Mathematical and Physical Aspects of Stochastic Mechanics

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

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



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

=

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

=

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 --co

= . 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)



]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 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

for

n

sufficient-

n

P(g(n,~)
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 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 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 O }

P[P (Yo)

k,Z 6 ~

(4.30) paths

> O] = 1

where

B = {S > O, inf P (Yt) = O } t 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<Sno}: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

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 > ~ - ~ -

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



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

, 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 + ~ll 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] = 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:~ ÷ ~ +

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 >


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]

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