equation.
are solutions of
It could be shown
that these equations can be related to a simpler one of the form
(see
[52] [53] for more details)
!
K(x,y)
~t
=
D(x,y)
with ~=2
where A g = q t.
1
ie (~J - - m
Aj) GJk(yR
is the vector potential and
Knowing
K
hence the drift
ie m
G
the
it is possible to obtain b+.
AR )
0
(x,y) part of
and
v
and
In the simplest case of a stationary process
one obtains an O r n s t e i n - U h l e n b e c k process namely x
dx t
xt
dW t
= A dYt
+ A Yt
dwtY
with A
Ii
and
X2
=
I I B0 2m e
-ll/l 2 1
-I -%2/~i
being the eigenvalues of
G.
One can deduce that the
radial d i f f u s i o n rate satisfies in the case of a strong m a g n e t i c
I~ <X2>T = F(ll'
12) - ~ - - 0
field
+ o( )
This result c o r r e s p o n d s to the simplest situation in which the stochastic d i f f e r e n t i a l e q u a t i o n is linear and can be e x p l i c i t e l y integrated. Thegn observed
behaviour of the radial d i f f u s i o n rate has been e x p e r i m e n t a l l y (Bohm's law).
APPENDIX
We
review
AI.
some
Notations
of the b a s i c and
this
This
that
means
space
book
is a m e a s u r a b l e
(~, 5
(~, F)
P)
on
(~, F)
F.
Fo
is a s u b - ~ - a l g e b r a
If
smallest
denotes
such
that
o-algebra
each
space
a given
is a m e a s u r a b l e
measure
the
of p r o b a b i l i t y
theory
below.
Conventions
A probability Throughout
notions
space
subset
of
F
containing
the
P(~)
= i.
probability
space.
and P is a p r o b a b i l i t y
of a
F°
(~, F, P) w i t h
complete
P-null
set
augmentation and
all t h e
in
F°
F
of
P-null
is in F°
sets
is in
F . O
Elements function
X
funtion ~n.
X
For
0-algebra
÷ ~n
is c a l l e d
any
the m e a n
of the
: (~,F)
random
measurable
variable
or e x p e c t a t i o n
E[X]
F are
is c a l l e d if
X
of
X
= /~ X(e)
events.
X-I (A) 6 F
/~ X(~) and
called
a n-dimensional
P(de),
is d e n o t e d
P(de)
A measurable
random
variable.
for all B o r e l if it exists,
by
E Ix].
sets
A A
is c a l l e d
Thus
=
E ~ n ] is c a l l e d the n th m o m e n t of X a b o u t zero and E[(X-E[X]) n] the th n c e n t r a l m o m e n t . The s e c o n d c e n t r a l m o m e n t is c a l l e d v a r i a n c e and w i l l be o f t e n d e n o t e d by o2
c2 = E [ ( X _ E [ X ] ) Z ]
For
any
random
variable
X
= E[(X_p)z]
: ~ ÷ ~n
the
function
~
: ~n
+ ¢
defined
by (p) = E [ e i P "x]
is c a l l e d
the
characteristic
function
of X.
Here
p
6 ~n
and p - X
=
n
i~iPiXi" Events
BI...B j
are
called
independent
41...j} k P[ l=Dl Bi I] =
k K i=I
P[Bil]
if
for e v e r y
{ i l . . . i k}
in
148
An a r b i t r a r y sub-family
family
of e v e n t s
is c a l l e d
independent
if e v e r y
finite
is i n d e p e n d e n t .
A filtration that
Fs c Ft
also
satisfied
is a f a m i l y
for all then
{Ft}tE I of s u b - g - a l g e b r a s
s < t in I.
If the
{Ft} t I is c a l l e d
(i) F t = Ft+ (ii)
the
standard
(right
= sot Fs
F ° contains
following
of
F
such
two c o n d i t i o n s
are
filtration:
continuity)
all of the
P-null
obligatory
but
sets
in F
(complete-
ness)
Conditions calities. gales
require The
{Wt}t
(i) and Indeed
(ii)
many
these
standard
are not
useful
theorems
many
techni-
parameter
martin-
hypotheses. filtration
I is d e f i n e d
simplify
for c o n t i n u o u s
{Ft}t61
associated
with
a Brownian
motion
by
F t = g{WslO~s~t} ~
where
the
inclusion
The p h y s i c a l events
A2.
meaning
occuring
of the of
P-null
Ft
sets
is the
up to time
t: the
in
Ft
ensure
following: "past
Ft
events
that
is the
F t = Ft+ g-algebra
of
up to t".
Conditioning Let
X :
(2,F)
÷ ~n
be a r a n d o m
Let
of
y
F be a s u b - g - a l g e b r a o : (2,Fo) ÷ ~ n such t h a t
for e v e r y
/A Y (~) P(d~)
is any
If ~=
Y
random
P - almost Y is c a l l e d
given)
Fo If
and F °'
variable
variable
F. T h e n
there A
= /A X(~)
with
the
in
such
exists
that
E [~
a random
< +~.
variable
F°
P(d~)
same
properties
t h e n we h a v e
everywhere. the c o n d i t i o n a l
is d e n o t e d
c Fo c F
by
are
expectation
of
X
with
respect
(or
E [ XlFo3.
g-algebras
then
EEEEXI o ll o I = EEXl;o The
following
jection
proposition
on a H i l b e r t
exhibits
space.
conditional
expectation
as a p r o -
149
Proposition algebra if
of
F.
Let
The
H = H* = H2
L2(~,Fo,P)
(~,F,P)
be a p r o b a b i l i t y
L2(~,Fo,P)
denotes
the
is a c l o s e d orthogonal
space
subspace
projector
and of
of
F°
a sub-a-
L2(~,F,P) L2(~,F,P)
and on
then
Hf = E l f IF ° ]
A3.
Stochastic
Processes
In a p p l i c a t i o n s saying
that
ness.
This
process ~+
the leads
we
to the
and
process
X
e6~,
are
called
cess
is
Given
(one says
be d e n o t e d
t £ I.
{Ft}t 6 I to
associated fying
a o'algebra
If
A
{~[T(e)
general
FT,
if
~i
set
and time.
is a s t o p p i n g
time
property. W
Loosely up to
o
in
t 6 I. M o r e -
(-) = x
a.s.
The
X(-,w) :t÷X(t,~),
on
~
X t 6 F t for e a c h
if and o n l y
T2
filtration
~ t}
in
Ft
~n
the prot 6 I
for e v e r y
stopping in F
with
t for e a c h
the c o n d i t i o n
sets
time
=
on
T is
time
T is
U F t satist6I
t 6 I.
A}
stopping
fixed
this
finite
all
{~(w)~t}£F
To any
of all
for
Brownian
properties
speaking some
then
a stopping
then
> 01W t £
are
if
t 6I.
consists
In p a r t i c u l a r
One of the b a s i c
motion
X
is c a l l e d
for e a c h
which
is a n - d i m e n s i o n a l
is a s t o p p i n g
if
T:~ ÷ ~ +
T A : inf{T
Wt
if
of o - a l g e b r a s
family
{Ft}t 6 I
N {~IT(~)
is any B o r e l
where
{Ft}t£ I
to this
is a s t a n d a r d
equivalent
A
for e a c h
~n
{Xt}t6 I. The m a p p i n g s
function
to a f i l t r a t i o n
If
X
stochastic
is an i n t e r v a l
non-anticipating).
An F - m e a s u r a b l e respect
by
family
to be a d a p t e d
sometimes
I
by
and r a n d o m -
A n-dimensional
where
value
is d e s c r i b e d
of time
of X.
an i n c r e a s i n g
said
definition:
is F - m e a s u r a b l e
initial
the p a t h s
of a s y s t e m
is a f u n c t i o n
: I x ~ ÷ ~n
= X(t,-)
X has
will
evolutions
system
following X
Xt(-)
say that
random
of the
X is a f u n c t i o n
= ~,+~)
over
the
state
if
motion
times, T
time
then
says
time.
In
T 1 ^ T 2 = mi~(Ti,T2)
is a s t o p p i n g
time
then
T ^ t
t.
of B r o w n i a n
stopping
is a s t o p p i n g
that time
motion
is the
given
the h i s t o r y
T
the b e h a v i o u r
strong
Markov
of a B r o w n i a n of
W after
150
that time d e p e n d s More precisely is a s t o p p i n g
with
o n l y on
if
f: ~ d
T
and the state
÷ ~
W
at time
~.
is a b o u n d e d m e a s u r a b l e
WT
of
function
and
time t h e n
E[l{mlT(m)<+~}
f(WT+ t) IFT ]
= l{mlT(m)<+~o}
EWT [f(W~]
E WT [-] = E['IWT] .
A4. M a r t i n g a l e s Let
{Ft}t 6 I
is c a l l e d
be a f i l t r a t i o n .
a martingale
i) S t 6 L 1
for each t
ii) M s = E[MtlF s] We call
M
a submartingale
supermartingale
significance
value
of
(ii)
s, the v a l u e
if the
M
and
"=" in
M
-M
The b a s i c
is r e p l a c e d by ~ and a
M = {Mt,Ft} t 6 I
at time
t
to its in a
for a s u p e r m a r t i n g a l e ) .
is c a l l e d c o n t i n u o u s
is a s t a n d a r d
ii)
{Mt}t6 I
has all p a t h c o n t i n u o u s
if and o n l y if
filtration
martingales is c a l l e d
up to
equal
of our f o r t u n e
{Ft}t6 I
I
The i n t u i t i v e
of e v e r y t h i n g
i)
M = {Mt,Ft} t
M is a m a r t i n -
is, on a v e r a g e
(the game w i l l be u n f a i r
source of c o n t i n u o u s
A collection
(almost s u r e l y P)
are s u p e r m a r t i n g a l e s .
at the future time
is fair to us
A martingale
(ii)
Vt)
by ~. In o t h e r w o r d s
at time s. T h u s M t is the v a l u e
game w h i c h
s < t
is that g i v e n the b a h a v i o u r
of
M = {Mt,Ft} t 6 I
(E [ IXtl ] < +~
for all
if it is r e p l a c e d
gale if and o n l y if b o t h
time
A collection
if and only if
is s t o c h a s t i c
integrals.
a local m a r t i n g a l e
if and only
if i) M
is an
o
ii) t h e r e that Mk is a m a r t i n g a l e . sequence
for M.
=
F-measurable o
is a s e q u e n c e T
k
+ ~
{Mt A T k
The s e q u e n c e
a.s MO'
random variable
{Tk} k
~
of s t o p p i n g
and for each
times
k
Ft}t ~I
{Tk}k £
w i l l be c a l l e d a l o c a l i z i n g
such
151
A stochastic is e x p r e s s i b l e
process
Xt
as a s u m o f a
is s a i d to b e a s e m i m a r t i n g a l e
(local)
martingale
and a process
if
Xt
of bounded
variation.
A5.
weak c o n v e r g e n c e It is o f t e n
useful
o f some m e a s u r a b l e properties. measure
Once
on the
is d e f i n e d Let
always
into
space
function
X
a space
space
Spaces
a stochastic
is c h o s e n
be a s e p a r a b l e sets.
continuous
is t h a t
to c o n s i d e r
space
the
on M e t r i c
process
of f u n c t i o n s the
a n d the
with nice
stochastic
0-algebra
as a m a p p i n g
process
over which
topological induces
a
the measure
is r e l e v a n t .
t e d b y the o p e n which
and M e a s u r e s
the
B
functions
0-algebra
be o n
(X,B).
metric
is a l s o
B
space
the
are m e a s u r a b l e . is g e n e r a t e d
Such
and
a measure
B
smalles
by
~
the
~-algebra
o-algebra
The
with
advantage
spheres.
is a l w a y s
genera-
respect
to
of s e p a r a b i l i t y
A measure regular
on
in t h e
X
will
sense
that ~(A)
= inf
~(G)
A c where
A is a n y B o r e l Very
on
often
(X,B)
blem
given
fully
Pn
will
if the
sequence suffice.
convergence Let
This means space
for e a c h b o u n d e d be denoted This
and by
notion
1
~ not
X, w h e r e a s It is u s e f u l
for n ÷ + ~
which
of p r o b a b i l i t y weakly
to
P
The p r o -
properties One
constructs and hope-
is w h a t w e want. subsequence
a certain on
measures
properties
a convergent
measures
n
tool.
certain
approximate
to d e f i n e
/f d P
notion
Even
will of
(X, B).
measures
on
(X,B).
A
if
= Jf d P
continuous
the
with
distribution,...).
converge
converges
explicitly
is a u s e f u l
that we have
of convergence
logy of
(X,B)
of p r o b a b i l i t y
P = w - lim
do not.
on
having
be the c o l l e c t i o n
{Pn}n6~CM lim n
will
p
dimensional
does
to c o n s t r u c t
convergence
{~n}n 6~
a limit
itself
on the M1
sequence
end weak
finite
of m e a s u r e s
often
necessary
a measure
a sequence
have
C c A, A c l o s e d
set.
a n d to t h i s
(like h a v i n g
G, G o p e n
it b e c o m e s
is to c o n s t r u c t
= sup p(C)
uniform
function P
n
f
on
X. T h e w e a k
convergence
.
takes
advantage
convergence
to k n o w t h a t w e a k
o f the u n d e r l y i n g
a n d the
convergence
strong arises
topo-
convergence from a metric
152
defined
on
MI(X):
d-convergence ful to h a v e K c M1
in
indeed M1
a condition
under weak
sequence
exists
{Pn}n 6~cK
a condition
through
Theorem
1
Let
be a c o m p a c t
a weakly
following
metric
on
M 1 such that
convergent space
It is a l s o u s e -
compactness
Such a condition
if the m e t r i c
the
d
convergence.
for the c o n d i t i o n a l
has
necessary
a metric
same as w e a k
convergence.
is m o r e o v e r
X
there
is the
X
of a set
ensuring
that
subsequence is c o m p l e t e .
every
exists We
and
give
such
theorems
space,
then
MI(X)
is c o m p a c t
under
weak
convergence. Theorem
2
Let
be a m e t r i c
X
(X,B)
such
space
and
~n(X)
ii)
for a n y
sequence
E > 0 there
3
Let
X
be a c o m p l e t e
pact
subset with there
For
a proof
Remark
A sequence
Press,
if for e a c h > i-~ For
fore
convergent
space.
Ks
{Pn}n £ ~
Let
of weak
K
s
of
subsequence.
K c MI(X )
be a c o m -
convergence.
For
any
such that
exist
tributions
exists
variable
measure
having
measures P
n
random variables and
measures
on metric
measures
a compact
subset
is s a i d to be K
such that
n.
any probability
for p r o b a b i l i t y
Probability
1967.
of p r o b a b i l i t y
E > 0 there
for a l l
a random
Parthasarathy,
New York
w - lim n there
subset
subset
> I-E
see K.R.
Academic
space
a weakly
metric
a compact
P 6 K.
spaces,
Pn(KE)
exists
) < E for all n.
to the t o p o l o g y
a compact
P ( K c)
tight
separable
respect
exists
Pn(X\K
has
{~n}n 6
Theorem
for all
on
is b o u n d e d
X such that
c > 0
of m e a s u r e s
that i)
Then the
be a s e q u e n c e
{Pn}n £
satisfying
on
~k
there
that measure
is on some p r o b a b i l i t y
as its d i s t r i b u t i o n .
There-
satisfying
= P
Xn
and
X
having
these measures
as d i s -
153
lim n According
to the
constructed
X
= X
n
following
(convergence
in law)
fundamental
on the same p r o b a b i l i t y
t h e o r e m the X and X can be n space and m o r e o v e r in such a way
that lim n a condition
which
X
n
(~) = X(m)
Ym
is of c o u r s e m u c h
stronger
than the c o n v e r g e n c e
in
law. Skorohod's
theorem
Let
P
and
n
T h e n there e x i s t s space P
(~,F,P)
P
be p r o b a b i l i t y
random vectors
such t h a t
X
n
measures
Xn
and
X
on
P = w-lim P n n on a c o m m o n p r o b a b i l i t y
has d i s t r i b u t i o n
P
n
~k
,
and
X
has d i s t r i b u t i o n
and lim
For a p r o o f New York A6.
see e.g.
(~) = X(~)
P.
In this
Billingsley,
function
where
~
be the
Wt(~)
of the
fo r m
denotes
for all
For e l e m e n t a r y
and e l e m e n t a r y
~
an d M e a s u r e ,
John Wiley
the e x i s t e n c e
~ e~(~)j X [ j . 2 - n j>0
for a
by
{WslS
S t}. We call a
(j+l)2-n)(t)
function
elementary
if
ej(~)
is
j.
functions
e(t,e)
fts e(T,~)
dWT(~)
following
important
=
we d e f i n e
the i n t e g r a l
e(~,~)
by
j=>0Z ej(~) [ W t j + l - W t j ]
(m)
observation:
is b o u n d e d
if e(t,~)
then EE(f~
of
Brownian motion
~.
generated
the c h a r a c t e r i s t i c
Fj2-n - m e a s u r a b l e
N O W we m a k e the
f : [o,~)x
=
discuss
is l - d i m e n s i o n a l
o-algebra
f(t,~) X
Probability
s e c t i o n we w i l l b r i e f l y
class of f u n c t i o n s Let
¥~ 6
Ito I n t e g r a l s
/to f ( s , ~ ) d W s ( e )
where
n
1979.
Stochastic
wide
X
dWT(~))2]
= EEf~ e(T,~) 2 aT]
154
From this basic isometry we get an indication of what functions we can extend the integration. To prove this fundamental relation let
AW~3 = Wtj+l-Wtj;
then we
have
Since
e e.AW. z ] z
ft e(Y,w) s
dW
and
are i n d e p e n d e n t for
AW
]
=
E ej(~) j_>0
T
AW. 3 i<j
it follows that
E [ e i e j A W i A W j] = E[ej 2 ] (tj+l-t j) 6ij which implies the basic isometry. Let
S
be the class of functions (i)
(t,~) ÷ f(t,~)
f(t,~)
is
(i.e. (iii)
E[/t
t
f
÷ ~
such that
B x F-measurable, where B denotes
the Borel u-algebra (ii) For each
: ~x~
on
the map
~+
~ + f(t,e)
is
Ft-measurable
is adapted)
f(~,~)2 d~3 < +~
s
I(f) f £ S
we will define the Ito integral I(f) = /t f(T,~) s
I(f) will b e
dW T
F - m e a s u r a b l e and
E [(I(f))~] = E[/~ f2dT] I(f)is a stochastic process called the
(stochastic)
Ito integral based
on Brownian motion. The idea of the c o n s t r u c t i o n is simple: We use the basic isometry to extend in several steps the d e f i n i t i o n for e l e m e n t a r y functions to functions in
S. An important p r o p e r t y of the Ito integral is that it is a
martingale.
For continuous m a r t i n g a l e s we have the following important
inequality due to Doob If
Mt
(see e.g.[92 a] ):
is a m a r t i n g a l e P [
provided
such that
sup O~T~t
E [ I M t I p ] < +~
IM
t ÷ Mt(e)
I~ I ] ~
is continuous
~pE[ IMt Ip]
i
Using this inequality and the fact that Mt(~ ) = /t o f(s,~) is a m a r t i n g a l e w i t h respect to
dW s Ft
we conclude that
a-s
then
155
P [
sup T 6[o,t]
J.,l >
<
f(s
Remark It is p o s s i b l e than We
to d e f i n e
ft f ( s , m ) d W o s
finish
this
section
with
extra
have
1 -~t
term
like Ito
an Ito
integral
have
shows
ordinary
of the
that
integrals.
integral
Wt
but
larger
/tdW
=
o
1 ~t the
Ito
stochastic
From this s
example
b y the m a p
a combination
of a
we
f(x)
dW s
integral
does
see t h a t
=
not be-
the
image
1 x2
is n o t
again
ds
integral.
and a
We
indeed 1 y W%
It t u r n s and a
out
ds
that
= ft 1 [+ W s d W 0 2 ds + 0 s
if w e
integral
A stochastic
define
then
integral
this
stochastic family
above
equation
integrals
is s t a b l e
is a s t o c h a s t i c
Xt = X o + ft B(s,~) o The
of f u n c t i o n s
an e x a m p l e :
/t W d W 1 2 o s s = 2WtThe
for a c l a s s
S.
process
under Xt
ds + /t o(s,~) o
is o f t e n w r i t t e n
in the
as a s u m o f a d W
of the dW
shorter
s
smooth maps. form
s
differential
form
dX t = ~ dt + o d W t Let
g t,x)
6 C 2 ( [ o , ~) x IR, JR) t h e n Yt = f(t'Xt)
is a g a l n
a stochastic
integral
and
~f
~f
dy t - ~t
(t'Xt)dt
+ ~
1
(t'Xt)dXt +~
~2f.
~--~Y(t'Xt)
(dXt)2
where dt This main
result
evaluating
Ito
The
stochastic
due
to K.
Brownian The
is c a l l e d
the
Ito
• dt = 0
formula,
dW t
which
• d W t = dt
is v e r y
useful
for
integrals. integral
Ito
(1941).
motion theory
a martingale
• dt = d t ' d W t = d W t
M
it o(s,w) d W (~) b a s e d o n B r o w n i a n m o t i o n is o s S t o c h a s t i c c a l c u l u s (Ito's f o r m u l a ) b a s e d on
is c a r r i e d of
out according
stochastic
is d u e
to
integrals
to t h e
rule
ft ~(s,~) o Kunita-Watanabe (1967).
(dWT)2 dM(s,~) They
also
= dt. based
on
develop
156
a stochastic calculus based on m a r t i n g a l e according to the rule (dMt) 2 = d<M,M>t, where
M
is the so-called quadratic v a r i a t i o n of
M
(see A p p e n d i x A 7 ). Among spaces of martingales, square integrable m a r t i n g a l e s
w h i c h may be studied,
the space of
is the simplest because of its Hilbert
space structure but also t h e r i c h e s t
to investigate.
Indeed the classical
types of stochastic integrals d i s c u s s e d in the literature had been introduced as isomorphic t r a n s f o r m a t i o n s of some special space of square integrable martingales. A7. D e f i n i t i o n and C h a r a c t e r i z a t i o n of Quadratic Variation For t 6 I c ~ + subset
a partition
It = {t0'tl't2'''"
We denote the m e s h of
If
{~tn}n 6 ~
of
tk} of ~,t]
~t
6~ t E
It
[o,j
such that 0 = t o < t I < ...< tk=t
by
max j=0,1,..k-I
Itj+ 1 - t I J
is a sequence of p a r t i t i o n of
the members of
~t n
is a finite o r d e r e d
will be denoted by
~,t],
tjn
then for each
j = o , l , . . . k n.
n
The main
result is the following t h e o r e m . Theorem Let t 6 I
and { ~ t n } n 6 ~
be a sequence of p a r t i t i o n of
lim ~ = 0. Suppose n÷+~ each n let n Zt
=
M
is a continuous
Z tjn£H~
(Mt(j+l)n
[o,~
such that
local m a r t i n g a l e and for
- Mtjn )2
Then i) if
M
is bounded
{~ n t }n 6
<M'M>t ~ M2t - M 2 o ii)
{Z~}n
We call
~
2
converges
in
L2
to
ft
o M d M
converges in p r o b a b i l i t y to <M,M> t
<M,M> t
the quadratic v a r i a t i o n of
<M,M> = {<M,M>t} t
I
M
at time
t
and
the q u a d r a t i c v a r i a t i o n process a s s o c i a t e d w i t h
{Mt}t 6I" A process
M
is a Brownian m o t i o n in
~
if and only if it is a con-
tinuous local m a r t i n g a l e with quadratic v a r i a t i o n <M,M> t <M,M> t = t
a • s
for all
t.
such that
R E F E R E N C E S
[I]
A B B O T T L.F., WISE M.B., "Dimension of a Q u a n t u m - M e c h a n i c a l Path", Am. J. Phys. (1981) , 37-39.
[ 2~
A L B E V E R I O S., B L A N C H A R D Ph. , COMBE Ph. , RODRIGUEZ R. , SIRUGUE M . , S I R U G U E - C O L L I N M., "Trapping in S t o c h a s t i c M e c h a n i c s and A p p l i c a t i o n s to Covers of Clouds and Radiation Belts" in "Quantum P r o b a b i l i t y and A p p l i c a t i o n s II" (ACCARDI L., von W A L D E N F E L S W., Ed., Lecture N o t e s in Mathematics, 1136 S p r i n g e r V e r l a g (1985), 24-39.
[3]
ALBEVERIO
49
S., B L A N C H A R D P h . , H O E G H - K R O H N R., a. "Feynman Path Integrals and the Trace Formula for S c h r ~ d i n g e r Operator", Comm. Math. Phys. 83 (1982), 49-76. b. "A S t o c h a s t i c Model for the Orbits of Planets and Satellites: An I n t e r p r e t a t i o n of T i t i u s - B o d e Law", Expo. Math. 4 (1983), 365-373.
c. " D i f f u s i o n s sur une v a r i ~ t ~ riemanienne: B a r r i ~ r e s i n f r a n c h i s s a b l e s et applications", Colloque en l'honneur de L. SCHWARTZ, A s t ~ r i q u e 132 (1985), 181-2oi. d. "Newtonian D i f f u s i o n s and Planets wi£h~a Remark on Nonstandard D i r i c h l e t Forms and Polymers" in "Stochastic Analysis and A p p l i c a t i o n s " , T R U M A N A., W I L L I A M S D . E d . , Lecture Notes in M a t h e m a t i c s 1o95 S p r i n g e r V e r l a g (1985), I - 24. e. "Reduction of Non Linear Problems to S c h r ~ d i n g e r or Heat Equations: F o r m a t i o n of Kepler Orbits, S i n g u l a r Solutions of H y d r o d y n a m i c a l Equations" in "Stochastic Aspects of C l a s s i c a l and Q u a n t u m Systems", A L B E V E R I O S., COMBE Ph., S I R U G U E - C O L L I N M . E d . , Lecture Notes in M a t h e m a t i c s 11o9 S p r i n g e r Verlag (1985), 189-2o6. [4]
A L B E V E R I O S., BIIANCHARD Ph., H O E G H - K R O H N R., M E B K H O U T M., "Strata and Voids in G a l a c t i c Structure", s u b m i t t e d to: The A s t r o p h y s i c a l Journal.
[4 b i s ] A L B E V E R I O S., B L A N C H A R D Ph., H O E G H - K R O H N R., S C H N E I D E R W., "Origin of Hetegonic Jet Streams in the S a t e l l i t e Systems: A Stochastic Model" BiBoS p r e p r i n t 1987
[5]
A L B E V E R I O S., B L A N C H A R D Ph., G E S Z T E S Y F., STREIT L., " Q u a n t u m M e c h a n i c a l Low Energy S c a t t e r i n g in Terms of D i f f u s i o n Processes", in "Stochastic Aspects of Classical and Q u a n t u m Systems", A L B E V E R I O S., COMBE Ph., S I R U G U E - C O L L I N M. Ed., Lecture Notes in M a t h e m a t i c s 11o9 Springer V e r l a g (1985) , 207-227
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