Stochastic Processes - Mathematics and Physics II: Proceedings of the 2nd BiBoS Symposium held in Bielefeld, West Germany, April 15-19, 1985 (Lecture Notes in Mathematics, 1250) 3540177973, 9783540177975

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1250 Stochastic Processes Mathematics and Physics II Proceedings of the 2nd BiBoS Symposium held in Bielefeld, West Germany, April 15-19, 1985

Edited by S. Albeverio, Ph. Blanchard and L. Streit

Springer­Verlag Berlin Heidelberg New York London Paris Tokyo

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

1250 Stochastic Processes Mathematics and Physics II Proceedings of the 2nd BiBoS Symposium held in Bielefeld, West Germany, April 15-19, 1985

Edited by S. Albeverio, Ph. Blanchard and L. Streit

Springer­Verlag Berlin Heidelberg New York London Paris Tokyo

Editors

Sergio Albeverio Ruhr-Universitiit Bochum, Mathematisches Institut Universitiitsstr. 150,4630 Bochum, Federal Republic of Germany Philippe Blanchard Ludwig Streit Fakultiit fUr Physik, Universitiit Bielefeld Postfach 8640, 4800 Bielefeld, Federal Republic of Germany

Mathematics Subject Classification (1980): 22-XX, 28-XX, 31-XX, 34BXX, 35-XX, 35JXX' 46-XX, 58-XX, 60GXX, 60JXX, 73-XX, 76-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

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 June 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 and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

PRE F ACE

The Second Symposium on "Stochastic Processes: Mathematics and Physics" was held at the Center for Interdisciplinary Research, Bielefeld University, in April 1985. It was organized by the Bielefeld - Bochum Research Center Stochastics (BiBoS). sponsored by the Volkswagen Stiftung. Our aim by choosing the topics of the conference was to present different aspects of stochastic methods and techniques concerning not only the mathematical development of the theory but also its applications to various problems in physics and other domains. The lInd BiBoS-Symposium was an attempt to provide an overview of these results, as well as of open problems. The success of the meeting was due first of all to the speakers: thanks to their efforts it was 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 Ms. M. Hoffmann. We are also grateful to Ms. B. Jahns, Ms. M.L. Jegerlehner and Dipl.-Phys. Tyll KrUger for preparing the manuscripts for publication.

S. Albeverio, Ph. Blanchard. L. Streit Bielefeld and Bochum, February 1987

CON TEN T S

Ph. Blanchard, Ph. Combe, M. Sirugue, M. Sirugue-Collin Jump processes related to the two dimensional Dirac equation

.

E. Briining A constructive characterization of Radon probability measures on infinite dimensional spaces..........................................

14

I.M. Davies A "Browni an motion" with constant speed

31

M.H.A. Davis, G.L. Gomez The semi-martingale approach to the optimal resource allocation in the controlled labour-surplus economy

36

R. Figari, S. Teta A central limit theorem for the Laplacian.in regions with many small holes..........................................................

75

M. Fukushima, S. Nakao, M. Takeda On Dirichlet forms with random data - - recurrence and homogeni zati on

87

R. Graham, D. Roekaerts A Nicolai map for supersymmetric quantum mechanics on Riemannian manifolds

98

Z. Haba Stochastic equations for some Euclidean fields

106

v Y. Higuchi Percolation of the two-dimensional Ising model

120

N. G. van Kampen How do stochastic processes enter into physics?

128

W. Kirsch Estimates on the difference between succeeding eigenvalues and Lifshitz tails for random Schrodinger operators

138

T. Koski, Loges On identification for distributed parameter systems

152

P. A. Meyer Fock space and probabi 1ity theory

160

Y. Oshima, M. Takeda On a transformation of symmetric Markov processes and recurrence property ............................................................................................................... 171 Y. Oshima On absolute continuity of two symmetric diffusion processes ..... 184 E. Presutti Collective phenomena in stochastic particle systems

195

Y. Rozanov Boundary problems for stochastic partial differential equations . 233 W. R. Schneider Generalized one-sided stable distributions

269

G. L. Sewell Quantum fields, gravitation and thermodynamics

288

VI

A. Stoll Self-repellent random walks and polymer measures in two dimensions

298

M. Takeda On 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 an 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

JUMP PROCESSES RELATED TO THE TWO DIMENSIONAL DIRAC EQUATION Ph. Blanchard Ph. Combe M. Sirugue M. Sirugue-Collin

Theoretische Physik and BiBoS Universitat Bielefeld Universite d'Aix Marseille II and CPT-CNRS. Marseille. and BiBoS CPT-CNRS. Marseille CPT-CNRS and Universite de Provence. Marseille

I. Introduction In four papers [1] written from 1963 to 1968, Symanzik indicated how to fit Feynman's formal approach [2] to Bose quantum field theory into a framework making possible mathematical control. The Euclidean strategy has become a central tool in the mathematical analysis of quantum field theory, which simplifies exact calculations and estimates of functional integrals (see e.g. [3] and the reference therein). The derivation of a path integral representation for the wave function of spin particles was solved by Feynman and Hibbs [4, ex. 2-6] and revisited many times [5], [6], [7], [8], [9], [10]. Recently, a probabilistic solution was derived for the Pauli equation [11], [12], [13] and an approach based on stochastic mechanics was formulated [14], [15]. There has been a revival of probabilistic representations of the solutions of the Dirac equation in the last few years with the work of Gaveau, Jacobson, Kac and Shulman [16] and the thesis of Jacobson [17] on the one hand, and a series of papers by Ichinose [18], [19], [20], [21] on the other hand. Ichinose proved the existence of a matrix valued countably additive path space measure on the Banach space of continuous 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 an external potential. The purpose of this lecture is to present for real time a probabilistic representation for the solution of the two dimensional Dirac equation in terms of pure jump processes, even in the presence of an external electromagnetic field. We refer to [22] for a generalization for the 3- and 4-dimensional cases. But briefly the deep origin for the derivation of such a probabilistic repre-

2

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 especially 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 Cdn be expressed in terms of the Wiener process or Brownian motion. In this lecture, we will show that other well­known 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 nonrelativistic limit c 00 can be studied. Finally, let us mention that lattice approximation in connection with path integral representation for the Dirac equation has been considered (17], (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 equation. This model has been considered in [24]. We have a one dimensional lattice and a particle starting from the origin x = 0 which always move with c either in the positive direction or in the negative direction. Each step is of duration 6t and covers a distance E (the lattice spacing). We have then C C6 t . At each lattice point we assume that a6t is the probability of reversal of direction, a being small. We introduce now the following dichotomic variable

°

J +1

with probability

1­a6t

1. ­1

with probability

a6t

(2.1 )

and consider a sequence 01 , ... on­1 of such independent random variables. Let Xn be the displacement of the particle starting from the origin after n steps. If the particle starts in the positive direction then Xn will be

and if it starts in the negative direction this displacement will be ­X n. Let be a smooth function. consider now the two following expectations (2.3)

Writing

3

and performing first the average over 01 we will obtain a recursion relation for + the tp;;' s , namely (x)

=

dtp[ x + cllt - ct:t (1 +

alit

L

-

+

alit tpn-l (x + cz t ) + (1 - alit )

X E(Xt.X

'

t) t)

2t 2 R

for small t,

2 R et -1)

for large t.

= R-4 . The velocity process is now running extremely fast and a

2

We have IEeXt)1 EeXt.Xt)


0 ,

0

K2

> 0 ;

(3-f) Y2 '

oK E lR+ 2

(3-g)

where sK and sL denote the fraction of capital and wage income saved (savings := z z investments) in the secondary sector, and 62 stands for the rate of capital depreciation. F(t;Kz,L z) is a neoclassic production function, i.e. is a smooth function .... lR+} of flows of capital K2 and labour services L2, homoof class geneous of degree 1 with respect to K2 and L2, the marginal productivities of both factors are non-negative but diminishing with successive increments of inputs, etc. Before we go over to formulate the control problem, we like to introduce some additional notations. Let us recall that the economy lives the j-th time period Tj which is the real interval [t j_ 1,tj) where t j is given by the function 8j .... t(8 j) whose characterization is part of otimization problem (P2). Thus, for the time being, we cons i der t j as given and set t* = t j. We denote the initial capital endowment K2(tj_ 1} by °K 2 and assume that it is fully utilized. Let C2 denote aggregate consumption, °L2 the total labour force available to the secondary sector during the j-th time period, L2 the employment level at which the wage bill exhausts output and min{oL 2 ,L2}· We assume that once full employment is reached the control board is able to require investors to save any fraction sk of capital income z as long as it remains below the upper bound sk ' Further, we assume that the capiz talists' control of the disposition of their income is limited to the ability to impose or, to put it less drastic, to negotiate with the control board the upper bound sK on the saving rate. Moreover, we assume a minimum wage rate W z , exogeneously z fixed, which remains constant over the time period. Finally, we introduce two strict concavefunctionals U(·) and g(.) of class .... :R_). U measures aggregate consumption utility by means of which the control board expresses its preferences and judgements with respect to social welfare. The functional g is a terminal pay-off and measures the utility value in terms of consumption of the capital bequeathed to the (j+1}-th time period. We shall understand U and g as a set of socio-econ-

43

omic alternatives and social utility values resulting from the government program by means of which the ruling party should have come to power and to its fulfillment it should have committed itself. Now we have all the economic elements we need in order to obtain an equation for the per capita capital accumulation. Let us recall a simple version of the Ito Rule. Let 1 be a Markov time, let be a process with values in a set of matrices of dimension dXd 1 , and let a(s) be a d-dimensional process. We assume that X < X < a(s) are progressively measurable. X < is a characteristic function, i.e. XS_1 < = 1 if S$1, elsewhere it is O. If the process s(t) satisfies the relation JP

{

sup[s(t) t$1

It o

+

_1 XS


1,

(9)

81

then the convergence of (9) would imply that

w'm

A) to (and hence of Gm

A -1 would follow. In fact, (A)

a

which gives immediately A

H m

00

(-

6 + aV + A)

-1 .

Notice that in the development of the are "many" terms involving (s+l) distinct value coinciding with the right hand side of then the convergence of the sum of all these

a ;A s-th power of the matrix m Go there wi' Each of these terms has an average (9). The law of large numbers guarantees terms to the right hand side of (9).

This suggests that what one has to prove is that the contribution of terms with repeated w. in (£ GA)s becomes smaller and smaller, on a set of configuration of 1 m 0 large probability, when m goes to infinity. It turns out that this is true in a very strong sense. This fact enables us to disregard terms with repeated wi even in the analysis of the fluctuations of HA around the limit operator. m

The final result we can get on the asymptotic behavior of

is contained in

Theorem 2: (a) For all the w(m) belonging to a set of measure going to O. goes to infinity IIG A(wmlx (w(m)) - AA 11m- mmt 1 (b) For any f,g E H (u) the random field

1 as m

00

(10)

converges in distribution to the gaussian random field variance: a 2[(AA fAA 9, AAf (here

(o,o)L2

v

=

(0, V 0), Xm

is extended to all

of mean 0 and co-

9\2 - (AAf,AA 9\2 (AAfl,AA9(2 v v v is the characteristic function of and IAA

u setting its value equal to

0 on

(11)

m

U B ) .

J;l

J

As a consequence of Theorem 2 a complete analysis of the asymptotic behavior is a bounded region, can be worked out. For the of the eigenvalues of 6m, when details see [12J.

82

Sketch of the proof: as a consequence of Theorem 1 it will be sufficient to prove the statements (a) and (b) with substituted with According to the intuitive picture we presented above, we introduce the following definitions m

GoA(w.1 ,w.1 )G 0A(w.1 ,w.1 ) ... G0A(w.1 ,w.1 )G 0A(w.1 ,wJ ) i 1,i , ... ,i s_ 1 =1 1 s_1 1 Z S­Z s_1 2 L

i

k '"

ik

i Q,

*i

\ *J

k",Q,

Vk vk

i '" J

{IA(S)}i

J

­ {NA(S)}i

=

­* -A ­ ­a G (x)

m

0

in such a way that

[

r

0>

L (-)

J

_)s

s

(lZ)

LS=O

A A A Nm(x,y) + Im(x,y) = Hm(x,y).

With and we will indicate the operator in spectively to the kernels NA(x,y) and IA(x,y) . m m A(i )... GA(i ,i 1) appearing in To any term G

corresponding re-

we associate the 1,i z . 0 0 s s+ =A or1ented of s steps {i 1 , i Z, ... ,i s +1 } ' By definition N (s) is the sum of terms in corresponding to non­selfintersecting graphs while jA(S) is the sum of terms corresponding to graphs with at least one repeated point. We want to prove Lemma: for A large enough, on a set of configurations of probability going to as m goes to infinity lim

m->-00

Proof:

m1/ 211IAII m

for any f,g E

1

0 we have from the definition (IZ)

Let us consider the set of all terms in 1A(s ) whose graph has the first repeated point after n steps and comes back for the last time to this same point

83

after a loop of steps (2:0; 2 :0; s-n). Their total contribution to st 1 a -A wi 11 be indicated by A(m)( IG-Ag I A(s) Gfl 2 , n g,f). Its explicit form is:

;;;s+m

s+l

= :::s+r72

x,

m

m

_

k±1 , ... , k -1

{G-A} g k

A

1

k {N (2- 1)}k1 , k2 {GA}k 0 2 3

Using Schwartz inequality one can separate the contribution coming from the "vertex" (consisting of the four steps leaving or reaching the first repeated point) from the rest to give the bound

The vertex part is bounded as follows )(m))

= m-7/ 2

r

p,k,r p*k p*r k*r

:0; m-7/ 2

1 r 1 1 + m-7/ 2 L 4 p,k,r IW p-wk l2 Iw k-wrl2 k,p Iw P-wk 1 k*p p*k p*r k*r The first term is a positive quantity whose average value is bounded by 1/ m- 2 E[IWp-Wkl-2IWk-WrI2]. In particular, it is going to a in probability as m goes to infinity.

For the second term we have: m-3/2

1 2 r 1 Min Iw -w 1 +£ m k,p

p*k

for any

> O.

p k

By the 1aw of 1arge numbers m- 2

1

k*p m L

Iwp-wk l3-£ i s bounded on a set

k*p of probability increasing to 1 as m goes to infinity. Moreover, the probability that m- 3/ 2 Minlw -W 1t £ > 6 is, for any fixed 6 > I), going rapidly to 0 as one p*k p kI

84

can immediately realize using our assumption of smoothness of the distribution density V(x). We have then shown that, in probability, lim n(w(m)) = O. On the other hand, from

mt

co

-

we have: L

i,G

hJ

w(m)

Taking S < 1/2 the sum in curled brackets is bounded uniformly on a set of of measure going to 1 when m goes to and we have finally (13)

The estimate (13) is independent of Q, and m . Moreover, the possible choices of of the couple Q"n are obviously bounded by s2 We have then

which converges to proved.

0 in probability for

A sufficiently large. The lemma is then

To prove Theorem 2 we are now left to prove statements (a) and (b) with replacing GA. In particular, instead of the random field g defined in (10) m we can analyze the large m behavior of the random field

-A N =A (s)G-Ag does not conta i n terms whose average We want to stress that G f In particular we have value is infinite (like e.g. lim =0 mtco Analogously, any product like NA(S)G A GAf NA(s' )G A does not contain terms g 9 with infinite average value. The computation of the covariance of becomes mainly a combinatorial exercise. We refer to [12] for the details of the computation which gives

85

(;)

8g(fl))

f

a 2 (AA f AAg,AAfl AA g) L

+

- (AA f, AA g) 2 (AAfl, AA g) Lv(rI)

]

.

An immediate consequence of this result is that VV
3.

does not vary too much at infinity, then the

recurrence situation would be the same as the constant case.

In fact we

have the following :

Theorem 2.1

5

d, R is a centered Gaussian random field.

X(x), x

E(X(x) 2 ) = D XW (x) 2p(dw).

Let

R(x)

(i)

If

d

If

d = 2

then (ii)

Suppose that

and R(x)

3

is transient for

recurrent for

Proof

and P-a.e.

a log lxi, Ixl > A, for some 2 log log

lxi,

(00

Hence we get

for some

l-d

r

a/2


1,

(00 )21f -X(ro) -1 (e do) dr ')A r-l(log r)-2U

(21f

(J

o

1

D.

o

e-x(ra) do) dr

1

(log A)

2u-2

Hence,

is

90

E(

{j: r- l

(a - 1)


8 c

= u-,S.h

if

13

u+,S.h

s

B or S>8 c c

and h '" O.

Percolation For every

wen,

let w- 1 (+1) be the set of all points xe2Z

2

, such that

w(x)=+l, and let E+ be the event that there exists an infinite cluster 1(+1). in wThe is the follOWing: Problem: about Remark:

For what value of (S.h) do we have 13 h(E+)? ,. ..,

+

+,

8 h(E..,)=l? And how •

From tail triviality of u+ and u_, these probabilities take

121

only values 0 or 1. Also, if by

then we simply write this measure

and we only have to look at the value

+

§2. What is known uo to now

Dual Graoh Argument 2

It is very convenient to introduce the dual graph JL of 2Z , JL consists of the same set of vertices as 2Z 2 , but the connection in JL is more 2 than in 2Z ; 1. e. two points x , yEJL are nearest neighbours in JL if 1.

JL (Fig. 1) First, note the following simple observation: Observation:

The following two statements are equivalent. 2 (L) There is an infinite cluster of w-1 (+1) in 2Z • 2 such that there is no JL-circuit of (ii) There exists a finite AC2Z w- 1 (_ 1 ) which surrounds A.

Now let us begin with the easier case; h = O. + By the above observation, if h = 0 we have (Eoo) =1, for otherwise a. s , we find a JL -circuit in w- 1 (-1) surrounding a given finite AC2Z 2 which, by Markov property.of implies that which is a contra-

122

diction to our assumption.

t 1)

+

that u, (E oo) = 0 by using Harris I argument ([2]) which can be avoided when we use so-called "Sponge PerAs for lJ._(E+), it was proved in 00

colation".

([5])

Once we have a result for h

= 0,

it can be carried into h = 0 case by

using FKG inequality. Namely it is a direct consequence of 1°) and FKG inequality that (i)

lJ.

+

for h > 0

(E S. h oo) +

o

(ii) lJ. • h (Eool S

,

for hSc' we have a complete description of the percolation region.

The argument is just the same as in the case 1°), and it is known that one has

+

lJ. .

3 0(E oo

=0

)

([1]). Also this can ([5]).

by using Harris' argument

be simplified by sponge percolation argument

By using Peyerl's argument, Kunz and Souillard [6J have shown that for every h >h

there exists sufficiently large h (S)

+

lJ.

0

(S)

such that for any

oo ) = l , which, in conjunction with 3°), implies the

o S• h(E existence of critical

+

such that

o

lJ.a.h(E oo)

we have a kind of phase transition in the parameter region where the free energy of

(1)

is analytic. This is just like as in the

Bernouilli case. In the Bernoulli percolation, one has a system of no interaction; i.e. the formal Hamiltonian of the system is given by (4)

,

H'(S)

x

where h is as before a real parameter. There exists unique Gibbs state lJ.

S. h

for given parameter

(B. h),

having the following form;

123

(5)

xE.A)

= (

JAI

1

1 +e

-26h)

For simplicity, let us write the right hand side of (5) by

piAl,

where

p=p(B.h) is just defined by

=

p

(1+e- 2 8h ) - 1 •

Then it is equivalent to say that we have i.i.d. {o(x)

, whose dis-

tribution is given by P(o(x)=+1) = p For Bernoulli percolation, we know that there exists Pc > l!S.h(E:) l!S.h(E:)

(l4

),[7]).

T6th

if

1 2

such that

p(S.h) >Pc

if

0

The best rigorous lowerbound for Pc is

obtained by

(9);

Now, there is a very simple and useful criterion to compare our two systems: Ising and Bernoulli. Lemma.

(Russo (8))

If for all a rI, l!13 . h (G (x ) =+ 1 I cr (y), y ,l{ x} ) ;: p , )

then we have S.h;:

; )

for all increasing functions f of crerl, where S.h and

stand for

expectations w.r.t. l!S.h and l!S.h respectively. Remark.

In [8], the statement corresponding to the above lemma is much

more general, but for simplicity we presented it in the above form. From the lemma, it is obvious that +

l!S.h (E,,) +

l!S.h(E",) =

«(4-) •

0

i f h

1 6

Pc

;::p c

4

Pc log - - + 4 1- pc

124

§3. Sketch of the proof In this section, we give a sketch of the proof of the statements we gave in § 2, 1 0 ) '" 4 0) • First we consider the case when h = 0, corresponding to 1 0 ) and 3 0) in §2. The only thing remaining to be proved is that for all S

(6)

>

0,

where of course lJ S, 0 lJ S, 0 the unique Gibbs state) for s;;; Sc ' As we mentioned, this can be proved by Harris' argument given in (21. r

,

Let V

n

{x 1 =n}

E

and

n

*

Ix 1,i

< _

Ix 2 I

n,

+* (En) be the event that

and let to

x E ZC 2 ;

= {

in V

::i n (x

1

},

=-n) is connected (*connected)

w-1 (+ 1), where *connection means the connection in JL.

(1

n

are the events defined by the same way for w-

1

(-1).

By duality of 2'l2 and JL, we have +) c (E n

(7)

where k : \I

+

\I

R(E n-* ),

=

foe n;;; 1,

is the rotation around the origin by 90 degrees, i.e.

1 2 :< 1 (Rw) (x ,x) = w(-x,x).

Since *connection is easier than the usual connection, we know that

By R-invariance and FKG inequality, we have

(0)

)

.

Here, we used the fact that lJ_,S,hoT = lJ+,S,-h for any S where T:\1 + (J reverses all the spins on 22

2

;

1=-3n} [x

2

l::in

)

f: 1/2.

is *connected to {x }()w

-1

1=3n}

in

(-1).

The key point is to show that there is a n-independent constant such that for

(9)

If

(9)

is proved, then putting

=

8

(0,2n)

*

n z rl , S>O.

8 : \I + \1 by (8 (y)=w(x+y) and xw) x -* -* F n , 2 = 8(_2n,OjRF n '

a>

0,

125

-*

F n,4

e (0,-2n)

-*

e (2n, 0)

we obtain (10)

there exists a *circuit surrounding V ;; ;;;

S, Cl

4

n °( 1;;;i;;;4

* ,i

n

in

w- 1 (-1)

)

)

The last inequality follows from FKG inequality and the invariance of 2 under {ex; x Ez::: } and R. As we have observed in §2, (10) implies that

which, by the tail triviality of

proves

(6),

Now it remains to show (9). Consider the event that {x 2=2n} is 2=n}, 1=2n} *connected to {x and {x is *connected to both {x 1=0, 1=0, and {x Denote this event simply by By FKG inequality and the invariance of 4

-1 r

S

°

-*

(E n

)}

3

we obtain •

But in Gn-* ' we know that the maximal *half circuit y(w) in vn+(n,O) 2 1=2nl = {O;;; xl;;; 2n, Ix 1 ;;; n I surrounding the origin is *connected to {x in (Vn + (n r 0)

(\ w

-1

(-1).

+* For each *half circuit y in vn+(n,O) surrounding the origin, let Hy (H-* ) be the event that the origin is surrounded by a *circuit which is y *connected to y (y) in e n w-1 (+ 1) ( e (l w-1 (-1) ), where is ' } ' 1 nd ' f y /' t { 1 t he ed by y an re y w.r. . x = 0 . By the symmetry of the conditional distribution of

we have

because -*

+*

Hy V Hy and both

+

S 0 and I

Hence if wEG

-*

Hy

I

-*

n

w(x)=+l

XE.y, w(y)

=-1

y6 y\y)

= 1,

S 0 have the same conditional distribution

' / "

given {w(x),

I ,

' by FKG inequality, the conditional probability that

y(w)

= y ) is not less than 1/2 for each y.

126

Multiplying

y(w)

y) and summing up for all y's, we get

1 {X =2n } is *connected to a *circu:t surrounding\ \.I-,B,O [ the origin in [-2n,2nJx[-n,nJi\ w 1 (-1) )

Reflecting this event w.r.t. {x ( 11 )

1=0},

and using FKG inequality, we get

r{X 1 =2n } \.I-,B,O

is *connected in (-2n,2nJX[-n,n](lw

l

-*

Denote this event by In

1=-2n} {x (-1)

1

Then from obvious inclusion (I

Jc

[e(n,o)

we obtain (12)

2

-25

,

which proves (9). For the case h

0, the argument is standard for the use of FKG inequality

in percolation theory. §4. Concluding remark. The known results up to now for the percolation of the two-dimensional Ising model is summarized in Fig.2, where we know that

By using FKG inequality, we know that for each B hc(B)

0, such that for

We expect that \.IB,h( E:O)

h

1 for h

B there exists c'

h (B) . c

< >

hc(B), too. and it is quite

interesting to know the critical value B defined by c' Bc' = sup{ S s Bc; hc(S)

>

° },

which we expect to be equal to Sc. If it is not the case, we have a new critical value for 6, whose physical meaning is not quite clear.

(e:. )

l'E.:') ':

=0 .. ..

-"."':. '

.

-': -"

..::

:

'.:

:;

:: .:

;'\\',--' ,'':'':-'', fA

:

(. E1"ao )

=1

h

o 1 4 log

.

Pc

T='P c

(Fig. 2)

Refe.!:'enees

(1) Coniglio, A., Nappi, C.R., Peruggi, F., Russo, L.: Cornm.Math.phys. 315-323

(1976).

(2] Harris, T.E.: Proe.Cambridge Philos.Soc. (3] Higuchi, Y.: (1982l.

13-20 (1960).

Z.Wahrseheinlichkeitstheorie verw. Gebiete

il,

75-81

(4) Higuchi, Y.: In Probability Theory and Mathematical Statistics. Proceedings of 4th USSR-Japan Symposium, 1982. Lecture Notes in Math. 1021, 230-237 (1983).

[5] Higuchi, Y.: A weak version of RSW theorem for the two-dimensional Ising model. Preprint. (6] Kunz, H,. Souillard, B.: J.Stat.Phys . (7

J

..!.2.,

77-106

(1978l.

Russo, L.: Z.Wahrscheinlichkeitstheorie verw. Gebiete (1981l.

(8) Russo, L.: i.b.i.d.

il,

129-139

229-237

(1982).

(91 T6th, B.: A lower bound for the critical probability of the square lattice site percolation. Preprint.

HOW DO STOCHASTIC PROCESSES ENTER INTO PHYSICS? N.G. van Kampen Institut fUr Theoretische Physik R.W. T.H. Aachen Templergraben 55 5100 Aachen F.R. Germany Abstract Fluctuations in non-equilibrium systems do not arise from a probabi lity distribution of the initial state, but are continually generated by the equations of motion. In order to derive them from statistical mechanics a drastic repeated randomness assumption is indispensable. One is then led to a master equation, from which both the deterministic macroscopic equation and the fluctuations are obtained by a limiting process. The approximate nature of the whole procedure makes the use of strictly mathematical delta-correlations and Ito calculus illusory. 1. Since the theory of Brownian motion was established by Einstein and Smoluchowski [1] the role of stochastic processes and stochastic differential equations in physics has grown into that of an indispensable tool. In many cases it is heuristically clear why and how this tool should be utilized, but in other cases it is not. For instance, in the theory of the laser [2,3] stochastic equations are used without the basic understanding that is needed to judge whether the result is reliable. Hence it is not just for intellectual satisfaction that the way in which stochastic processes enter into physics should be investigated. Of course, like everything in physics, the stochatic description can only be an approximation, but it is necessary to understand precisely which approximations are involved. A test for real understanding is that one can indicate how higher approximations should be obtained. Unfortunately insufficient effort has been devoted to the analysis of these questions. 2. Consider a closed, isolated, classical physical system described by canonical variables (Q1'" .,QN,P1 , ... ,PN)' and a Hamilton function H(q,p). The equations of motion define a family of trajectories defining a flow in the 2N-dimensional phase space f. In an alternative notation: every point xEf is carried by a flow into a uniquely defined point XtEf after a time t. The flow preserves the phase space volume. If x is the initial state of the system at t = 0, its state at t is xt = X(t,x) and the Jacobian equals unity: ]dX(t,x)/dx! 1 (t fixed). Statistical mechanics tells us that a physical system in which the number N of degrees of freedom is large should be described by an ensemble of identical replicas. Accordingly, rather than a single initial state x one introduces a probability density p(x) of initial states, to

129

be determined by physical considerations. This turns x into a stochastic variable and x t = X(t,x) into a stochastic process. Its single­time probatility density is t ­t f o(x 1 ­ x 1) p ( x )dx = o l x , 1) and the entire hierarchy of joint probability densities is

f

n

i

1

olX

t. 1

-

x.JP(x)dx. r­

A physical quantity is a function A(x) in phase space. Its value A(x t) at time t has become a stochastic process, fully determined by the choice of ensemble . A special choice for p is a stationary ensemble, that is, a distribution having the property (x t) = (x); for instance, p (x )

Z = J e­SH(q,P)dq dp,

with arbitrary positive parameter B. With this choice our random functions X(t,x) and A(x t) are stationary processes. 3. All this is exact, but of little use when dealing with actual many­body systems since the mapping x + x t = X(t,x) is much too complicated to be determined explicitly [4]. What one does in practice is the following. One selects somehow a set of "relevant" variables Ar (x) and . makes suitable assumptions concerning the stochastic properties of the associated processes Ar(x t). These usually amount to assuming that they obey a multivariate Langevin (or ItB)equation. For instance, in the popular Projection Operator Technique [5] one first formally derives an equation for the Ar alone by eliminating all other variables; this is done purely mathematically and inevitably the equation involves an integral over the preceding values of A from the initial time up to the time considered. In addition there is a term involving the initial values of the eliminated variables. This integral equation is called the "generalized Langevin equation", but is actually merely a different form of the exact microscopic equations of motion. One then turns the additional term into a random force by assuming some probability distribution for the initial values of the eliminated variables. Nobody asks what this special initial time is. Subsequently a "Markovian assumption" is used to get rid of the integral and obtain an actual stochastic differential equation of the Langevin type. In Linear Response Theory [6] the initial time is shifted to _00, but

130

it is again true that the randomness enters only through the assumed initial probability distribution [7]. Thus in these and similar approaches the essential difficulty of statistical mechanics is resolved by assumption; in the absence of anything else one cannot complain, provided that no claim is made that something has been derived. 4.

However, the whole idea is wrong. This is not the way in which As an illustration take a Brownian particle; together with the surrounding fluid it constitutes a closed, isolated system. The "relevant" variable A is the position of the particle, and constitutes a stochastic process (approximately a Wiener process). Obviously, this is not due to our ignorance concerning the state of the system at some initial time. Rather, it is due to the fact that the single variable A does not really satisfy a closed differential equation, but interacts with all fluid molecules. Their variables are not present in the equation for A but their effect shows up in the random Langevin force. Fluctuations in A are constantly being generated by the collisions, and would be there just the same even if I had been able to start the system off in a precise microscopic state & t = O. Another illustration is shuffling a deck of cards. Generally, the evolution of a many­body system is described exactly on the microscopic scale by the flow x + xt. Experience has taught us that there is also a macroscopic description in terms of a few, suitably chosen, macroscopically observable quantities Ar(x), which obey a closed set of differential equations. These equations are not exact, however. The enormous number of eliminated microscopic variables makes itself felt by causing the actual values of the Ar to fluctuate about the values given by those macroscopic equations. The actual values are extremely complicated functions of time, which cannot be found without solving the microscopic equations, but their short­time averages (and other moments) do have simple predictable properties. In practice one replaces these time averages by ensemble averages for convenience. Summary. Stochastic processes describing fluctuations in physics do not arise from a probability distribution of the initial microstate. Rather, they serve as a tool to describe the irregular motion of the actual trajectory about the smoother evolution determined by the macroscopic equations. That is how stochastic processes enter into physics. Our next task is to describe in somewhat more detail how this happens. stochastic processes enter into physics.

5. The macroscopic variables Ar determine a "coarse­graining" of the phase space r by cutting it up in phase cells defined by

131

(all r), where is roughly the lack of precision of the observations. An observation or measurement tells me in which cell the point x lies, but no more. The basic assumption of statistical mechanics of time-dependent processes is that I don't have to know more, but that I may replace the precise point x by a probability distribution in the cell ­ with constant density in that cell and zero density outside it. The flow in r carries this density along and after time t a fraction of that density lies in phase cell a'; this fraction may be denoted

It represents the probability P(a',t) z a ' that a system starting at a will be found (at time t ) in phase cell ls e ' at a In our formulation of the basic assumption no restriction has been imposed on how the system originally had arrived in the cell at a. Hence we may apply it to the time interval t,t + and find P(a" , t

+

f

la')P(a',tlda'.

Thus the motion among the cells is small M one obtains

= f {W(ala' )P(a' ,t)

a

(1 )

Markov chain. In the limit of

W(a' la)P(a,t)} da',

(2 )

where W(ala' )da is the transition probability per unit time from a into da. This is the differential form of the Chapman­Kolmogorov or Smoluchowski equation, now usually called the master equation [8]. It describes the evolution of the system, as seen by a macroscopic observer, in terms of a Markov process. 6. With respect to our drastic assumption the following remarks can be made. (i) All existing treatments relating macroscopic equations to the microscopic ones use such an assumption in the form of a "Stosszahlans a t z :' , "molecular chaos", "random phase", "Markov assumption", or "repeated randomness assumption". Rather than to hide it one should make it explicit so that its validity can be investigated. (ii) The picture is that the microscopic trajectory is so complicated that it practically covers the whole phase cell during the short time

132

t. Thus one implicitly uses a kind of local ergodic theorem: the time average during 6t equals the phase cell average. (i i i ) As a consequence, (1) cannot be va 1 i d when 6t is too sma 11 and (2) is not really a limit 6t O. Rather, (2) holds approximately when it is possible to pick a 6t that is large enough for (1), but still so small that the values of the Ar do not change appreciably. (iv) It follows that any process in physics is stochastic only if the variables are measured with a sufficient margin. [9J It is Markovian only if one does not look at too small time intervals. In particular, in the Langevin equation the random force is never strictly delta­correlated; at best its auto­correlation time is short compared to the other relaxation times in the system [10J. From a physical point of view the Ito calculus is based on a misconception, since it requires strict delta correlations. (v) The validity of the basic assumption depends on a proper choice of the Ar. This choice is not determined by what the experimenter wants to observe. Rather, it is determined by the requirement that the Ar should incorporate all slow variations; they must account for all longtime correlations, since otherwise the Markov property used in (1) cannot be valid. (vi) Whether or not such a separation of time scales is possible depends on the system, i.e., on its Hamilton function. For some many­body systems a reduced description in terms of a few A may be impossible, r such as self­gravitating systems, e.g., stellar clusters. [11J (vii) In actual physical cases one is often able to guess (on the basis of experience, intuition, or trial and error) what the correct choice is for the A In a simple fluid they are the local density, ver. locity, and energy density. But the lack of an actual criterion makes it impossible to judge whether higher approximations can be obtained by merely adding terms to the equations for the same Ar [12J; or require the addition of new Ar into the macroscopic equations, such as the heat fl ux, as in "extended thermodynami cs" [13 J. 7. We have arrived at the master equation (2), which does describe the evolution of the system on a macroscopic scale, but as a stochastic process. The familiar deterministic macroscopic equations (Navier­Stokes, Ohm, etc.) can be extracted from it in the following way [14J. In general, W(ala') involves a paramter D with the property that for large D the fluctuations are relatively small. D may be the size of the system, the mass of the Brownian particle, or the capacity of a condenser. Expansion of (2) in D­ 1 gives to lowest order the desired deterministic

133

equation

a

Thus the rate of change r equals the average of the jump per unit time. We therefore ca 11 the process "j ump-dri ven" . The next approximation is of order Q-1/2 and gives the fluctuations in Gaussian approximation. Let be the solution of (3) with some given initial value, and set l'Ia r = a ; Then one finds < i'la r > a and the matrix rs < i'la > obeys r6a s (4 )

where the matrices A and B are given by

J (a r l

-

a r )(a's - as)W(a' [a l da

!

•

For the detailed derivation we refer to the literature [8J. For some systems, however, to be called "diffusion driven", the lowest order (3) happens to be zero. In that case the next order is the leading one. It turns out that this order consists of a Fokker-Planck equation [15,8J: aP(a,t) at

=_I

a Ar(a)P r aar

+

i

r ,s

da r aa s Brs

(a) P •

(5 )

Here Br s is as before and Ar(a) is what remains of (3) in the next order of Q , the lowest being zero. An example is an electron in a semiconductor (possibly inhomogeneous), subject to an electric field (possibly not constant). Evidently for this class of systems the 0-expansion does not yield a deterministic macroscopic equation, but again a stochastic description. All it does is to substitute a Fokker-Planck approximation for the master equation (2). In order to obtain a deterministic equation an additional expansion is needed, for instance in powers of the temperature. [8,16J One then finds again for the noise in lowest order a Gaussian distribution. Summary. Statistical mechanics leads, on the macroscopic level, to a stochastic description in terms of the master equation (2). Subsequently, deterministic equations plus fluctuations can be extracted from it by suitable limiting procedures. The view that one should start from the known macroscopic equations and the fluctuations should be somehow

134

tagged on to them is responsible for much confusion in the literature. 8. I call a stochastic differential equation any differential equation whose coefficients are random constants or functions of time with given stochastic properties, i.e., Fr(a;

with given random

(6 )

In the mathematical literature one usually restricts the name to the case that is Gaussian and delta-correlated (derivative of the Wiener process), but that is clearly inappropriate in physics, since such a is only an approximation of an actual random force. How do stochastic differential equations arise in physics? Consider a closed, isolated many-body system consisting of a small subsystem S with few variables Q,P and a large "bath" B with many q,p: (7)

An example is a single nuclear spin S in a crystal B; the lattice vibrations are described by the q,p. [17] For the interaction we take the simple, but not unrealistic, form

? h(qi)'

HI(Q,q) = g(Q)

1

The idea is that one first imagines the motion resulting from HB in isolation to be determined, giving rise to complicated and rapidly varying functions qi(t). Their explicit form is of course unknown, but one puts

I

h(qi(t)) =

1

Then is regarded as a stochastic process, of which the properties are guessed. Usually one assumes to be Gaussian, as it is the sum of many terms (although not independent); and delta-correlated, as the qi(t) vary rapidly and irregularly (although determinstically). With these assumptions one obtains a stochastic Hamilton function for S H(t)

=

HS(Q,P)

+

(8 )

leading to stochastic equation of motion for the subsystem Salone. One recognizes the same philosophy as before: the Q,P are our previous Ar , the q,p are the eliminated variables, and the fluctuations in the A are continually generated by the q,p. In addition, however, r

135

one has introduced a second assumption, namely that the motion of the q,p may be determined from HB alone, that is, that the reaction of the subsystem 5 on the bath is negligible. This assumption reduces the interaction with the bath to an external random force with given properties and thereby reduces the master equation (2) to a stochastic differential equation (6). 9. The plausibility of this assumption depends on the system con­ sidered. The earth's troposphere is turbulent so that its density, and hence its refractive index, is a random function in space and time. An electromagnetic wave propagating through it obeys Maxwell's equation involving a random dielectric constant. [18] Clearly the effect that passing radio waves have on the turbulence is negligible, so that in this case a stochastic differential equation is justified. For a spin embedded in a crystal lattice, however, the assumption cannot be true. (Of course one has to replace Q,P with spin operators, but the argument remains the same.) The reason is that (8) leads inevitably to an ever growing value of the energy HS' as if the bath had an infinite temperature. What is missing is the spontaneous emission, the fact that transitions from a high level to a lower one are more likely than vice versa. Hence one is forced to add a damping term in the equations of motion, corresponding to the friction term in the equation for the Brownian particle. The magnitude of this term is related to the random term in (8) by the requirement that the outcome of the competition between both must be the known equilibrium distribution ("fluctuation­dissipation theorem"). Yet this damping term is entirely ad hoc and demonstates the heuristic nature of stochastic differential equations in physics. Incidentally, for some simple systems, such as the Brownian particle, a more sophisticated approach is possible [19]. One determines q(t) not from HB, but from HB(q,p) + HI(Q,q), regarding Q as slowly varying and in lowest approximation as constant. As a result the stochastic properties of now depend on Q. It turns out that this automatically leads to the desired damping. The result can again be expressed as a stochastic differential equation; however, it is obtained without the "second assumption" in section 8, but by working out the program of section 5. Summary. The master equation (2) describes a closed, isolated sys­ tem. If this system can be subdivided in a subsystem and a bath as in (7), it may be possible to write stochastic equations of motion for the subsystem alone. The fluctuations are then caused by an external

136

random force caused by the bath. This random force is not affected by the subsystem, any effect of the subsystem on the bath must be accounted for by a deterministic damping term in the equations for the subsystem. 10. In systems that cannot be so subdivided the noise is intrinsic, it is part and parcel of the evolution itself. Examples are radioactive decay and chemical reactions. They can be described only by the original master equation (2). Yet such systems are often treated in the literature in terms of a Langevin stochastic differential equation. How can this be understood? First we have seen that there exist diffusion-driven systems, in which the dominant term of the has the form of a FokkerPlanck equation (5). As any Fokker­Planck equation is mathematically equivalent to a suitably chosen Langevin or Ito equation, one is free to use these as a formal device instead of (5). Higher approximations in D­ 1, however, give additional terms to (5), which cannot be incorporated in the Langevin equation. Secondly, jump-driven systems are described in the first two orders of the by the deterministic macroscopic equation (3) plus Gaussian noise determined by (4). It is possible to construct a FokkerPlanck equation whose solution in the same order of produces the same macroscopic behavior and noise. Thus to this order in such tems can be described by a Fokker­Planck equation and hence by a Langevin equation. [20] It is not clear, however, why one should do this, in particular since these equations are much harder to solve than the first and second order approximations (3), (4) themselves. REFERENCES [1] A. Einstein, Ann. Physik (4) 17, 549 (1905); 19, 371 (1906); M. v. Smoluchowski, Ann. PhysTk (4) 2 , 756 (1906). [2] H. Haken, in Encyclopaedia of M. Sargent, M.D. Scully, and W. ley, Reading, Mass. 1974).

cs

c (Springer, Berlin 1970); Physics (Addison­Wes­

[3] H. Haken,

Synergetics (Springer, Berlin 1976, 1978); C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, Berlin 1983).

[4] With the exception of a few soluble cases, viz., the linear harmonic chain, see e.g. G.W. Ford, M. Kac, and P. Mazur, J. Math. Phys. 504 (1965); P. Ullersma, Physica 2,27, 56, 74, 90 (1966).

137

[5 ]

S. Nakajima, Prog. Theor. Phys. 20, 948 (1958); R. Zwanzio, J. Chem. 1338 (1960); M. Mori, Prog. Theor. Phys. 33,423 (1965).

[6 ]

R. Kubo , J. Phys. Soc. Japan 11,570 (1957).

[7]

N.G. van Kampen, Physica

279 (1971).

[8 ]

N.G. van Stochastic Processes (North-Holland, Amsterdam 1981).

in Physics and Chemistry

[9 ]

The vital role of such margins was forcefully argued by P. and T. Ehrenfest, in: Enzyklopadie der mathematischen Wissenschaften 4, Nr. 32 (Teubner, Leipzig 1912); translated by M.J. Moravcsik wlth the title Conceptual Foundations of the Statistical Approach in Mechanics (Cornell Univ. Press, Ithaca 1959).

[10] The founding fathers were of course fully aware of this: G.E. Uhlenbeck and L.S. Ornstein, Phys. Rev. 823 (1930). [11] T.S. van Albada, Bull. Astr. Inst. Neth. 19,479 (1968); W. Thirring, Z. Physik 235,339 (1970); R. in: Advances in Chemical Physics (W1Iey, New York 1970). [12] N.N. Bogolubov, Problems of Dynamical Theory in Statistical Physics, in: Studies in Statistical Mechanics I (G.E. Uhlenbeck and J. de Boer eds., North-Holland, Amsterdam 1962); G.E. Uhlenbeck, in: Probability and Related Topics in Physical Sciences I (Proceedings of the Summer Seminar in Boulder, Colorado in 1957; Interscience, London and New York 1959) p. 195 ff.; E.G.D. Cohen, in: Fundamental Problems in Statistical Mechanics II (E.G.D. Cohen ed., NorthHolland, Amsterdam 1968). [13] R.E. Nettleton, J. Chem Phys. 40,112 (1964); I. MUller, Z. Physik 198, 329 (1967); L.S. Garcla-COTin, M. Lopez de Haro, R.F. Rodriguez, and D. Jou, J. Stat. Phys. ll, 465 (1984). [14] N.G. van Kampen, Can. J. Phys. 39, 551 (1961) and in: Advances in Chemical Physics 34 (Wiley, NewYork 1976); R. Kubo, K. Matsuo, and K. Kitahara, Statis. Phys. 51 (1973). [15] N.G. van Kampen, Phys. Letters 62A, 383 (1977). [16] H. Grabert and M.S. Green, Phys. Rev. A19, 1747 (1979); H. Grabert, R. Graham, and M.S. Green, Phys. Rev. A21, 2136 (1980). [17] C.P. Slichter, Principles of Magnetic Resonance (Harper and Row, New York 1963; Springer, Berlin 1978). [18] V.I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York 1961); U. Frisch, in: Probabilistic Methods in Applied Mathematics 1 (A.T. Bharucha-Reid ed., Acad. Press, New York 1968); V.I. Klyatskin and V.I. Tatarski, Sov. Phys. Usp. 16, 494 (1974). [19] P. Mazur and I. Oppenheim, Physica quoted there.

241 (1970), and literature

[20] T.G. Kurtz, J. Appl. Prob. 7, 49 (1970); 8, 344 (1977); J. Chern. Phys. 57,2976 (1972); Z.A.-Akcasu, J. St a t i s . Phy s . 16,33 (1977); N.G. van Kampen, J. Statis. Phy s . £2,431 (1981). -

Estimates on the difference between succeeding eigenvalues and Lifshitz tails for random Schrodinger operators Werner Kirsch Institut fUr Mathematik Ruhr-Universitat D-4630 Rochum, W.-Germany

In this note we give estimates on the difference of eigenvalues of second order differential operators. We both treat the one-dimensional and the multi-dimensional case. Moreover, we apply our estimates to certain concrete problems of theoretical physics. In particular we prove Lifshitz behavior of the density of states for a broad class of random Schrodinger operator. Let us denote by H the second order linear differential operator H

I

(x )

i,j

d iiX:

+ Vex)

J

where the matrix

(x ) ] . . i.s positive definite for every x , l,J

(x) and vex) are

realvalued functions. To avoid technicalities we assume that a .. and V are bounded, continuous functions an, i j

a .. has continuous partial derivatives. The results below, however, are true under i.j

much less restrictive assumptions. since the operator H is bounded below we may arrange the eigenvalues of H below the essential spectrum in increasing order. We denote by Eo' E

1,

••• the eigenvalues of

H and may suppose that (2)

The chain (2) may, of course, be empty, finite, or infinite. Each eigenvalue

occurs in (2) according to its multiplicity. For the operator (1) is always non degenerate, i.e. that

it is known that

while for d = 1 no eigenvalue is degenerate, that is:
denotes the inner product of

-cf ,

2(dx». L So, we have: inf .


0 such that p' E Lloc(R ),

and 1 E' (u , v) = "2

I

u ' (x ) v' (x ) p (x ) dx.

Fukushima [3] treated the case that Brownian motion and

M'

M

is a multi-dimensional

is a conservative diffusion process.

Our

case is a generalization of his case.

§2.

Relation of the Dirichlet forms

Let

m

and

M =

m'

be everywhere dense positive Radon measures and

and

1

M'

be

Co-regular diffusion processes on forms

(E,F)

and

(2.1)

E(u,v)

(2.2)

E' (u,v)

D

m

(E' ,F'), respectively, where d

1

"2 i, z j=l 1

d

"2 i, l:j=l

ID ID

dV Clv ClX

and

m'-symmetric

associated with the Dirichlet

j

i j

dVij

and

186

for

1

Our basic assumptions are the followings.

u, v E CO(O). Hypothesis M and

are irreducible and conservative. P '\, P' for M-q.e.x. x x m (resp. m') does not charge the M'-polar (resp. M-polar)

(i) (H) (Hi) sets.

M'

(V.. ) is locally uniformly elliptic, that is, for any compact (iv) :l-J set K there exist positive constants Cl and C2 depending on K such that C

(2.3)

d L

l i=l

:l-

v

H

Lemma 1.

The measures

Vij

L

i, j=l

d . , Sd) E R

for all

d

d
0, aij(x) (D) and (aij(x)) is locally uniformly elliptic in D. As in §2, let M be an m-symmetric irreducible conservative diffusion process associated with the 2 L (D; m) - Di r i c h l e t (3.2)

E(u,v)

form determined by the smallest closed extension of 1

"2

d

fD

E

i,j=l

a .. (x)

1J

Under the hypothesis in §2, M' is

m(x)dx,

1

u, v E Co (D) •

p(x)m(x)dx-symmetric and whose

Dirichlet form is given by the smallest closed extension of (3.3)

E' (u,v)

1 "2

d

E

i,j=l

fD

a .. (x) 1J

p(x)m(x)dx,

1

u, v E Co (D).

In this section, under the above hypothesis, we shall derive a regularity property of

p.

When there exists a version

p

of

p

such that, for every k and for a.a. fixed (xl' ... ,x k_ l' d-l xk+l' .•. ,xd) E R , p is absolutely continuous in x k such that (xl' ... ,x d) E D, we shall call p is differentiable. Theorem 7. and

Under the above hypothesis, p 2

{a/axi)logp E L ({Kn};dx).

is differentiable

Moreover, the density

L

t

is given

192

by 1 exp( 2

(3.4)

d

x

s

ox.

J

Proof.

Let

U

and

(D)

E

a

a i l d ft ax.logp(Xs)dMs - 8 L o 1. .i , j=l

fD

Z

i=l

II E

1

Co (D), then

un E

ax. 1.

l ogp

)ds).

F n F'.

According to (3.1),

and N[Ull] = ft t 2 0

(3.5)

d

a (

1

ill ax. ma i J· 1.

i,j=l

These equality combined with (2.6), we have

k i,j=l Z d

=

(3.6)

+

Let any (3.7)

(x)

v

1

1

2

t

fo .

a oX

.. -"+.:.:...:...:_) 1.J

i

(X )

s

ds

d

I:

1.,J=l

be the mollifier and set

ps(x) = P*Xs(x).

Then, for

CO(D),

a( d 1 .. lim E(PsV,Ull) = l'rm 2 I: J a 1.J i,j=l D s .... o s··o

a

(u n )

m(x)dx.

J

On the other hand, by (3.5) and (3.6) , we have (3.8)

lim E(P v,Ull) = lim lim f E d [Ull(X O) - un s .... O s s .... o t .... O psvm x

(X

t

)]

d a a (u n) lim lim -1 E [N[u n ] ] = -llim fp v I: -a-(maiJ.-a--)dx p s vmdx t 2 s .... O E e.... O t .... O t i,j=l xi xj d -1 f pv (rna. . a (u ll) ) dx I: T 1.J ax. i,j=l J d -1 1 a . a (Ull») (X ) ds] ft E' [ lim I: pvmdx o 2 i,j=l m ax. J ax. s t .... O t l J -1 E'

lim t .... O t

pvmdx

[N[urll

t

]

1 im -1 E I [N I [u n l pvmdx t t .... O t

-

f to

d I:

i,j=l

.f.a(Ull)(X )ds] s

J 1. ax. J

193

j

1

d

"2

I:

i,j=l

d I:

+

E' (v u n)

fo

i,j=l

a .. v f. a ( un ) pmdx 1J 1

8Xj d I:

a .. 1Jaxi

Jo

Jo

i,j=l

a .. 1J

pmdx.

1 aXj

Therefore d

J0

lim I: E:-+O i,j=l Taking

d lim I: E:-+O i,j=l Set

v __ E:

fo

2

mdx

ax.

I:

i,j=l

1

f0

vf . a(un) 1 aX

pmdx.

j

Suppv, we have

on

n = 1

such that

n

d

ap

a i J·

apE: a .. v ax. 1J 1

d

mdx

2

l:

i,j=l

fo

a .. v f. 1 1J

au

pmdx. J

then u = x j' d lim l: E:'''O i=l

fo

1

For

E CO(O),

j

apE: mdx aX i

v

taking

v

(a

d

2 -1

I:

i=l

fo

1

) jk m 1)1

j

v f. pmdx. 1

and summing relative to j,

we have 2

J

f

p dx.

1)1

k

Hence

- Jo

p(x)

aXk

for all

dx = 2

f

f

k

1)1

p dx

Which implies that

p

is differentiable and

(3.9) Combining (3.9) with (2.5), we obtain the result. Remark.

By (2.16),

d

a.e. P'. l: X i=l d 2 Hence I: f (x ) (x) o (x) m (x) dx is a smooth measure of M'. i i=l implies that the nest {K } in Theorem 7 can be chosen as n (3.10 )

(x)

a aX

i

logp E L 2({K };pdx). n

This

194

Remark. By a similar argument to Fukushima [3J, we can show the following converse result of Theorem 7. Let

M and

M'

be the irreducible conservative diffusion

processes associated with (3.2) and (3.3), respectively. Suppose (i) a sequence {K} is M-nest iff M'-nest, that they satisfy; n 2 and (ii) p is differentiable and (3/3xillogp E L ({K n}; (l+p)dx). Then P 'V for q.e. x (M). x Acknowledgements.

This work was done during the stay at the

Centre for Interdisciplinary Research (ZiF) of University of Bielefeld and the Bielefeld-Bochum Research Centre of Stochastic Processes (BiBoS).

The author would like to thank Prof. S. Albeverio, Prof.

Ph. Blanchard and Prof. L. Streit for the kind invitation and warm hospitality there.

References. [lJ [2J

[3J

[4] [5] [6] [7] [8] [9]

M. Fukushima; Dirichlet forms and Markov processes, Kodansha and North Holland, 1980. M. Fukushima; On a representation of local martingale additive functionals of symmetric diffusions, Proc. of the LMS Durham Symp. on stochastic , Lecture Notes in Math. 851, Springer, 1981, 112-118. M. Fukushima; On absolute continuity of multidimensional symmetrizable diffusions, Proc. of the Symp. on Functional analysis in Markov processes, Lecture Notes in Math. 923, Springer, 1982, 146-176. H. Kunita and S. Watanabe; On square integrable martingales, Nagoya Math. J., 30, 1967, 209-245. S. Orey; Conditions for the absolute continuity of two diffusions, Trans. Amer. Math. Soc. 193, 1974, 413-426. S. Orey; Radon-Nikodym derivatives of probability measures: Martingale methods, Tokyo Univ. of Education, 1974. Y. Oshima; Some singular diffusion processes and their associated stochastic differential equations, Z. Wahr. verv. Geb. 59, 1982, 249-276. Y. Oshima and T. Yamada; On some representation of continuous additive functionals locally of zero energy, J. Math. Soc. Japan 36, 1984, 315-339. M. L. Silverstein ; Markov processes, Lecture Notes in Math. 426, Springer,

COLLECTIVE PHENOMENA IN STOCHASTIC PARTICLE SYSTEMS

Errico Presutti Dipartimento Matematico, Universita' di Roma

ABSTRACT Collective phenomena in stochastic particle systems are surveyed. Macroscopic field equations describe the evolution of some of these systems just

like the hydrodynamica1 equations describe the dynamics

of real physical fluids. Shock wave phenomena

and metastable behavior are also investigated.

Research partially supported by Nato grant n. 040.82 " Consiglio Naziona1e delle Ricerche".

and by the

196

1.

Introduction

In this article I will discuss some collective phenomena appearing

in the evolution of systems with many components.

In par-

ticular I will consider features like the establishing of the hydrodynamical regime, as ruled by the hydrodynamical equations ( Euler and Navier­Stokes equations), the formation and propagation of shock waves, the existence of metastable states and their transition to equilibrium. All the above phenomena are experimentally observed in physical systems.

They are therefore believed to be present in Hamilto-

nian models of interacting particle systems, but a mathematical derivation seems beyond the reach of the present techniques.

I will

consider the above questions in the frame of random interacting particle systems, where some progresses have been achieved in the last years.

Such systems may exhibit a behavior analogous to that of re-

al physical systems: hydrodynamical equations, fluctuation and dissipation theorems, shock waves, metastable states will emerge from our analysis.

Precise mathematical properties are the counterpart

of such phenomena;

the law of large numbers, the central limitttID­

rem (invariance principle) and large deviation results. Same or similar models as those which I will consider are introduced in such different frames as in genetics, chemistry, economics, statistics ..... the terminology changes but the basic phenomena are

the same. Before describing models and results I will recall in Section 2 the scheme proposed by Morrey for

hydrodynamics.

The

197

crucial point is the notion of "local equilibrium", assuming local equilibrium one can derive the Euler equations.

A critical discus-

sion of such assumption is first given in Section 2 and will keep showing up in the other sections of this paper. recall the way the Hilbert and Chapman ­Enskog

I will also briefly expansion in the

Boltzmann equation fits in the Morrey's approach to hydrodynamics. In Section 3 I will describe stochastic models where the Morrey's scheme can be applied.

Results refer to particular models

except for the theory of space­time fluctuations in equilibrium (the Fluctuation­Dissipation theorem) which has been established for a large variety of models. In Section 4 I will discuss models where shock waves are present, the shock wave front will be characterized by the spatial region where local equilibrium fails, "dynamical phase transition". Finally, in Section 5, I will discuss some models which exhibit a metastable to stable transition and describe the so called "pathwise approach to metastability".

2.

Local equilibrium and Euler equations.

We are in the setup of classical mechanics, we consider a system of identical point particles, having mass m and interacting via a smooth pair potential

V(IXj), which only depends on the distance

198

IXI between particles.

We assume that the initial state of the sys-

tern is described by the probability

d/

L

exp ( -

+

If

(, ),

v(. )

>. (. )

and

q) [

(q,v)

A( E q)

])

IT

(v-v( £. q))2 +

1 L

V( l q-q ': )

q1q'

(2.1)

dq dv

were constant, then

Gibbs measure at inverse temperature and chemical potential

1m

(kT)

-1

d

t

f.

would be the

,mean velocity v

; suitable assumption on the interaction V

are required, like that V is superstable. To avoid ambiguities let us imagine that the particles are confined in some, very large, region. If

then

I"(. ),

v(, ) and

),(,) are "slowly varying" functions,

describes approximate equilibria and it is called a local

eqUilibrium distribution. Slowly varying of course means that

f (. ),

v(. ) and A(') are essentially constant over distances of the order of the length of the correlation functions in the measure To make the statement sharp I fix once for all the functions v( - ) and

f (. ),

(, ) ( assumed to be suitably smooth) and then I take the

limit of small E , the size of the region confining the particles being increased by

0

Given any

and a cylin-

drical function f, if the temperature,defining the Glauber dynamics, is small enough, then there is T such that) for A suitably large)

219

P({w

1 T-

ttT

S

f(w(s»

ds

t

[t_(f)-O

O}

of a variable

=

29, + 1 where /: ,

t

(t

1,t 2)

on the plane,

Levy Brownian Motion

in

Rd with an odd dimension d

is the Laplace operator,

Generalized Stationary Field

= n(t),

t E Rd ,

with the operator L = IakOk which corresponds to a spectral density

14e would like to note that all these random fields represented by the corresponding equation (1) enjoy the so­called Markov property which despite of different approaches has in common that the random field in a domain S == T , is conditionally independent of its part outside of S under fixed boundary conditions on a boundary f = oS given by means of all events from a o­algebra A(r) = nA(

where the o­algebra A(f ) the ­neighbourhood of f.

represents all events generated by the random field in

What kind of events associated with a boundary behavior of do form the a­algebra A(f)? One can look for some explicit form of these events as they are in case of Brownian motion t E when for a "future" domain S = (t,o» wi th a current time­point t > 0 the corresponding boundary conditions can be set in a form of a fixed boundary value at r {t}. A problem suggested seems to be

235

the most interesting in a case when there are no values of

s

essentially is the generalized field and

which can be treated somehow as boundary values on the boundary

r.

The generalized equations (1)-(2) give us nothing but a linear functional

s = (x,s), x = L*

,

which is continuous with respect to a norm in a standard L2(T)-space because of the very equation (2). One can get an idea to treat x = as the Schwartz distribution and to extend s over all distributions x L*f with f E L2(T) . It turns out that this way leads to some remarkable results. Namely, assuming the fact that L* is the non-degenerate operator which for any non-zero f E L2(T) gives us non-zero distribution L*f

=

*f)

=

E Co(T) ,

let us introduce a Hilbert space (3)

with an inner

product (L*f, L*g) X = (f, g) L ,f ,g E L2 (T) . 2

The continuous (in mean square sense) linear functional

given for all x E L*C;(T) can be extended by continuity over all C;(T) is dense in L2(T)-space and

is a

unitary

operator so we can set

(x,s) for

x

lim

x E X because

lim(L*

represented by the limit of the corresponding

(4)

E C;(T).

236

The corresponding values (x.s).

supp x

r

(5)

over all distributions x E X with supports on the boundary r = 3S of the domain S T have proved to be the natural boundary values of the random field s; in particular they generate the very a­algebra A(r) which was above the starting point of our interest. These boundary values can also be used as boundary data in a FORECAST PROBLEM which requires to find the best linear extrapolation of the field s over the domain S by means of its values outs i de of S. The result is that the corresponding FORECAST (t). t E S. can be made by means of a partial differential equation L*Ls(t)

(6)

O. t E S •

A

with the boundary conditions (x. )

(x.s). supp x

r ;

(7)

as a matter of fact the stochastic boundary problem (6)­(7) for the linear differen­ . operator L* L 0 is of the Dirichlet­Sobolev type and it has the unique sol­ tlal ution = (t). t E S . Our approach to the stochastic model (1)­(2) is based on the proper continuity of the corresponding generalized source n

=

(tIl.n).

til

E C;(T) •

which has been treated as the white noise in L2(T)­space. This approach can also be applied to another case. for example. to the well­known Markov Free Field ( 1 - t,) q t )

= n (t) • t

E

d R •

with the white noise n = (tIl.n). til E C;(T) in the Sobolev space W which d) is a closure of the test functions space C;(R with respect to the inner product (u.v)W

d (u.Lv). u.v E Co(R ). 00

associated with the operator L = 1 ­ ts, To apply now our approach we have to substitute L by Sobolev's W­space and to introduce the corresponding space of our 2­space test distributions x E X

237

X = L*W This change with respect to (3) is reflected for example in the FORECAST PROBLEM which solution = t S. in the domains S c Rd can be given by means of the corresponding differential equation L (t)

= 0,

t ES •

with the boundary conditions (x. ) = on the boundary r

=

as

supp x

r •

(cf. (6)-(7)).

Let us consider the Equation (1) in some domain

T0 cT. -

= n(t). t E To •

(1' )

wi th the random source n ,

continuous in the standard wide sense with respect to II\PII L • The Equation (1') has 2 which are well-defined over all «test distributions» a variety of solutions x E X with the corresponding values

in the closure To of the domain ary values

on the boundary gives all values

fo

To continuous with respect to ll xl! .

The bound-

aT o can be partly determined by the very equation (I') which

= (f.n). x and in particular the boundary values

238

where supp x c-

f } 0

represents a generalized boundary formed with the generalized variable x E X on the boundary f o of the domain To . It is remarkable that on another part X+(f)

of the generalized boundary

in the direct sum indicated the boundary values (x.E;)

=

+

(2' )

). XEX (fa)

a

can be given arbitrarily by means of any random function

of a type considered.

Theorem. The equation (1' ) with the arbitrarily given boundary conditions (2' ) has the unique solution which can be represented over all test distributions in the closure of the domain To as

= (Gx.n)

+

o

)

where TI is the projection onto the boundary subspace X+(f) in X parallel to L* L2(T o ) and G is the «Green operator» giving the solution g = Gx E L2(T o ) of the conjugate equation L*g

x -

TIX •

It turns out that the solution of the stochastic boundary problem (1') - (2') enjoys the Markov property in the case of the boundary conditions (2') independent of the random source n inside of the domain To represented by a generalized random field with independent values. Namely, the following result holds true. Theorem. Consider the random function in any set S' conditionally independent of its part outside of S' conditioned with its boundary values supp x on the boundary r of all boundary data

as'

=r

of S'. The best 1inear FORECAST of

supp x

=

f

,

can be given as the unique solution of the equation

in

S by means

(3' )

239

L*L u(t} in the interior domain 5

o

=

0, t E So

of 5 with the boundary conditions (3') .

These results can be extended on a very general case in a framework suggested in [1], [2]. §2. General Boundary Conditions for Linear Differential Equations

•

We are going to consider here a general linear differential operator

in L2 (T)- space and the equation Lu (t)

=

f(t), t

E 5 ,

(8)

in the domain 5 T with respect to u E W from a certain functional class W where the operator L will be determined in a very direct way. We do not assume anything about coefficients of this operator (they could be even Schwartz distributions) so from the beginning L is well-defined on the corresponding Schwartz space of infinitely differentiable test functions E ,

One can take W sponding norm we take

E

in a way that L be continuous with respect to the correso it can be extended to the closure in W;

W as a completion of

with respect to a semi norm L 2

where any u

E

W can be treated as a 1imit u = 1im

of the corresponding

and we set

E

Lu

= 1 im

.

240

Of course F = LW is a closure of product

in

and W is the Hilbert space with the inner

L 2(T)

(u,v)W = (Lu,Lv)L

2

Considering u E W in the domain SeT we are dealing with a fairly rich functional class and there is a variety of solutions u E W of the Equation (8) for any f E F . It turns out that the solution corresponding boundary conditions

u E W is determined uniquely by means of the

(9)

which prescribe the boundary values (u,x) of u E W over certain Schwartz distributions x with supports supp x r on the boundary r = as. Namely, let X be a space of all Schwartz distributions ,

x = (o,x}, (() E

continuous with respect to IIL(()II L 2 by the corresponding limit

•

All of them are well­defined on the space W

(u,x) = lim (((),x) for any u

1im (().

We suggest to treat

u E W as a function

u = (u,x), x EX, of a generalized variable X with values (u,x) which are the result of application of the corresponding Schwartz distribution x E X to u E W. Let us call X the test distributions space. • To avoid a confusion with complex conjugate values let us assume that we deal a with the real L2(T)­space and the conjugate operator L* is introduced as follows: L*f for any f E L2(T) is a Schwartz distribution

241

given by means of a standard inner product in LZ-space; of course

in a case of L with infinitely differentiable coefficients, say. Lemma 1.

The test distributions space can be represented as

X = L*F = L*LZ(T) . Proof. Any x f X gives us a linear functional respect to in the Hilbert space

and it can be represented by means of some f

so this x coincides with the distribution distribution

f

which is continuous with

F in a form

L*f; moreover, for any f

f

F the

is continuous with respect to f L so it belongs to our space X. For any Z' f f LZ(T) the distribution L*f coincides with L*pf where P is the projection of in because operator onto the closure F = o 0

*f) = =

Z

*Pf),

f

=

Z

= . Co(T)

The proof is finished. Thus we deal now with the very same class of the Schwartz distributions x f X as it was suggested in (3). The only change must be done with respect to the inner product in X which we set now as (L*f,L *g)X

=

(Pf,Pg)L ' f,g Z

f

LZ(T) .

By the very definition of our Hilbert space W any test distribution x = (u,x}, u f can be treated as a 1inear continuous functional on W. On the

242

hand, any linear continuous functional x = (u,x) on W is continuous with respect ' u E ColT), so it gives us some test distribution x EX. Thus to IluII W= Illull b X is a dual space to W. Moreover, taking into account the representation (u,x)

=

(u,l *g)

(lu,g)L

=

(10)

2

of the bilinear form (u,x), u E W, x EX, we see that any u E W is a linear functional u

=== (x,u), x E X (u,x) def

On the Hilbert space X with

Obviously any linear continuous functional represented as (g(lu)L

2

=

x

l *g,g E F .

u

(x,u)

on x

l *g, g E F can be

(Lu,g)L ,g E F , 2

and it can be identified with the corresponding u E W for we have one-to-one correspondence in the relationship lW = F by means of the unitary operator l: W+ F. One can easily verify that Wand X are the dual Hilbert spaces conjugate to each other with the bilinear form (10),

and

ll u II W = sup I(u ,x) I Ilx II X=1 Ilxli x

sup

uu Il w=1

l(u,x)1

We see that any function u E W is uniquely determined by its value (u,x) over all distributions x E X. It will be beneficial to treat u E W as the linear continuous functional u = (x,u), x E X, with values ( x,u ) def ===

(u,x).

•

• Going back to the Equation (8) in the domain S let us appeal to the corresponding boundary values

243

(u,x), supp x

c f,

over our test distributions x E X with supports in the boundary f 8S. Obviously a collection of all these distributions is a certain subspace X(r) eX; let us call X(f) boundary space. One has to expect that some of the boundary values are determined uniquely by the very equation (8) for any solution u E W. Namely, the equation (8) gives us the function Lu in the domain S by means of the corresponding f E F and according to (10) we have

for all x = L*g with g E L2( S) so the values (u,x) over all test distributions x E L*L2(S) from a closure of L*L2(S) in X are determined by the very equation (8) and it holds true in particular for all generalized boundary points x E X­(f) from the boundary subspace

Thus all boundary values (u,x), x E X­(f) , are uniquely determined by the very equation (8). It is remarkable that the other part of the boundary values can be arbitrarily described for the solution u E W of the equation (8); to be more precise, this is true for the boundary values (u,x), x E X+ (f) , at all generalized boundary points from any forms a direct sum

boundary subspace

X+(f)

X(f) which

This fact is reflected in the boundary conditions (9) given by means of the corresponding values

of an arbitrary linear continuous functional

244

on the boundary subspace

x+(r).

Let us introduce the space ports supp x S in a closure

X(S) of all test distributions x E X with supS = sur of the domain S. Let

be an orthogonal complement to a subspace

of all

g E F with supports The following result

supp g

[1]

SC

in a complementary SC = T\S .

holds true.

The differential equation (8) with any source f E F has the unique solution u E in S with the arbitrarily given boundary conditions (9) and it can be represented over all test distributions x E X(S) as (12)

IT;

where is a projection operator onto the boundary subspace the subspace in the direct sum

. and G+r 1S an operator giving us g differential equation L*g

with

+ E x - 'TTrX

G;x

x+(r) parallel to

by means of the conjugate

E

+ x ­ 'TTrX

C .L L*F(S) .

The linear continuous form c (g,f), g E F(S)

,

which appears as the first term in the formula (12) can be determined by means of f E F in the very domain S because according to Lemma 2 the projection PL 2(S) is dense in and we have

245

(Pg,f) = (Pg,f)LZ(T)

(9,f)LZ(T)

(g , f) L (S) , 9 E LZ(S) Z

as a matter of fact the first term in the formula (12) represents the values

of the solution u = u1 with zero boundary conditions so the operator G; plays here a role of a generalized Green's function; correspondingly the second term in (12) gives us the values

of the solution u Proof.

= uz of the equation (8) with zero source f = 0 .

As has been mentioned, the equation (8) gives us all values

remember, P is the orthogonal projection onto F = LC;(T) gonal complement

is formed by all

L*­harmonic functions

in

x = L*g with 9 E L*9

9 E LZ(T), i.e.

Lemma 2.

are zero distributions =

(tp,L *g)

=

(Ltp,g)

The subspace

coincides with the closure PLZ(S)

in

LZ(T)

and

0,

over

LZ(T) . The ortho-

L*g(t) = 0, t E T , for all

(u,x)

tp E C;(T).

246

Proof.

The intersection

has its orthogonal complement in L2(T)-space generated by the sum of the corresponding orthogonal complements F.L and L2(S) = L2(Sc).L, namely L2(T) e F(Sc) Taking into account that F(Sc)

= F.L

+

F we obtain

L2(T) e F(Sc) = [F.L F.L

L2(S)

$

$

F) e F(Sc)

[F e F(Sc)) = F.L

$

F(Sc).L

and F(Sc).L = [L

2(S)

P[L 2(S) + F.L)

* F For the unitary operator L: *

L L2(S)

+

F.L) e

=

PL 2(S)

X we have =

L*PL 2(S)

-*.,.--L PL 2(S)

= L*F(Sc)

That is all. To prove our theorem, let us appeal to the space X of our test distributions . h g E F and to L* : F X as the unitary operator. We have x = L*g Wlt (j),X) = (L(j),g)L

2

= (L *L(j),L *g)X =

"" . (L*L(j),x)X ' (j) E Co(T) Let us consider the subspace X(S) c X of all x E X with supports in S and the boundary space X(r) X(S). Obviously the subspace X(r) X(S) is formed by all x E X(S) which vanish in the domain S,

Thus we have the orthogonal sum X(S)

*-- $

L

X(n

247

where and

Remember in the direct sum (11),

we have

so obviously the following direct decomposition holds true:

(13)

As has already been emphasized, the equation (8) with f E F treated in the very direct way gives us nothing but values (u,L *g)

= (Lu,g)L

= (f,Pg)L

2

2

(f,g)L

2

(Pg,f)L' 9 E L2(S) , 2

of the bilinear form (10) so by the very equation (8) we are given all values def (x,u) =

(u ,x )

which determine its solution x

(f,g)L

2

= (g,f)L

2

c .L ' 9 E F(S) ,

u E W over all test distributions =

* def * C.L L 9 E X-(S) = L F(S) .

With the arbitrarily given boundary conditions (9) we can determine a linear continuous functional u = (x,u) with the prescribed values

def (u,x), x EX,

248

and (u.x ) = (f,g)L

Z

'

x = L*g E X- (S)

which are uniquely determined on the direct sum

for according to the direct decomposition (13) we have

This ends the proof. One can treat (8) as the generalized equation (L*

=

co E CO(S)

for the corresponding u E W as the function u = (x,u) over the test distributions x EX. It is easy to verify that in this way we get the very same result for the boundary problem (8), (9) as was given by our theorem. •

Considering the generalized differential equation (L*c.u)

= , E Co(S)

=

we can in our framework treat the source f =

f E F as a generalized function ,

E

(14)

which is continuous with respect to

Z

=

ilL *

.

(15)

By some reason we would like to indicate the case when the equation (8) does which is continuhave the solution u E W for any generalized function f ous with respect to the norm L in LZ­space. Obviously it holds true if and 2 only if the orthogonal projection operator

249

on the sUbspace L2(S) in L is invertible, i.e. 2(T) bounded. This can be characterized as the following

p- 1 does exist and it is

COMMON CASE: either we have F = L2(T) or the orthogonal compiement F.1 = L2(T) e F forms a non-zero angle with L2(S) in L2(T)-space. (Roughly speaking, if any L*-harmonic function g E H belongs to the subspace L2(S) over SeT then being zero in the neighborhood T\S of the boundary of the domain T it is the solution of the differential equation L*g(t) = 0, t E T, with zero boundary conditions so gO; thus H

n L2(S)

=

a

and one can verify his feeling about our "COMMON CASE" being common case by an example of the Laplace operator L = 6.) In the COMMON CASE, our theorem on the boundary problem (8) - (9) can be improved in the following way. There is no necessity to introduce the bilinear form (Pg,f)

(g,f), g E L2(S) ,

=

on Pg E F(Sc).l which allows us to define the first term in the formula (12), for in the COMMON CASE we have one-to-one correspondence in the relationship PL 2(S) = F(Scf, so in the formula (12) we can substitute 9 = G;X E F(Sc).l by the corresponding -1 -1 P 9 E L2(S), (P g,f) = (g,f) and introduce another operator

which for any x E X(S)

gives us 9

* =x

L 9

-

= G;x by means of the conjugate equation +

ITrX

with the unique solution 9 E L2(S)

(

16)

•

• One can be interested in the equation (8) with the source f being a vector function in a Hilbert space, say. To deal with this case, let us introduce f E F as a class of all generalized vector functions in the Hilbert space considered which are continuous with respect to IIP\jJ11 L as was indicated in (14-(15). For any 2 f E F we define (P\jJ,f)

def

(\jJ,f) , \jJ E

,

and by continuity it can be extended to the vector linear function

250

c J.. , (g,f), g E F(S)

on the subspace

- see Lemma 2. We have to introduce u E W as a class of vector functions which are well-defined over all test distributions x E X(S) in the closure S of the domain S with the corresponding vector values (x,u) ,. (u,x) which form a linear continuous vector function on the generalized variable x E X(S). In a similar way the boundary conditions can be represented by means of the arbitrarily given linear continuous vector function

on the boundary generalized variable x E x+(r). It is easy to see that our approach to the boundary problem (8)-(9) can be applied to the vector functions and the following result hodls true. Generalized Theorem. For every vector source f E f the generalized equation (8) has the unique solution u E with any arbitrarily given boundary conditions (9); this u E can be represented over all test distributions x E X(S) by the corresponding formula (12). (Of course in the COMMON CASE the first term in the formula (12) can be ob+ tained by means of the conjugate equation (16) for Grx = g EL • 2(S).) • We would like to note that our approach to the boundary problem of the (8) - (9) type can be applied to the operator L in any functional Hilbert space which is local with respect to L, i.e., any elements l.o , u , f are orthogonal in this space if tp E and f have disjoint supports (see [2]). As an example, our Hilbert space X of the test distributions is local with respect to the operator L*L , (L *Lp,x ) = (tP,x)

if tP E

a

x E X are with disjoint supports and the following result holds true:

For any generalized vector function

251

which is continuous with respect to 1I\jJ!I W = I!L\jJI!L

ut. *L\jJII X

2

the generalized equation L*Lu(t) has the unique solution u E

=

y(t), t E S ,

with arbitrarily given boundary conditions

(x,u) = (x,u r), x E X(r) ;

(18)

this u E W can be represented over all test distributions x E X(S)

as (19)

where TIr is the orthogonal projection operator onto the boundary space X(r) and Grx = v E is the unique solution of the equation

=X

=

in S) W. The first term in the formula (19) represents the values of the solution u E W with zero boundary conditions (18) by means of the linear continuous function (v,y), v E over the closure of in our functional space W= defined from the very beginning; the second term represents the values over the test distributions x E X(S) of the solution of the equatien (17) with zero source y = O. (One can exercise that the solution u E W of the boundary problem (8)-(9) in the same time is - with the source y = L*f and the corresponding the solution of the equation (17) boundary conditions (18).) • • Exercise. Let us consider L = d/dt in the finite interval approach we get the corresponding Sobolev spaces

w= Wi2

'

X

T c R1. By our

W- 1 2

The space F = LW is formed by all functions f E L2(T), IT f(t)dt = o. The corresponding H = L2(T) e F formed by all L*-harmonic functions g(t) = const., t E T,

252

gives a trivial illustration to our COMMON CASE when the subspace H has a non-zero angle with LZ(S) in LZ(T)-space. The generalized equation (8):

gives us by the limit LZ(S) the values

l(t.b)

to the (t.b)-indicator function

u(b) - u(t) = lim

a




Ix[O)j

(1) < _I x[O)

I

for

x[+)

for X[-)

}

A conjugation J is an anti linear transformation of [g,f) •

[11)

4f,

such that J2

I and [Jf,Jg)

292

?( t o}

c

" -c0.,,,

I, ('hl(- )

We denote the boundaries /1)

x(O)

of these wedges by H(±) (cf. Fig. 1).

The wedge X(+) can be parametrised in terms of Rindler coordinates

x

(1)

T,X

T, x(2), x(3))

(0)

with 1;, T running through

,

[12 )

and

1R,

respectively.

Thus, by equations

(10)

and

(12) the metric in X(t) is given by (13 ) T

is therefore a time coordinate for X(+).

Furthermore, the trajectories of uniform

acceleration along axel) in X(+) are the curves, C, on which 1;, constant (cf. Ref. [11], Ch , 6).

) and x(3) are

From (12) and (13), i t follows that the proper time

and the acceleration on C is given by T

and

prop

= TI;/c

(J.

(14) (15)

res pecti vely • We note here that the boundaries H(±) of X(+) are horizons in the following sense.

293 The past light cone of any point P in X(+) does nct cross the boundary H(+) (cf. Fig. 1), and therefore no signal can reach X(+) from anywhere across H(+).

signals emitted inside X(+) can cross the boundary termed

H( -).

Likewise, no (+)

For these reasons, H -

are

evenr horizons.

We now make the important remark that time-translations for a uniformly accelerated observer in X(+) correspond to Lorentz boosts for an inertial one.

To prove this, one

simply notes that time-translations for the former observer correspond to the transformations --

+ s,

->- T

By equations

(12),

x(O)

these induce the space-time transformations ->-

x(O) coshs + x(l) sinhs,

(IS)

which are just the Lorentz boosts for velocity ctanhs along axel). Curved Sp ece-vt Irnes , wise x

(u,

Consider now a class of space-times of the form

X = tR 2

x Y, point-

y), with metric given by

VI

2

2

2

2

flu -v Ll dv -du ) -

2

g Iu

2

-v )

with f and g smooth positive-valued functions on _ (-1")

u, yare spatial ones.

We define X -

Thus, v is a time coordinate and -

to be the open submanifolds of X given by

I vi

u >

(18)

and

u

(x), a Hermitian operator-valued distribution, representing the quantum field. P, a vector in

H,

representing the vacuum state, that is cyclic with respect to the

algebra of observables!

generated by the 'smeared field'

with f run-

ning through the Schwarz space The theory is centred on the Wightman functions (22) which are taken to be tempered distributions, so as to cover the requirement that the canonical commutation relations involve 6-functions. to the algebra

Since 'I' is cyclic with respect

of the smeared fields, it follows that the distributions W carry all

the information concerning the states given by the vectors and density matrices in

H.

The Wightman axioms concerning (1+, U, cj>, '1'), and thus W, are the following.

(IJ cj>(x)

and cj>(x') intercommute if x and x' have spacelike separation.

This is the

requirement of relativistic causality. tII)

and

Uf g l cj>(x) U(g-l)

cj>(gx)

(23)

(24)

U(g)P

which are the requirements of a relatiVistically covariant field and invariant vacuum, respectively.

It follows from the latter condition that the Hamiltonian H, defined as

(-in) times the generator of the time-translational subgroup of U(G), annihilates the One could regard this algebra as a representation in in Section 2.

of an abstract one, as discussed

295 state'!', Le.

o.

(25)

lIII)The Hamiltonian H, which we have just defined, is a positive operator.

In view of

(25), this is the condition of stability of the vacuum against creation of particles. The essential power of these axioms stems from the fact that they imply that the Wightman distributions Ware boundary values of analytic functions of complex variables W(zl' •• , zn)' over certain domains in(:4n (cf. Ref. [12J).

Here the analyticity

properties arise from the positivi ty of H in the same way they do for the correlation functions of ground states, discussed at the end of Section 2.

Furthermore, the

axioms (Il and (II), of local commutativity and relativistic covariance, serve to extend the domains of analyticity of the functions W.

In particular, they permit an

extension of the one-parameter s ucgroup Lv(s)/selK.{Of U(G), representing the Lorentz boosts (16), to imaginary values of s , where the resultant transformations then correspond to rotations in the plane Dx(D) x(l) and, in the case s = irr, to space-time " " W) (0) (ll 0) ->- -x ,x ->--x

a nve rs a ons x

The following two key theorems have been derived from the Wightman axioms by methods involving the analytic extensions of the distributions W, and in particular of the Lorentz transformations V. PCT Theorem

12

There is a conjugation J pCT of H such that (26)

This theorem tells us essentially that the laws of Physics are invariant under the combined action of space-time inversions and particle-antiparticle interchanges.

The

next theorem concerns the action of the Lorentz transformations on the algebra of observables for the Rindler wedge X(+), defined as the algebraJf,(+) generated by the smeared fields Jo/(x) f(x) dx, with f running through in X(+). Bisognano-Wichmann Theorem

13.

functions with support

Let iK be the infinitessimal generator of the one-

parameter unitary group [V(s)/silR'lof Lorentz boosts.

Then (27)

where J is the conjugation of J

given by the formula (25)

and R is the representation in N of the spatial rotation through rr about Dx(l) • " f th t f l.e. 0 e rans orma t"lon x (2) ->- -x (2) ,x (3) ->- -x (3)

296 5.

THERMALISATION OF QUANTUM FIELDS BY GRAVITATION It is now a simple matter to infer from the results of the previous three Sections

that the gravitational field corresponding to a uniform acceleration in flat spacetime thermalises any ambient quantum field. Section 2, time­translations (+)

Rindler wedge X

(T 7 T

The argument runs as follows.

As noted in

+ s) for a uniformly accelerated observer in the

correspond to Lorentz boosts for an inertial observer.

follows from the specification of K, in Section 3, that

Hence it

is the Hamiltonian govern-

ing time­translations on the '(­scale for the accelerated observer.

Furthermore, by

the Bisognano­Wichmann theorem, the restriction of the vacuum state

to the observa-

of X(+) satisfies the KMS condition (7) with respect to Lorentz boosts. Hence, as viewed by a uniformly accelerated observer in the Rindler wedge, this is a KMS state.

Furthermore, on comparing the formulae (7) and (27), one sees that the

temperature of this state, on a scale for which T is the time, is

Therefore,

by (14) and (15), the temperature corresponding to the proper time­scale of the accelerated observer is Ta

1'lal2nKc,

in accordance with Unruh's result.

By the Principle of EqUivalence, this result implies

that the gravitational field corresponding to

uniform acceleration

0(

in flat space­

time thermalises any ambient quantum field to the temperature T Since this result a• is a consequence of the Bisognano­Wichmann theorem, its connection with the PCT theorem is evident. Comments.

signi fi can C8

(1)

The essentiai);;'"f the result described here is that a state that is a

vacuum for an inertial observer is thermal for a uniformly accelerated one.

In other

words, it corresponds to a relativity of temperature with respect of acceleration. It does not imply that the accelerated observer

ees effects corresponding to what the

inertial observer would regard as excitations (or particles), when the system is in its vacuum state:

for the Hamiltonians, 1'lK and H, of the two observers are quite different.

(2) From an operational point of view, this result signifies that a localised thermometer carried by a uniformly accelerated observer will register the finite temperature To-+ W(t,w) -W(s,w) . Obviously, every where Z: M x II ---+ IR other Wiener process

n Llx

, let

IN

n

0-

/d6t' , n

1 (s, 't )

Z

E

k 1,··· ,k d E

W'

T

k E IN O

o-

n

a -n

) x

n

T

T

a

I(.

n

an arbitrary function, and dom variables on

lln

,t

r

)

T

with

< t

< b

n

a < s -n

n

Q . For each

leads to the same distribution LIt

be a probabili ty space,

(lln,Oln'P n)

t

n

.-

n

1 (k 1 LlX n,··· ,kdLlx n) I

Tn

< b < b a < a -n n -n n

e : Tn n

n

Tn)

a positive real,

x

T

n

---+

0-

[0,1]

a sequence of independent ran-

with distribution

P (X t = x ) = 1/(2d) for all n n, x e 1 +e1,-e1, ... ,+ed,-e ,where el, ... ,e is the standard base of d d) d IR For each n E IN , define the processes W T x II IR d , P : r n n n n n

I

SETn:s

IR) • ("Reasonable" essentially means that

x

S - continui ty for every infini te

Proof.

A vaguely, n,e n n-+ oo 2

11).

Then

'eO(M x IR

*e

(or equivalently,

2

n+ oo

e

..} , f , e.

*1

preserves

11

11. )

It is sufficient to show that for every infinite 11, we have l O. But this follows immediately from (7) and theorem 3. a

L(*Ol1) • st-

Now let us return to the proof of theorem 2. According to the nonstandard version of Kolmogorov's continuity theorem, it would be sufficient to find positive reals

", 8 , C

E;.[ Ipe(x) -Pe(y)I" J for all

x, y

ns( f)

Ei:

C

such that

I x-y Id + 8

(8)

Unfortunately, such an estimate cannot be true,

0 $ 0 ) , because then Pet f = 0 . But since e e o is fair, we need the estimate (8 ) only in the case that x - y E , be-

e

e

if e. g.

(and

cause then a similar argument as in the proof of theorem 3 shows that Pet r e

Pet r 0

and

have the same standard part

1 'V = "2

with

ee

,..,

e

o Our proof of the estimate (8) is based on a discrete version of the Fourier K be the smallest even

inversion formula. To this end, let

.

(K/2)tlx > (l v dw)/ tlx

that

,

kK/2

(11,=1, ... ,d) nil, E *Z Note that w (t) 11 / (1 " dw )

Lemma 5.

Let

&(y).-

a:

t XE

where

x·y

a (z )

f

-- *C

l

Let }

.-

f

k (k E IN)

w (s)

to

f

1

*integer such

(111tlX, ..• ,ndtlx) I -kK/2 < nil, and put y .- 211/[K(tlX)2] < 0y for all

be internal. Define

s &:

I

t f

E

1

T

with

-- *t

S,t

by

(tlx)d exp j v I x-y ]

r1

denotes the scalar product of the vectors (2Yl1)d

t

y. r 1

Vz

(tlx)d &(y) exp[ -yi y.z]

Proof.

I XEf

a(x) (

1

l yc;:f

exp[yiy·(x-z)] 1

) .

E I' 1

x, y •

f

f

1

• Then

w

304 So it is sufficient to show that

1

,(x,Z).-

{

exp[yiy.(x-z)]

y er1

K/2

if

o

x

z

otherwise

d

1

rIx z) j

"d=1-K/2

e xp ]

I

e xp j v Ln e x

wi th

= 0 , i. e.

*;;

1;

x, z

d = 1 . Fix arbi trary

such that

-(K-l)

(K-l)

1;

E

. Hence

r

l K[,

. Then , iff

x = z . This implies K/2

1

dx,z)

)J,

-

j

we only have to consider the case 1; fiX

I

9,=1

K/2

d II

1;

d

). Since

(x , z E:

x - z =

K

n=1-K/2

{

K/2

2 exp[yi",(flx)]

K , if

o ,

I

n=1 K/2

K[I; , i. e.

K

x = z • t1

otherwise

By lemma 5, we have Y

I

(2 ,, ) d

(lIx)d Pe(y,w) exp(-Yi y.z )

(z

r 1 ' co

E

Q)

YE:f l

with

a

b

s=a

t=b

I'!It I'

c t. e t s t ) exp(yi[w(t) -w(s)].y) , by (6). c

Therefore, we get for every

k

and

u a exp(-yiu:y)])( I' J s =a 1 -

[ w(

)

-

to

b

a

I'

b

I' ... I'

t =b 1 -

t =b k -

e

x, y

L k

f

l

(IIX)

1

(lIt)2k[

: dk

k

(II

j=l

[exp(-yiu:x) J

k

k e(s.,t.)]E[exp(yi L j=l J J j=l II

(s . )]·u . ) ] )

J

Using the fact that

J

(w(t»t E T

has independent, identically distributed

305

increments and employing Holder's inequality, we after some substitutions arrive at

I

(6X)

dk

k

(IT

VkEf k k (v.-v. l)·xJ -exp[-yi (v.-v. l)'yll) J

J-

J

I

(6x)

dk

Vk"f k

k (IT

-v.

lexp[-yi (

J-l

j=1

k

b-b 2 [FBtIE(exp[yiw(t).v.J)IJ j=l t=O J

1/2

x - Y

1/2 IE(exp[yi w(t).v.))1 J2} J

).xl-exp[-yi(v.-v. J

, where

IT

If

a-a

(F s t

j=1 t=O

J-

v

O:

, we can choose a suitable transformation

Iii

lexp(-yi

j=l

J-l

)·yJI)

O.

v

v

such that

we can use the estimate

(a E JO,I[)

in order to obtain the following lemma after some very tech-

nical manipulations: Lemma 6.

If

a " [0, 1 [ , and

ktlN

(1 - e xp [

where

v0:

0,

r.-

0

o

T [v j I 2

: = 1 ,and

2

.s ] )

r: =

S

E

let

*IR

+

? .

{x/ 12' I x E r 1 }

if

d

=

2

, re-

{(x / 12 , ... ,x

, i f d is I (x •.. ,x ) E r 1 1' d d_1/I2,xd) = c (d,k,w) E IR+ k E IN , there exists a constant 1 with the following properties: spectively

1

odd. Then for every

(i)

Whenever

a E JO, 1 [

and

x, y E r 1

wi th

X -

Y E fe'

then


,n

Moreover

:5.

exp(Gc

3vn)

, by lemma 12, where

c

(w)

3

with

w

O(v+1)

.

Thus it remains to show the finiteness of - 2G [a

2 n_1 II

11=0

Fix

11

¢,n -

0

,n])

v exp (- 2G [ T e (t, (n))

n'

{O, 1 , ... , 2

n_1}

J

¢ -

• Put

T

v 0 (t, (11»)

n'

v.- 2

K

-

¢ ]

nt,t

)

J • Note that

(20)

312

-(n-l)

(n 2) vGl the class of quasiinvariant probabality measure on Q' satisfying the above condition. In this case, Hamiltonian is written as 1 00 2 (1.5) (Hf)(U ="2[ v i=l 1 1 1 where s(e i) = 2i(1T(e i ) 1 ) (t). From now, we assume that v belong to

11'1'

Next, we must ask the following uniqueness question.

(I) All closed extensions of

coincide?

This is equivalent to the question whether H self­adjoint operator on

or not.

v

already essentially

But in this note, we treat the

Markovian uniqueness question which is weaker than the above, that is, (II) For which v do all Dirichlet extensions (cv,FC

O) Namely, when do all self­adjoint extensions of S

v

rating Markov semi­groups coincide ?

coincide? = H tFCOOO gene­

v

In §3, we give two classes of quasi­invariant probability measures such that S

v

has the Markovian uniqueness.

In particular,

Theorem 3 gives an extension of the result of [8J.

§2.

The characterization of maximum Dirichlet space We denote by

sions of S. v

(2.1)

Al

v

) the totality of Markovian self­adjoint exten-

Let us introduce a semi­order

< A2

C

in

v

) by

l)V­ A2)

(/­A l u,,I­A l u)v

V­A 2u,,I­A2U)v for uE

Duqi) .

In order to show the Markovian uniqueness, we usually need two steps. The first step is to show that there exists a maximum element, say AR, in Att(sv) and to characterize the space i)V­A The second step is R). to identify the operator A with the Friedrics extension H Recently R v' for a fairly general class of quasi­invariant measures, Albeverio­

321

Kusuoka [4] settled the first step. If v belongs

Theorem 1 (Albeverio-Kusuoka)

and strictly

positive, there exists the maximum element of*n(Sv), say A and the R, space D(/-A is represented as R) u is stochastic H Gateaux differentiable} 2 with respect to V, ray absolutely conti(2.2) uEL (v) ;nuous and the stochastic Gateaux deriva{ 2 tive DU of u satisfies that U'Du (v)VrrEL (v) To prove Theorem 1, they prepare the next lemma. If v belongs to

Lemma 1 (Albeverio-Kusuoka) that for any A _

JllASv) ,

1)(r-::A)

a,

then

ii)

has a unique Markovian self-adjoint extension.

Here,

is Sobolev space of order r and degree p on the Wiener space (see [6]) •

fying that

i)

a

1

00

sequence of Co(R )-functions satis-

Proof < 1

ii)

(t)

={

on 1 0

t

2

on

t

< 1

2




322

on

la,

iii)

(t)

I

{:

,where c is some constant. 2£+1

on

Put £ (x) = a£o1jJ(x) and set 2 {u E L (1jJ2)1) ; 41£' u E U Dl for any £ and \(DU,DU) H1jJ2 d)1 < oo}. 'Y l 1. Actually, in this case an algebraic lemma 131 says that for generic pairs of Cartan subalgebra tl, t2 then tl, t2 and [tl, t2] generate L£neanly the Lie-algebra sU(N) of SU(N) (a simple counting of dimensions makes this result plausible). This algebraic lemma is the infinetesimal version of i) (see 131). Clearly for other classical groups, the dimension argument does not work and Lemma 77.3. cannot be true. G Now, taking A a Borel set in Ce(T'/T) one sees that for PI e A and PI AC , the spectral measures of uP1 and of uP2 are Gaussian measures supported by disjoint sets only depending on A. This means the disjointness of the direct integrals (13).

The proof of Theo/tem 7.2.

is now easy.

The commutant of

A = {UL(T S), UR(TS) } II (13)

only contains decomposable operators of the direct integral e e or, in other words, A is maximal abelian in the bounded operators on H. Considerfng the operator, for y (W(Y)F) (E;)

acting on F e L2(C e(G/T),lll) W= f

ffi

T; , I:;

Ce(G/ T)

( 1 1/2 (dll1 Y- U) F(y- 1 1:;) d III (E;)

and

W(Y) dll T (a)

340

(the integral of the constant field of operation a + W(y)), one easily sees that and UR(T eS). It follows from the simplicity of spectrum W commutes with UL(T S) e of A,

that:

for

all

a and consequently,

this with Lemma 71.3. ii)

GeS :

¥W

implies that UCt(G S)" contains the multiplication by e

all measurable bounded functions on Ce(G/

T)

.

By the ergodicity of the left action is maximal abelian in

UR(T S)" in e Irreducibil ity of Ua,

The same happens for ducing 11

110 1

+

11-

1

•

UR( GS)" e

for

UL(G S)". e

by conjugacy by the mappi ng i nall

a comes now from

(TheoJtem 8-32).

For the factoriality of UR, takin!:J Z in the center, one has that is decomposable in the decompositi on of TheoJtem 1. 1. . By irreducibil ity of for almost all a, Z is diagonalizable.Using a conjugacy and ergodicity as the case of W before, one obtains that Z is a scalar.

The proof of TheoJtem 7.2.

is completed.

REFERENCES

III 121

S. ALBEVERIO, R. J. Funct. Anal.

i!.,

D. TESTARD. 378,

(1981).

§l, 115,

(1983) .

S. ALBEVERIO, R. J. Funct. Ana 1.

D. TESTARD, A. VERSHIK.

Z a U in

341

I

I

31 41

I 51 I 61

S. ALBEVERIO, R. J. Funct. Anal.

D. TESTARD. (1984).

49,

T. DUNCAN. Brownian motion and Affine Lie Algebras. Lawrence, Kansas, (1984). I. FRENKEL, V. KAC. Invent. Math. 62, 23,

Preprint University of Kansas,

(1980).

Ya. GELFAND, M. GRAEV, A. VERSHIK. Uspehi. Mat. Nauk, 28, 5, (1973). (Translation: Russian Math. Surveys,

83,

(1973)).

1 71

V. KAC. Funkt. Analys i ego prilozh, 252, (1969). (Translation: Funct. Anal. Appl. 3 252, (1969)).

I 81

J. MARION.

Anal. Pol. Math., 43, 79,

191

110I

G. SEGAL. Commun. Math. Phys.,

(1983).

80, 301,

(1981).

M. TAKESAKI. "Theory of Operator Algebras, Vol. I". Springer-Verlag, New-York, (1979).

Proof of an algebraic central limit theorem by moment generating functions Wilhelm von Waldenfels Institut fOr Angewandte Mathematik Universitat Heidelberg 1m Neuenheimer Feld 294 0-6900 Heidelberg 1

J

In a previous paper [ 2 for any integer s > 0 a non-commutative central limit theorem was established. For s 1 the theorem was the analogue to the weak law of large numbers. for s = 2 it was the analogue to the usual central limit theorem. For s) 2 there does not exist a classical analogue because the vanishing of all the first and second order moments implies that for probability measures the theorem is trivial. The proof in [2] was based on combinatorial considerations. We establish a new proof by the use of moment generating functions. The proof uses simi l ar- ideas as [3]. Consider a set of non-commuting indeterminates and the free com-

f:: ([; (X.),"

plex algebra

X'I'!)

J

generated by them.

of the complex linear combinations of form e.g ••

x. 1

x.1 •

xix j ! xjx i if i f j. Assume a s-dimensional matrix

functional ;}'Q on

661

(A)

aQ

into

...

(Xi .. ,

"

{SA} '"

where p

0 =

cco.1

is a linear function

-= 0

R

of . . 1

.

••• 1

a:

)1. The Gaussian 5

given by

k

is not a multiple of

-=

'-to-

L

{ S'''') ") Sf}

Q.

... '1

runs through all partitions of

with exactly

Q.

'=

Q.I . . .

J"

s

I"

x.

s, ]

if

Q.

x·

sets

's

and of monomials of the

-=:-{

rQ &'Q

f

t x.... x· I

'V

consists

Remark that the order of factors is important.

k

1

tF

s

,.

J-)

elements and where

Q. t

s ,..

{1) ... }p4}

343

ct

Let

be an algebra over

be a family of elements of

en

terminates of

11:

and let

on

1.

Let

(o., L.: :r

indexed by the same set as the inde-

a:

or.. -7

at @N

Consider the tensor algebra

r... @N

([ with unit

be a li near funct i ona 1.

and the linear functional 11" @N

given by

Vl

(f1@

lr®N

Define the imbeddings ®

1-) 1-)

Central limit theorem of degree

IT(o,

"'J

FE e

t: (t. 1

Xi + ... + til X, )

determines all the moments of the stochastic variables provided the moments exist. As pointed out in

[2.1

X

A> • , >

X11

the theory presen

346

ed here is a theory of moments. One is tempted to try

J:

J : (tlj), ") t-n )

1--,

11

(e l

-+ .,

+ 6",

x.. )

)

as generating function for all the moments, i.e. the expressions

1('1'

X,,

I...,

k

This, however, is not convenient. Firstly, because no convergence assumptions have been made and secondly, because, even when every-

f

thing converges, we can hope to get out of are symmetric in the if

x.X. J

1

i ! j.

A generating function

:f

+ X.X.,

J

1

only expressions with but not

wi 11 be a formal power series in

f[[{]l.Le.

i '" f

l'

0

(we assume

11

f + II. t

1

-t . .

f

t

to commute with the elements of

where

are linear combinations of the

f(

that the functional

alone

r ) of the form

xi' It is clear

is determined by all the power series of the

form

11" ( e No

t

convergence problems arise because of the introduction of the

indeterminate

t.

Let

:f = then (4)

6 f

'=

J@ J

347

Hence by (3)

ioing back to

(IT N

e

(1)

N

we obtain 0

01 ( N - 1;'

)

)

(

f)

1f

®N

(lr (

)f

((0< 0(

(N- 1 / 0

is a homogeneous polynomial of degree

• So

)

) ®N )

f ))

N

We have

where

1ft

ex (N- 1!/;l ) f"= -1 + N- 4 " t. 11 + N- 2//J ttl?-t Taking into account the assumption of the theorem

IT \ d. (N - 1{,

J )}

'=

-1 + -t

Hence (6)

As

equations (5) and (6) prove

(1).

l i t era t u r e [1J

[2 J [3J

N. Bourbaki, Elements des Mathematiques, Algebre, Chap. III, Hermann, Paris 1970. N. Giri and W. v. Waldenfels, An algebraic version of the central limit theorem. Z. Wahrscheinlichkeitstheorie verw. Gebiete 42, 129-134 (1978). M. Schurmann, Positiv definite Funktionan auf der freian lieGruppe und stabile Grenzverteilungen. Diplomarbeit Heidelberg 1979.

AVERAGING AND FLUCTUATIONS OF CERTAIN STOCHASTIC EQUATIONS Hisao Watanabe Department of Applied Science Faculty of Engineering Kyushu University 36 Fukuoka 812, Japan

§

1.

Introduction Let us consider ordinary differential equations with parameter E

dx ill = EF{t,x E ),

(l.I)

E

>

E

o.

In the study of asymptotic behavior of trajectories of (1.1), the averaging method was introduced by Krylov and Bogolubov. Let

F{t,x)

be a real valued function of

and satisfied a Lipschitz condition in

(x,t)G[O,ro)XR d, uniformly bounded,

x , with constant independent of

t

and

x.

If the limit

fT

1 lim T T-+ro

0

F{x,t)dt

F{x)

xE R d, then the trajectory of (l.I) is in some neighborhood of the

exist uniformly trajectory of (l. 2)

0 EF(x (t)

on

L

O:St :SL/c, where

is an arbitrary constant.

Also if

xE{O) - xO{O)

O{E),

then it holds O:St:SL/c The assertion that the trajectory aging principle.

x E(t)

is close to

x O(t)

is called the aver-

In this paper, we discuss the averaging principle in some

stochastic equations. §

2.

Stochastic ordinary differential equations If the equation (1.1) is disturbed by noises, we describe

F{t,x)

as a random

function.

By making the change of variable, we can write (l.l) as follows:

(2.1)

dx (t) dt

where

E

W

E

[l

and

= F{!E' x E (t) ,W ) ( [l , 13 , P)

is a probability space.

aging principle for (2.1) is obtained by Khas'minskii.

The first results about averHe assumed that F satisfies

349 uniform Lipschitz conditions; namely, for any F(t,x

IF{t,xl,w) K

where

2,w)

is independent of

I

:£

t , x, to •

I SUPt >0 E IT

°

for any

x ER

formly for

d,

Jto+ to

O:£t:£L

as

E

+

s, with probability one.

F(t,x, w)dt - F(x) I

then he showed that

F(s,x,oo)

Also, he assumed that

grable in finite interval with respect to

T

t > 0,

for any

K! xl - x21 ,

xE(t)

0, where

+

0,

(T +

00

If )

converges in the mean to

xO(t)

is inte-

xO(t), uni-

is the solution of the equation

(2.2) Furthermore, under the more restrictive conditions he showed that (xE (t )

xO(t»flE

converges weakly to a Gaussian process yO(t). Roughly speaking, it implies that O xE(t) can be approximated by x (t) + IE yO(t). After Khas'minskii [6], there appeared several papers, namely, Brodskiy and Lakacher [2], Geman [4], and Kushner [8]. Consider the following example,

where

f;A (t)

covariance (t)

-Cae

°

is a stationary Gaussian processes with means E (f;A(t) ) and A! = etl, a >0 and b>O. Now if we put a(t,w) =

E(f;A(t)f;A (0» + be

1 (t)

use of the relation dxA(t)

0

2

1

are bounded and continuous with

6)

xER .

(B.VI) {ai/t.x,w)}, {bi(t.x,w)} processes, for each fixed

:s

such that Ie 2:

(i,j=I, ... ,d)

are stationary correlated stochastic

Rd.

E

= o{aij(t,x,w),bi(t,x,w)

(i,j=I, ... ,d[s:Su:S t , xER d}

and S(t)

sUPS20 sup

s

00

AE lll'O,BE

ffC s + t

[P(M"IB) - P(A)P(B)

I·

Then

J: (B. VI)

SI/\t)dt
-a=- '

Xi

1 E

,d

+ L i=l

-

d

,

1

and E

F (t,x,w) = (A Therefore, yE:(t,x)

-

0

- Ax)u (t,x).

can be expressed in the following way (cf. Eidel'man [3])

yE(t,X) = Now we split

E W

t: x

yE(t.x)

1 Jt ds

o

J

p E: ,w (t,x;s,y)F E: (s,y,w)dy.

Rd

into two parts

where

and 1 f

J:r: d

O E : (s,y,w)dy. (p E, w (t,x;s,y) - p (t,x;s,y»F

{the space of rapidly decreasing functions}.

We define

353

for Lemma 4.1.

For

2

E( I D C( z t:(s , x)

sup XER

I C( 1 ;:

d

2

1

) < 00 •

t:>0 O;:s;:t

By definition, we can easily show that

satisfies the following equation

where

and

J

F(t) cp

Rd

F(t,x,w)