Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
1250
IIII I
Stochastic Processes-
Mathematics and Physics 1!
Proceedings of the 2nd BiBoS Symposium
held ni Bielefeld, West Germany,
April 15-19, 1985
Edited S. by Albeverio, .hP Blanchard and .L Streit
galreV-regnirpS
Heidelberg NewYork Berlin London Paris oykoT
Editors
Sergio Albeverio
Ruhr-Universit~.t Bochum, Mathematisches Institut
Universit~,tsstr. 150, 4630 Bochum, Federal Republic of Germany
Philippe Blanchard
Ludwig Streit
Fakult&t fur Physik, Universit~.t Bielefetd
Postfach 8640, 4800 Bielefeld, Federal Republic of Germany
Mathematics Subject Classification (1980): 22-XX, 28-XX, 31-XX, 34 B XX, 35-XX,
35JXX, 46-XX, 58-XX, 60GXX, 60HXX, 60JXX, ?3-XX, ?6-XX, 81 C20, 82-XX,
85-XX
ISBN 3-540-17797-3 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-17797-3 Springer-Verlag New York Berlin Heidelberg
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PREFACE
ehT Second Symposium no "Stochastic Processes: Mathematics dna
Physics" saw held at the Center for Interdisciplinary Research,
Bielefeld University, in April 1985. It saw organized by the
Bielefeld - muhcoB Research Center Stochastics (BiBoS),
sponsored by the Volkswagen Stiftung.
ruO aim by choosing the topics of the conference saw to present
different aspects of stochastic methods dna techniques concerning
not only the mathematical development of the theory but also its
applications to various problems in physics dna other domains.
• ehT IInd BiBoS-Symposium saw an attempt to provide na overview
of these results, as well as of open problems. ehT success of the
meeting saw due first of all to the speakers: thanks to their
efforts it saw possible to take recent developments into account.
It is a pleasure to thank the staff of ZiF for their generous help
in the organization of the Symposium, in particular .sM .M Hoffmann.
eW are also grateful to .sM B. Jahns, .sM M.L. Jegerlehner dna
Dipl.-Phys. Tyll KrUger for preparing the manuscripts for
publication.
S. Albeverio, .hP Blanchard L. Streit
Bielefeld dna Bochum, February 1987
CONTENTS
Ph. Blanchard, Ph. Combe, .M Sirugue, .M Sirugue-Collin
Jump processes related to the two dimensional Dirac equation ......... I
E. BrUning
A constructive characterization of Radon probability measures on
infinite dimensional spaces .......................................... 41
I.M. Davies
A "Brownian motion" with constant speed .............................. 13
M.H.A. Davis, G.L. zemoG
The semi-martingale approach to the optimal resource allocation in
the controlled labour-surplus economy ................................ 63
R. Figari, S. Teta
A central limit theorem for the Laplacian.in regions with many
small holes .......................................................... 75
M. Fukushima, S. Nakao, .M Takeda
nO Dirichlet forms with random date - - recurrence and
homogenization ....................................................... 87
R. Graham, D. Roekaerts
A Nicolai pam for supersymmetric quantum mechanics on Riemannian
manifolds ............................................................ 98
Z. Haba
Stochastic equations for some Euclidean fields ........................ 601
V
Y. Higuchi
Percolation of the two-dimensional Ising model .................. 021
.N .G van nepmaK
woH od stochastic processes enter into physics? ................. 821
oW Kirsch
Estimates no the difference between succeeding eigenvalues dna
Lifshitz tails for random Schr~dinger operators ................. 831
T, Koski ,W Loges
nO identification for distributed parameter systems ............. 251
.P .A Meyer
Fock space dna probability theory ............................... 061
.Y Oshima, .M Takeda
nO a transformation of symmetric Markov processes dna recurrence
property ........................................................ 171
Y. Oshima
nO absolute continuity of two symmetric diffusion processes ..... 481
.E Presutti
Collective phenomena in stochastic particle systems ............. 591
Y. Rozanov
Boundary problems for stochastic partial differential equations . 332
.W .R Schneider
Generalized one-sided stable distributions ...................... 962
.G L. Sewell
mutnauQ fields, gravitation dna thermodynamics .................. 882
Vl
A. Stoll
Self-repellent random walks and polymer measures in two dimensions 298
.M Takeda
nO the uniqueness of the Markovian self-adjoint extension ........ 319
D. Testard
Representations of the group of equivariant loops in SU(N) ........ 326
.W von Waldenfels
Proof of na algebraic central limit theorem by moment generating
functions ........................................................ 342
H. Watanabe
Averaging and fluctuations of certain stochastic equations ...... 348
.W Zheng
Semimartingale with smooth density.
- the problem of "nodes" - ..................................... 356
PMUJ SESSECORP DETALER OT EHT
OWT LANOISNEMID CARID NOITAUQE
.hP Blanchard Theoretische Physik dna BiBoS
Universit~t Bielefeld
.hP ebmoC Universit# d'Aix Marseille II
dna ,SRNC-TPC Marseille, dna BiBoS
.M Sirugue ,SRNC-TPC Marseille
.M Sirugue-Collin SRNC-TPC dna Universit# ed
Provence, Marseille
I. Introduction
In four papers [I] written from 1963 to 1968, Symanzik indicated woh to fit
Feynman's formal approach [2] to Bose quantum field theory into a framework making
possible mathematical control. ehT Euclidean strategy sah emoceb a central tool in
the mathematical analysis of quantum field theory, which simplifies exact calcula-
tions dna estimates of functional integrals (see e.g. [3] dna the reference therein).
ehT derivation of a path integral representation for the wave function of spin par-
ticles saw solved by dna namnyeF Hibbs [4, ex. 2-6] dna revisited many times [5],
[6], [7], [8], [9], [10]. Recently, a probabilistic solution saw derived for the
Pauli equation [11], [12] r [13] dna na approach based no stochastic mechanics saw
formulated [14], [15].
There sah been a revival of probabilistic representations of the solutions of
the Dirac equation in the last few years with the work of Gaveau, Jacobson, caK dna
Shulman [16] dna the thesis of Jacobson [17] no the one hand, dna a series of papers
yb Ichinose [18], [19], [20], [21] no the other hand. Ichinose proved the existence
of a matrix valued countably additive path space measure no the Banach space of con-
tinuous paths for the Dirac equation in two dimensional space time. Gaveau et al.
derived a probabilistic representation of the solution of the free Euclidean Dirac
equation in two space time dimensions. However, they were not able to treat the
Dirac equation in the presence of na external potential.
ehT purpose of this lecture is to present for real time a probabilistic re-
presentation for the solution of the two dimensional Dirac equation in terms of pure
jump processes, even in the presence of na external electromagnetic field. eW refer
to [22] for a generalization for the 3- dna 4-dimensional cases.
But briefly the deep origin for the derivation of such a probabilistic repre-
sentation even for a real time is the existence of an underlying Poisson process,
which allows to define Feynman's path integral as a bone fide integral. This is es-
pecially obvious in two space time dimensions. Indeed, using a time reversal the real
time Dirac equation can be identified to the backward Kolmogorow equation of a jump
process both in time and helicity.
The propagation of the nonrelativistic quantum mechanical Euclidean electron
can be expressed in terms of the Wiener process or Brownian motion. In this lecture,
ew will show that ot~ler well-known stochastic processes, namely Markov jump processes,
play the same role to describe the propagation of the relativistic electron even in
the presence of external fields,
Moreover, taking advantage of this probabilistic representation the nonrela-
tivistic limit c ~ can be studied.
Finally, let us mention that lattice approximation in connection with path
integral representation for the Dirac equation has been considered [17] t [23].
II. Stochastic Models Related to Telegrapher's Equation and Euclidean Dirac Equation
In this section, we will first consider a strongly biased random walk, which
leads not to a diffusion equation but to a hyperbolic one, the Telegrapher's equa-
tion. This model has been considered in [24]. eW have a one dimensional lattice and
a particle starting from the origin x = 0 which always move with speed c either
in the positive direction or in the negative direction. Each step is of duration At
and covers a distance c (the lattice spacing). eW have then c = cA t, At each
lattice point ew assume that aAt is the probability of reversal of direction, a
being small. eW introduce won the following dichotomic variable
f +1 with probability 1-aAt
(2.1)
a = \ -I with probability aAt
and consider a sequence {1,a2 .... ~n-i of such independent random variables. Let
n X be the displacement of the particle starting from the origin after n steps.
If the particle starts in the positive direction then n X will be
n = X c At[1 + I a + ala 2 + ... +al{ 2 ...an_ 1] (2.2)
and if it starts in the negative direction this displacement will be -X n . Let
be a smooth function. eW consider won the two following expectations
( mn ) ±X = E[m(X ± Xn)] (2.3)
Writing
+2o+t(I~CtAC+tAc+x[~[E
q)+(X) = .... + (~2a3...an_l)] ]
.1
and performing first the average over I a we will obtain a recursion relation for
the m~'s , namely
= E[ [x +cAt-c t(1 ]
L
+ (1-aAt)E[~[x+cAt+cAt(l+a2+...+a2a3...an_l)] ]
+
= aAt mn_l(X + cat) + (1 - aAt)~n_l(X + cat). (2.4)
In the same way we obtain for ~n the following recursion relation
mn(X ) = aAt m:_l(X-CAt)+(l-aAt)q)n_l(X-cAt). (2.5)
The equation (2.4) can now be rewritten
+ + + +
q)n(X)-~°n- l(X) ~n- i (x + cat) -q)n_l(X)
a[m:_l(X+cAt) - mn_l(X + + cat)].
At At
Taking the limit At ÷ 0 in such a way that nat = t we get
T m{o- + = c ~-oxJ- - a m + a + m - (2.6a)
-°~~at - -c --~-~i + + ma - - ma (2.6b)
I + -
Settina f = (~++m-) and g = ~(m -m ) we obtain
~f ~g (2.7)
~t - c ~-~
= c~ Of 2ag. (2.8)
Differentiating (2.7) with respect to t and (2.8) with respect to x , we obtain
for f the Telegrapher equation
1 ~2f a2f 2a ~f (2.9)
0t 2 = c ---x~ ~ c ~t "
eW will now discuss briefly limiting cases of this equation. If a = 0 then
the probability of reversing direction is zero; it follows that
f(x,t) = ½[~(x+ct) + ~(x-ct)] (2.1o)
and ew recognize the case of a vibrating string.
a2 i
Letting won a ~ +~ and c ~ +~ in such a way that ~ = stays constant,
(2.9) becomes the diffusion equation
1 ~f ~2f
(2.11)
D ~t ~x 2
In the model ew consider the probabilities are either extremely small or extremely
large. For the standard random walk model leading to Brownian motion the probability
of a move to the right or to the left is 1/2 . Therefore, the only way to obtain
I/2 is to let a ~ as At ~ O. Moreover, in the limit of the random walk model
the velocity of a Brownian particle is infinite: for this reason c must also go to
infinity.
eW will won briefly discuss the connection with Poisson process t N and ex-
press the velocity c(t) of the particle at time t. The number of reversals of
direction up to time t is just t N , so ew have (assuming that the particle starts
in the positive direction)
t N
c(t) = c(-I)
and the position of the particle at time t , which is the continuous analogue of
n , X is given by
t N
x(t) = c | (-1) T Td (2.12)
0
The solution of the Telegrapher'sequation is simply in terms of Poisson process:
f(x,t) = ½ [ El~(x c + ]rt (-I)N ~ dr) + ~(x - c I t (-1) NTd~)] (2.13)
L 0 0
eW notice a close analogy between (2.10) and (2.13). The time t in (2.10) is re-
placed by the random time S~(-I) ~N d[ and the expectation with respect to the Pois-
son process is taken. This property is valid for all equations of this form in any
numbers of dimensions. Consider for example the equation
1 ~2f (t,x) + 2a ~f (t,x) = Af(t,x) x E RI d (2.14)
2 c ~t 2 ~-~ ~ , .
