Stochastic Analysis and Applications: Proceedings of the International Conference held in Swansea, April 11-15, 1983 (Lecture Notes in Mathematics, 1095) 9783540138914, 3540138919

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

1095

Stochastic Analysis and Applications Proceedings of the International Conference held in Swansea, April 11-15, 1983

Edited by A. Truman and D. Williams

Springer-Verlag Berlin Heidelberg New York Tokyo 1984

Editors Aubrey Truman David Williams Department of Mathematics and Computer Science University College of Swansea Singleton Park, Swansea SA2 8PR Wales

AMS Subject Classification (1980): 60H05, 60H10 ISBN 3-540-13891-9 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13891-9 Springer-Verlag New York Heidelberg Berlin Tokyo

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210

PREFACE

This volume contains a number of papers presented at the Workshop on Stochastic Analysis and its Applications, held in Swansea from 11 April to 15 April 1983, together with some more recent research papers by the Swansea school.

The applications include such diverse

topics as stochastic mechanics and the Titius-Bode law, non-standard Dirichlet forms and polymers, statistical mechanics, quantum stochastic processes, the applications of local-time to proving path-wise uniqueness of solutions of stochastic differential equations and its application to excursion theory, Bessel processes and pole-seeking Brownian motion, queues, potential theory and Wiener-Hopf theory. Some new results for Brownian motion on how one process determines another are also given.

The applications to Mathematical Physics

appear first, followed by the papers on local-time, Bessel processes and queues.

The papers of the Swansea school are collected together

at the end of the volume.

We are grateful to SERC for financial support

through research grant GR/C52162 and we are especially indebted to James Taylor for invaluable help and advice during the conference. Finally we should like to record our thanks to Mrs E. Williams, Mrs M. Prowse and Mrs M. Brook for making such an excellent job of typing the Swansea contributions.

A. Truman D. Williams Swansea April, 1983

TABLE OF CONTENTS S. ALBEVERIO, PH. BLANCHARD, R. H0EGH-KROHN, 'Newtonian diffusions and planets, with a remark on non-standard Dirichlet forms and polymers'. J.T. LEWIS, J.V. PULE, mechanics' •

'The equivalence of ensembles in statistical

· 25

F. PAPANGELOU, 'The uniqueness of regular DLR measures for certain one-dimensional spin systems. .

36

R.L. HUDSON, K.R. PARTHASARATHY,

45

'Generalised Weyl operators'.

J.F. LE GALL, 'One-dimensional stochastic differential equations involving the local-times of unknown processes'.

• 51

P. McGILL, 'Time changes of Brownian motion and the conditional excursion theorem' • .

· 83

M. YOR, 'On square-root boundaries for Bessel processes and poleseeking Brownian motion' • .

.100

P.K. POLLETT, 'Distributional approximations for networks of quasireversible queues' • .

.108

J. HAWKES,

.130

'Some geometric aspects of potential theory'.

G.C. PRICE, L.C.G. ROGERS, D. WILLIAMS,

ISAdS '.

G.C. PRICE,

'BM(JR3)

and its area integral .155

'The unique factorisation of Brownian products'.

N. BAKER, 'Some integral equalities in Wiener-Hopf theory'. L.C.G. ROGERS, D. WILLIAMS, theory' •

'A differential equation in Wiener-Hopf

.166 .169

. .187

NEWTONIAN DIFFUSIONS AND PLANETS, WITH A REMARK ON NON-STANDARD DIRICHLET

AND POLYMERS

by

S. Albeverio Mathematisches Institut Ruhr-Universitat Bochum

Ph. Blanchard Theoretische Physik Universittit Bielefeld D-4800 Bielefeld

R. H¢egh-Krohn Universite de Provence Centre de Physique Theorique, CNRS F-13288 Marseille and Matematisk Institutt Universitetet i Oslo Blindern, Oslo

Abstract We discuss diffusion processes on Riemannian manifolds, for which a Newton law holos

(in the stochastic sense). We

the existence

of a general mechanism for the formation of impenetrable barriers for these processes, corresponding to the nodes of the density of their cistribution. We discuss some applications to natural phenomena like the formation of planetary systems; the morphology of galaxies, the formation of zones of winds in the atmosphere and

the formation

of spokes in the rings of Saturn. \1e also relate the recent hyperfinite theory of Dirichlet forms with the theory of local times of Brownian ')

motion, polymer measures and thp. (¢l theory.

')

cf quantum field

2

1. Introduction

In this lecture we shall discuss two topics, which are connected by the theory of diffusion processes. In the first part, consisting of Sections 2, 3 and 4, we shall discuss a class of diffusion processes, which we call "Newtonian diffusion processes", which show the remarkable phenomenon of "barrier formation", leading to a possible explanation of a large class of natural phenomena. In the second part we shall briefly discuss the concept of local time of Brownian motion and a new hyperfinite version of the theory of Dirichlet forms and apply this to the study of polymer measures associated with certain quantum field theoretical models. In Section 2 we give the definition and basic properties of Newtonian diffusions on manifolds. This theory has been introduced by E. Nelson in connection with stochastic mechanics [ 3 ], [ 4 l . [29] and developed further particularly by Dankel [ 1], Dohrn and Guerra [ 2] and for the case of manifolds recently by Meyer [ 5] and Morato [19]. We review the basic formalism and discuss the stationary case in particular (this case has been discussed previously by ourselves in nection with Dirichlet forms, in [ 7],

[13], [28] and, in con-

[10] and by Nagasawa [16]).

In Section 3 we discuss the general mechanism in the symmetric case for the barrier formation in Newtonian processes, using previous results obtained in ref.

[7] and [10], in the context of the "Dirichlet ap-

proach" to quantum mechanics

(see e.g. [42],

[31], [34],

[50], [52], [57]).

In Section 4 we discuss the mentioned applications to natural phenomena like the formation of planetary systems, the morphology of galaxies, zones of winds circulation and the formation of spokes in the rings of Saturn. In Section 5 we briefly discuss some problems in connection with the so­called polymer measures. This involves the study of "times spent at intersections of Brownian motions", quantities that also have arisen in the very stimulating lecture of Prof. E.B. Dynkin. We mention a couple of central problems in this area and can partially

indicate how in dimension 4 we

solve these by using a non standard

theory of Dirichlet forms.

2. Newtonian Diffusion Processes In this section we shall briefly describe how an important class of Markov diffusion processes, called "Newtonian processes", shows an

3

interesting phenomenon of formation of barriers on the nodes of the solution of a linear equation of elliptic or Schrodinger type, and how this remarkable property can be used to describe situations in nature, in which regular patterns of "confinement" arise. Let

M be a smooth oriented Riemannian manifold of dimension

Let

X t ETc JR+ ' be a diffusion process with values in M t, analytic description of X is by its infinitesimal generator t which we assume of the form

..:LA 2'-' + where

SiD.

S.D

0 fJ· i

l

d The L

D

'

(2.1)

1r

=

t

•••

is a (non random) COO vector

d,

field (the "drift"), which might depend explicitly on the is the covariant derivative and

D

M

tor on S(Xt,t)

The connection between

6

time t . is the Laplace-Beltrami oper a-:

S

and the process

is the (mean) forward derivative of

X t

sense that i

S (x,t) where

E[. IX

t

6t from

X t

to

=

x]

lim

x]

(L,t)

means conditional expectation with respect to

is the vector attached to Xt +lI t

X t

6tl

' wi th length

X + lIt and X S(Xt,t) t t. derivative in the sense of [5 ] . dx

(2.2)

6t+o

distance of

Let

X is that t at time t in the

tangent to geodesics

equals to the geodesics

is also the forward stochastic

be the Riemannian volume element on

M. Due to the assumpp (x,t) of the law

tions we know that there exists a smooth density of

Xt

Let

f

with respect to

f E

, then

dx, i.e. E[foX

t]

f

=

t

E dx)

f(x)p(x,t)dx M

ap

M

dP(X

p (x,t)dx. and

dt

E[f

0

] =

f(x)3t(x,t)dx . On the other hand, by the definition of

left hand side is equal to

f

(L M

t

f) (x,t) p(x,t)dx . By partial integra-

tions we arrive at the Kolmogorov forward equation (Fokker-Planck equation) 1

26p - div(Sp)

(2.3)

4

- div(B· ) being the adjoint of ator on vector fields on

L

M

and

t

div

means divergence oper-

Let us now denote by i.e. that

X , t E -T, the time reversed process to X f t t It is well known, see e.g. [27], has the same law as is again a Markov process with infinitesimal generator

X t

- i3 . with

S. D

(2.4)

D

being the "backward drift" defined, for t E T,

Si

by lim (H)

x]

lIt+o i

(2.5)

i

with Y -lit defined as Y lit with -lit replacing lit. Then B is the backward stochastic derivative of X . By the same procedure as t above, one arrives at the Fokker-Planck equation for the reversed process,

t E T : d

ItP Set now

u

ity and

v

1

2 (B-S) A

and

P

V"

+ di v (i3

1

2(

).

u

(2.6)

p)

is called the osmotic veloc--

is called the current velocity. Inserting these expressions

in the Fokker-Planck equations (2.3),

(2.6) we get the "continuity

equation" -div(pv)

P

(2.7)

and the "osmotic equation" 1

2'''; P

=

As remarked first by Nelson in the case u

(2.8)

div(Pu) M

d :IR

we have also

21 v log P .

This follows, see [29], by computing for

(2.9)

f,g E C: (T

x

M) :

5

where and

a 11 + B.V (the operator of mean derivative on functions) 0+ - at + .1. 2 a D - at - .1.11 + B. V . Using partial integrations to bring 0 + to 2

act on

gp

and using Fokker-Planck's equation (2.3), we arrive easily

a

1

-0_ ""-at - B.V+V(logP)V + 2 11 · we then get (2.9). FroD. this equation (2.9) we have,

f r om this to the conclusion that

B "" D_Xt taking the time derivative and using the continuity equation (2.7)

Using

- grad div v - grad (v.u)

(2.10)

.

We shall now define the mean acceleration associated with the process

X . To do this we would like to have the concept of mean forward and t backward derivatives of vectors on M . The appropriate definition has i been given by Dohrn and Guerra ( 2 ], (5]. Let F "" F (x,t) be a vec-

M, then the mean forward derivative of

tor field on

P

is defined by (2.11)

0+ F (x , t) '" lim ( 11 t) - 1E ( T X lItto t' where T

+' P is the vector at T , M obtained from the vector y,y "y y+"y PET r1 by Oohrn-Guerra's stochastic parallel transport along the geoY

desics from

y

to

y + lIy . We recall briefly the definition of this

transport, for more details see (2 ], ( 4] and [ 5 l : Let So

s

s1

' be a segment of geodesics on

Ty(So)M . Let

hIt)

M. Let

, t E (0,1], be a curve on

M

Let us transport in a Levi-Civita parallel way getting a vector field Yt(so) {Yt(s)

h(t) , So

B(s) '" : t

G(t)

Yr(so) s

s1

r

Yt(s)

• Let

Yt(S)

G(t), s < s1

s . This then gives, for

Ty,y+lIy F

y

"" P. h tt )

o

. The family of geodesics

=

B(s)

differs from the Levi-Civita displacement of in

such that

y(s )"'G(t) along

s"" So

this is a vector field along ) ,y(s)F

Y(s) , be a vector in

be the geodesic such that

t E (o,1J) is parallel for

B(So) = F . By definition

pi

and F

Let

Y(s) T

with

y(so)'Y(s)

F

by second order terms

y(so)' y + 11y = y(s)

the transport

needed in (2.11). One computes easily ( 2], (19] 0+

+ S.V + 21 £l DR (2.12)

D £lOR '" £l-R

+

13.\7

1

2 £l OR

being the Laplace-de Rham-Kodaira Laplacian on

the Ricci tensor, acting on vectors.

M, R

being

6

Let us now define the

a(xt,t)

a(xt,t)

by

- 1(0 D + 2 +,t)

we get

(2.13)

hence dV

a + U'vu - V'Vv +

at

21

6

DR

(2.14)

u .

Let us also remark that a purely probabilistic description of the process is given by the solution of the stochastic differential equation (in

Ito's sense)

with

W the standard Brownian motion on M . Of course this is not t an intrinsic description, for such see e.g. [5], [53]. Given S we X and get P, hence i3 t satisfying (2.10), (2.14). Moreover, given X t P we can get Sand i3 as mean forward resp.

can, under suitable assumptions, construct and hence

u

and

v

and its distribution

backward derivatives and then get with

a

u, v

defined as mean acceleration of

satisfying (2.10),

(2.14),

X t.

For a class of diffusions diffusions"

X which we shall call here "Newtonian t' (they coincide, under regularity assumptions, with Nelson's

conservative diffusions [4 ]), we shall show that one can recover X t from (2.10), (2.14) and the initial conditions u(x,o), v(x,o). Definition:

A Markov process

X is called a t satisfies "Newton's law in the mean", in the sense

if X t that there exists a positive constant m and a real--valued function V

on

Mx T

such that

7

and such that in addition the corresponding current velocity is a gradient field. Ne then say that toni an diffusion.

a

!ve shall see below that there exist

is the acceleration of the New-:

conservative Newtonian diffusions.

We shall first discuss the forms of their distributions. Let be any function on

Mx T

such that

is only defined modulo functions of

v(x,t) '" grad S (x,t) t

alone). Let

S(x,t)

(clearly

P (x,t)

S

be the

density of the distribution of

X , as above. Under our assumptions t P is smooth and strictly positive; hence log P is well defined. R+'S 1 Setting we have easily e 1 , so that = P, R = 2 log 0 from m a -'ilV and (2.14):

div u (because by taking grad div and

v

from

'

on this and using

grad R

grad lul

2

=

2u.'ilu,

div grad

grad Ivl

2

=

(2.15 ) 6

both u R being gradients, we see that this is just (2.14». Moreover, 60

=

u

grad

o

+

=

2v''ilv,

we have

e

2R

- div (e 2 R 'ilS)

hence

o

R + } div grad S + grad R.grad S From (2.15),

(2.16 )

(2.16) we see that

2.± _ _ .l2 3t -

A ,I,

V

,"

(2.17)

Thus we see that for conservative Newtonian diffusions the probability distribution and all

p(x,t)

t ET

by

of the process p

=

II/!

i 2

,

where

X t

is given, for all

is a solution of the Schro-

dinger equation (2.17), with initial condition 11jJ(x,o)

1

2

and

v

such that

u

1jJ

is a solution of (2.17) and if we write II' = e and v by u = grad R , resp. v grad S ; then

satisfy (2.10) resp.

particular

1jJ(x,o)

gives the initial distribution of the process.

Vice versa if and define

x E M

B

(2.14). From

u

and

v

R+iS u

we can get in

and thus the stochastic equation for a proces

Xt , II/!(x,t)

2 the distribution of which is then for all times 0 (x,t) = if at time t 0 it has distribution 11jJ(x ,o) [2 Moreover, the 1

process statisfies Newton's equation in the mean.

,

8

Remark:

It has recently been shown by Nelson that the stochastic New-

ton law in the definition of conservative Newtonian diffusion can be replaced by a variational principle [ 4]

(see also [30],

[58]).0

We are particularly interested in the case where there exists a station­' ary distribution

P(x,t)

=

P(x)

for the process, i.e.

this case we have from the above, that

11ji(x,t)

1

2

,

=0

and hence

In R,

is

independent of t . Assume 1ji satisfies (2.17) with initial condition "'(x,o) (x)+iS(x,o) • Th en we h ave, u s i nq 33R '¥ t ­­ 0 e that 31ji lll1jJ + V 1jJ Le . ­}ll1ji + V 1ji is equivalent with . 1ji t i 3t 2

e

R+iS

[­

Hence for all

x

+ (vS)2 + 2ivR.vS + V] e

­(VR)2 + such that

eR(x)

R+i S

(2.18)

*0 (2.19 )

llS + 2 vR.vS

o .

(2.20)

This system of coupled partial differential equations with initial con' dition

=

S{x,o)

So(x)

has a solution of the form

S(x,t) = iff

1 ­ '2ll1jJ + V 1jJ

­ E1jJ

has a solution of the form

R( x j ­iEt iSo(x) e e e

1ji (x,t)

(2.21)

Et + So(x)

.

Splitting in real and imaginary parts

we get ­

II R ­

(vR) 2 + (VS ) 2 + V o

(2.22)

E

o . We set

P

= e 2R

and remark that (2.23) can be written in the form llSO +

i.e.

(2.23)

12 22.· vS P 0

o

(2.24)

*

V (PVS

= 0 for all x such that P 0 0'1e set v VS o o) is just the continuity equation div (Pv). Hence if, l O R ­iEt lSo 1ji is such that­'2ll1jJ + V1jJ = E1jJ has a solution of the form 1jJ = e e e

then (2.24)

then (2.22),

(2.23) are satisfied. Setting

u =

P

in this case

we then have that the system of coupled equations is satisfied. From

9

u

and

v

we can compute

This process has stationary case

for all times

S

and hence the process

p as invariant distribution. Note that in the S

d08s not depend explicitly on

Remark: The case where

VS

= 0

ving for the process

(J o at =

0,

hence

P

t

is the only case where

situation (2.23) is c Le ar Ly satisfied. Setting

X t.

u =

v

= 0

log

. In this

P and sol--

K we find from the continuity equation that t, is stationary. v 0 is equivalent with the sym-·

metry in

(M, pdx) of the Markov semigroup P giving the distribut tion of X [12]. In fact, this is equivalent with time reversal int variance [11].

3. Barriers for Newtonian Diffusion Processes In the case of a symmetric diffusion process a general theory of formation of barriers for the process (non trivial decomposition into time ergodic components) has been given in [13], [16], [7], [10], [28], [51]. Let us take the opportunity to recall here the main facts of this theory. Let

E

L 2 . Let

be a regular Dirichlet form on

M be a locally com-

pact space with a countable base for the topology and let Radon measure on

m

be a

M, strictly positive on every non void open subset.

It is well known that there exists a one-to-one correspondence between submarkovian semigroups on L 2(M; m) and Dirichlet forms (i.e. positive symmetric closed bilinear forms E on L 2(M; m) with the contraction property with

f

#

_

(f v

0)

A

1

Vf for which E(f,f) < + 00). The correspondence is such that if -L is the infinitesimal generator of P t then E (f, f) (L1/2 f, L1/2 f) , where on the right hand side we have the scalar product in L 2(M; m). Moreover, diffusion semigroups are in one-to-one correspondence with local Dirichlet forms, local meaning

E(f,g)

=

0

whenever

f,g

have

disjoint supports. On the other hand, to any regular Dirichlet form (regular meaning that the continuous functions with compact support in the domain of the form i.e.

D(E) n Co(M)

are dense both in

and in (Co(M) ,I: 011=), where El (f,f) '" E(f,f) + (f,f) and l) 11.11 00 denotes the supremum norm) by a construction of Fukushima and Silverstein there is a Hunt process with m-symmetric transition

(D(E) ,E

function precisely Pt ' Pt being the symnetric sernigroup associated with E . X can be taken to be a diffusion (in the sense of having continuous t

10

(X continuous in t E [o,!; J) 1) iff E is local in t addition to being regular. One can show [54] that any regular local

paths,

pX

Dirichlet form can be written as closed extension of the form E(f,f) for some Hilbert space (Jf"

ae,

f being smooth cylinder functions on

v , V o being positive Radon measures with values in the cone of positive self­adjoint operators resp. in JR+ • If

M is an oriented Riemannian manifold and

on

M then the closure of

2

2

L (M;m) xL (M;m),

E(f,f)

=

f

m

is a Radon measure

df(x).df(x) m(dx)

in

M

1 o

first defined on C (M)­functions, is a local Dirich­

let form. Sufficient conditions for the existence of the closure can be extracted from work by Fukushima

8 J, Albeverio, H¢egh­Krohn and

Streit [31] and Rockner and Wielens

9

(the latter reference gives

also a survey of this type of results). In particular, if a density

P

with respect to the volume measure

dx

m(dx)

and if

p

has is

strictly positive on compacts and locally Lipschitz, then there exists only one closed extension of E [33J. Other results yielding closability (but in general not uniqueness) involve e.g. conditions of the type pl/2

p > 0 , dx a.e and that the distributional derivatives of with respect to local coordinates are locally in L 2(dX) , on

any open subset of [9J, [46J.

M whose complement has m­measure zero ([31], see also E on L 2(M;m) , (M being

For any regular Dirichlet form

again a locally compact second countable Hausdorff space and

m

a

Radon measure strictly positive on non void open sets) one defines the capacity of U by Cap U

inf [ E (f r f) + (f, f) J

the infimum being taken over the set L u of functions f in the domain of E, which satisfy f 1 m­a­e on U (the infimum is taken to be

+=

if

L

u

¢). By standard methods one can then extend the

definition to any subset quet capacity. L unique element

A c M as an outer capacity yielding a Cho-

u being closed and convex for open U there exists a in L u which minimizes E(f,f) + (f,f) , this el-

eU

ement being the equilibrium potential of Cap (U)

U. One has

0 s e

U

s 1

and

11

of

is actually a version of the hitting probability ,


.{"'" is identified with a ?/A/jt-o//.d= pha-se-Laan.si.t.con.; from (2.9) it is clear that, in terms of the free energy densi ty f >fI, there is a non-empty phase-transi 'ti on segment [Xl, X2] . From our present point of view the interest lIes in his method of proof. He used the f'o l.Low i.ng result of Berezin and Sinai [2]: Berezin-Sinai Lemma: In onden. .tha.: a non empty pha-1e-uan.ji.t.Lon -1egment 1lJi.;th ch.emcccu. potenti.a1 '" 0= exi..j;/: to//. -some (/> , it .L.j -1utf-i-uent ;that tu//. -some. I> > 0 and 1';> 0 and all -1utf-i-uentJ..i/t. , (3.3)

In other words, a first-order phase-transItion can be detected as a violation of the law of large numbers in the grand canonical ensemble. (Griffiths [5] showed that for sufficiently large and .(. the mean-value of X is less than S from which

t

1-

28 (3.3) certainly follows.) Dobrushin gave a proof of the Berezin-Sinai Lemma which is simpler than the one given in [2]. He deduced it from the following Dobrushin Lemma:

we tiov e

Fo». S;'O and

Wlc:lX

-

mllx

\

/

(3.4)

»he».« The function .x: I-t is convex; in the particular c as e of the La t t i.ce-cgas model it satisfies the symmetry Gondition (3.5)

i t follows that the right-hand side of (3.4) is equal

implies that 14.

to is a phase-transition segment.

- gli) .

Then (3.3)

The Bose-Einstein Phase-Transition

The traditional description of Bose-Einstein condensation is this: in a system of non-interacting hosons in thermal equilibrium the excited states saturate at a critical value Pc. of the density; when the density P is increased beyond this value the excess p - Pc. goes into the zero-energy state. The phenomenon is sometimes described as 'condensation in momentum spaGe'. The condensate has zero entropy as well as zero energy, and so makes no contribution to the pressure. Consequently, the pressuredensity isotherm has a flat part: the pressure increases with increasinp, density for densities below p... and thereafter remai n s constant. There is a basi c d i.f'f icu I ty which we have to face if we attempt a rigorous proof of these statements: a phasetransition manifests itself sharply in the mathematical behaviour of thermodynamic functions onl yin the bulk 1 i mi t, but in thi s 1 i mi t there is no uni que preci se formulation of the zero-energy state. For non-interacting particles in a box of finite volur.1e, the single-particle energy-levels are well-defined and there is a unique ground state; as the volume increases, every energy-level tends to zero; for the infinite system, the si.ngle-parti c Ie energy-spectrum is a continuum fi 11 ing the ha If-line but there are no eigenstates. There are two good candidates for the concept of macroscopic occupa t i.on of the zero-energy state: mac.aoocopi.c. occupation of- the ;ywund -di;a;te is said to occur when the number of particles in the ground state becomes proportional to the volume; conden-dai;,Lon is said to occur when the number of particles whose energy levels lie in an arbitrarily small band above zero becomes proportional to the volume. ObViously, the first implies the second. However, the second can occur without the first; this is called conden-dat,Lon. These matters are discussed in [6] where it is proved that there are, in general, two critical densities: there is (>, whi.ch is the density at which singularities in the t.her-modynam ic functions occur, there is Pm which is the rai n i Dum dens i ty for macroscopi c occupation of the ground state. Generali.zed condensation oC'curs whenever p is greater than Pc. macroscopic occupation of the ground state occurs if and only if the weak law of large numbers for the particle number density is violated. As far as we know, the first rigorous proof of the macroscopiC' occupation of the ground state of the Laplacian when the bulk-limit is taken by dilating an arbi.trary star-shaped regi.on was sketched by Kac in 1971; his manuscript remained unpublished until 1977 when it was incorporated in the review by Ziff, Uhlenbeck and Kac [7]. The mathemaUcal details were ied in the thesis of PULE [8] and in the papers of Cannon [9] and LEWIS and [ 10 ] ; the connect i on wi th the work of Araki and Vioods [lJ_] was di.scussed by LEWIS [12]. Kac obtained the limiting distribuUon Kl7C;P> (now known

29

as the Kac d.is t.r-Lbu't lon ) of the particle number- density density p by conputing its Laplace transform:

Xt

= N/IALI at :fixed mean

He found that, when P exceeds P.. , the distribution is exponential; details may be found in [6] where it is shown that, in genera], the distri buti on is infinitely divisible. In the mean-field model of a s y s t.era of interacti.ng bosons, the interaction energy is represented by a tern 11. N7./'2.ll\t l wh i.ch is added to the hamiltonian of the free boson gas, where Q. is a strictly positive constant representing the strength of the interaction. This crude model of a system of interacting bosons is conmonly called the .i..m.pVl.f-ee..t bO-1on [}a-1; it is of interest because the patholOGical aspects of the free boson gas are removed by the mean-field interaction: the grand canonical parti ti.on function converges for aII real values of the chemical potential [4]; the weak law of large numbers holds for the particle nUr7lber density for all values of the chemi.cal potenti.al [13] (see also [14] and [15]). However, it is proved in [16] that generalized condensation persists in the imperfect boson gas: generali.zed condensatIon i.s stable with respect to a r7lean-field perturbation of the free-particle hamiltonIan. §5.

An Extensi.on of Laplace's tlethod__for Integra Is

In t.h i s secti.on, we present a version of Dobrushi n ' s Lemma whi ch holds under condi ti.ons wh i ch are satIsfied by a wide c lass of conti nuous systems in stati s ti.c a l. mechanics, both classical and quantum. do this by means of a version of Laplace's method for i.ntegrals whi ch, un Li.k e the standard treatments (see Copson [17], for example), makes no hypo t.he s i s of d if'f'er-en t i ab i Hty concerni ng the integrand. Lemr7la 1

(Laplace's method)

Lei be a -1equence of- .Lowe//. »emi.s con.cinuou» Loncci.on.s, f", suppose :thai:. on each compae..t :the -1equence {of..i i-1 bounded below and conVVI.[}e-1 unif-oJUrt1!f i:.o f, and :thai:. f( 0 )=0. Let {to" lI\ '" be a of- po(t for For each

e:-'t(Xt£krex)

dktex)

t t. tp .. ,P .. i-Et) sufficiently large, so that

t

Pt.J

- A'l- J

the inequality (6.9) follows in an analagous fashion.

Putting together Lemma 5 and Lemma 6, we have Lemma 7:

U"" n( ) Suppose that " I-i Pc,1) ;: ("'0;1'( f exists, is strictly convex and differentiable except possibly at one point fJ; then there exist strictly posi.tive constants A I A1.. (given by (6.5) and (6.7)) such that

1

35

References 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

\L Feller, An Introduction to Pr-obab i.I ity Theory and its ApplIcations

(two vo1umes) (Wi1ey and Sons, New York, 1966). LA. Berezi.n and Ya. G. Sinai., Trudy l1osk. Mat. Obshch. 17, 197-212 (1967). R.L. Dobrushi.n, Proc. Fi.fth Berkeley Symposi.um, III, 73-87 (1967). D. Ruel.le, Stati.stical Mechani.cs: Rigorous Results (New York, Amsterdam, Benjamin, 1969). R.B. Griffiths, Phys. Rev. 136, 437-439 (1964). r1. van den Berg, J. T. Lewi s and J. V. Pul.e, J. Math. Anal. and Appl. (in press). R.!·\' Zi.ff, G.E. Uhlenbeck and M. Kac, Physics Reports 32C, 169-248 (1977). J.V. D. Phil. Thesis, (Oxford, 1972). J.T. Cannon, Commun. Math. Phys. 29, 89-104 (1973). J.T. Lewis and J.V. Commun. flath. Phys. 36, 1-18 (1974). H. Araki. and E.J. J. Math. Phys. 4, 637 (1963). J •T. Lewis, The Free Boson Gas, tn llathemati.cs of Contemporary Phy s i cs, Ee. R.F. Streater, (Academic Press, London, 1972). E.B. Davies, Commun. Math. Phys. 28, 69 (1972). M. Fannes and A. Verbeure, Phys. Lett. 76A, 31 (1980). E. Buffet and J. V. J. Ilaths. Phys. 24, 1608 (1983). 11. van den Berg, J. T. Lewis and P. de Smedt, J. Stat. Phys. (to appear i.n Dec., 1984). E.T. Copson, Asymptotic Expansions, (C.U.P., Cambridge, 1965). D. Ruelle, Lectures in Theoretical Phy s i.c s , Ed. W.E. Br-i t t.i.n and VI.R. Chappell, VI, 73, (Univ. of Colorado Press, Boulder, 1964). R.B. Griffiths, J. r,lath. Phys. 5, 1215 (1964).

THE UNIQUENESS OF REGULAR DLR MEASURES FOR CERTAIN ONE-DIMENSIONAL SPIN SYSTEMS F. Papangelou Random fields are not always uniquely determined by their specifications, i.e. their systems of conditional distributions.

A general result is presented here,

giving sufficient conditions under which a one-dimensional specification admits at most one random field (up to equivalence in distribution), within a specified class of such fields.

In a specific application this result implies that certain one-

dimensional spin systems with long range interaction admit unique regular DLR measures, regardless of "temperature". The present article has been written in the informal style of the talk given at the conference and begins with a brief introduction to some known results from the literature, selected for their immediate relevance to our subject. line of the proofs of new results is given in the last section;

A sketchy out-

for full details

the reader is referred to [9J and [lOJ. §l.

The background. Although the systems to be considered here are ordinary stochastic processes

parametrized by the integers, it is useful to think of them as one-dimensional random fields. ables

Xi

elements

By a d-dimensional random field we mean a collection of random varidefined on the same probability space

i

of the d-dimensional lattice

as "sites", then

Xi

and parametrized by the If one thinks of the elements of

is some variable (say a spin) associated with site

i.

The speeifieation of a random field is the system of conditional distributions QA(B\r;.,jiA) J

(B

E

where

P((X.)., 1

1EJl

E

slx,

J

= r;.,jiA) J

A ranges over the finite subsets of

Whenever we refer

to a specification, we will assume that the versions of all conditional distributions in it are regular probability kernels satisfying in a strict sense the obvious consistency conditions implied by the definition. In statistical mechanics and other areas, where random fields have been widely employed, it often happens that the nature of the inter-relationship between the Xi's

is best described in terms of the specification.

However, a specification

does not always uniquely determine an (unconditional) distribution for a random field and, given a particular specification, two natural questions immediately arise: (i) is there a random field admitting the specification?

(ii) if there is such a

random field, is it unique up to equivalence in distribution?

If we identify a

random field with its distribution, these questions reduce to the existence and

37 d

uniqueness of a probability measure admitting the specification.

IT,

on the product

a-field

of

,

If there are two or more probability measures admit-

ting the same specification, we say that phase tpansition occurs. This is in summary the approach adopted by Dobrushin and by Lanford and Ruelle in the late sixties, in their treatment of equilibrium states for infinite Gibbs systems. Before describing the particular class of statistical mechanical models we will be concerned with, we state three early theorems due to Dobrushin, which give sufficient conditions for uniqueness of the measure admitted.

The theorems are not

stated in full generality here but only in a form which is relevant to our subsequent discussion.

In particular, we only consider translation invariant specifica-

tions. Suppose that a translation invariant specification is given. Take d d where o is the null element of and for k E "{O} define

II-II

where

denotes total variation and the supremum is taken over all pairs J J"'O

1.1

{O},

A

that differ only at site

Theorem ([4]).

I

If

k#o

random field.

Pk


t,

and

if either

) j iA'

or I:; j

'1;j

1:;. =

J

'1;.

J

t+1,t+2, ... ,t+k

for

then

Then the specification admits a unique random field. If the true state space of a random field arising from the given specification is a compact subset of

and the conditional probabilities have continuous densiIn the unbounded case

ties, the conditions of Theorem 1.3 are not unreasonable.

however they are too severe, even for Markovian specifications. So


0,

=

a

G(n)dn

function, and analogously for

Q{O}(.

the distribution

2.1

Theorem

cr 2 J) .

where 0,


n

n

h, x, x, z, z

where«

denotes absolute continuity and

Qh(·lx,z)

and

Q\·lx,z),

r

h

(·1 x , z ;

i,z)

is the "overlap" of

i.. e. their greatest lower bound in the space of measures.

For the remaining two hypotheses IV and V we postulate the existence of certain special sets

c

v = 1,2, ..... ;

=

1,2, •••

below. IV. A

+-

T

For any and any

v

E >

1

0,

any (metrically) compact set

there exists

1

such that

with the properties described +

C c T,

any compact set

41

Q[s,tJ(lR i- s x M (j-i+l) x JR t- j !x,z) > 1u

whenever

s < i

j < t,

E Mv(k) ward sequence (The case k = 0 V. j.l z

1

z

is a forward sequence, say

for some

k

E:

>

0,

Z k

=

that for some

k

E C, E Mv(k)

X

E A).

any compact sets

there exists an integer

Cr,l'

k

x

->-

C c T

and

\1

3.1 E:

>

on

i.

IB(JR7L»

and any

v

=

C••• ,

1, If

are such

,1;-1)

M)k)

( .•.• (-k-2.(-k-l) E A

A = [s,tJ,

«.)

nMj.l(t-s+l) Ix,z)

1- -\1

Ql s ' t J (C.) nMj.l(t-s+l) x,z)

[

Q s t; J (M (t-s+l) Ix.i:)

J CM (t-s+l) 1x z ) u

j

j

there exist a compact set

for all

EA.

u

I


- 1

an integer

there is a set

II

is a random field admitted by the specification and let

s > 0

such that for every pair of integers -+

- 0

and

>- 2k + 2

with

and r

[s+l t.r-L] t-s-ll - ' (JR x,z; x,z) >-

e

The important point in this assertion is (x,z) EG and (x,z) EG • s,t s,t that e does not depend on t - s , One begins by restricting ( ••• ,X and (X ••• ) t, s) -n)

= 0

=

IT(D)

follows from this, once it is shown that two

tame probability measures admitted by the specification cannot be mutually singular. The translation invariance of

IT

is a consequence of its uniqueness.

Turning now to the spin systems considered in §2 above, suppose (3) holds and define

\P (i)

IJ(k) I,

i

=

1,2, ..• ,

so that

Hi)


0,

there is

n

L

i=l for arbitrary

sand

£.

i

1,2, ... ,£)

?;

l-E

This implies condition (i) of Theorem 2.2 and can also

be shown to imply condition (iii).

The implication (iii)

=>

(i) is proved similarly.

It is worth mentioning at this point De Masi's result ([3J) that translation invariant DLR measures for the spin systems considered here are tempered. assertion is contained in the implication (iii)

,=>

This

(i).

Finally the uniqueness part of Theorem 2.3 is an application of Theorem 3.2 one can show that the specification (1) satisfies hypotheses I-V. References 1.

Benfatto, G., Presutti, E., Pulvirenti, M.: DLR measures for one-dimensional harmonic systems, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 305-312 (1978).

2.

Cassandro, M., Olivieri, E., Pellegrinotti, A., Presutti, E.: Existence and uniqueness of DLR measures for unbounded spin systems, Z. Wahrscheinlichkeitstheorie und verw. 313-334 (1978).

3.

De Masi, A.: One-dimensional DLR invariant measures are regular, Phys. 43-50 (1979).

4.

Dobrushin, R.L.: The description of a random field by means of conditional probabilities and conditions of its regularity (in Russian), Teor. Verojatnost. i Primenen. 13, 201-229 (1968). (English transl.: Theor. Probability Appl. 12, 197-224 (1968».

5.

Dobrushin, R.L.: Prescribing a system of random variables by conditional distributions (in Russian), Teor. Verojatnost. i Premenen. 15, 469-497 (1970). (English transl.: Theor. Probability Appl. l2, 458-486 (1970».

6.

Kesten, H.: Existence and uniqueness of countable one-dimensional Markov 557-569 (1976). random fields, Ann. Probability

Comm. Math.

44 7.

Lebowitz, J.L., Presutti, E.: Statistical mechanics of systems of unbounded spins, Comm. Math. Phys. 195-218 (1976). Erratum: ibid. 78, 151 (1980).

8.

Papangelou, F.: Stationary one-dimensional Markov random fields with a continuous state space. In "Probability, Statistics and Analysis" (ed. J.F.C. Kingman and G.E.H. Reuter), London Math. Soc. Lecture Note Series 21, 199-218. Cambridge: Cambridge University Press 1983.

9.

Papangelou, F.: On the absence of phase transition in one-dimensional random fields. (I) Sufficient conditions. Submitted.

10.

Papangelou, F.: On the absence of phase transition In one-dimensional random fields. (II) Superstable spin systems. Submitted.

11.

Ruelle, D.: Superstable interactions in classical statistical mechanics, Comm. Math. Phys. 18, 127-159 (1970).

12.

Ruelle, D.: Probability estimates for continuous spin systems, Phys. 50, 189-194 (1976).

Comm. Math.

GENERALISED WEYL OPERATORS by and

R L Hudson Mathematics Department University of Nottingham University Park Nottingham NG7 2RD Abstract

K R Parthasarathy Indian Statistical Institute 7, S.J.S. Sansanwal Marg New Delhi 110016 India

Using the quantum Ito's formula of [5J we construct operators satisfying a

generalisation of the Weyl commutation relations, in which scalar-valued test functions are replaced by operator-valued ones. §l.

Introduction Let

H denote

the Boson Fock space f(L 2[0,=)) over L2[0,=) [2] and for each

f E L2 [0, =) let I/J ( f) be the corresponding exponent ial vector [1J, I/J(f)

in If.

z:

0, f, C2!)

_;

(3!)

The Weyl operators W(fl, f

_1

E

2fQfGH, ... )

L2[0,=) are the unitary operators in H "hose actions

on exponential vectors are 0.1 )

W(f)'l'(g) They satisfy the Weyl relation W(f)W(g) ::: exp i i Im}W(ftg)(f,g EL 2[0,=)).

0.2)

v

Introducing the mutually adjoint annihilation and cr'eat ion operacors a( f),

f)

by

means of their actions on exponential vectors a(f)l/J(g)

= l/J(g)

d a t (f)l/J(g) ::: dtl/J(gttf)

I

t:::O

and noticing that at (f) - a( f) is essentially skew-self-adjoint, we may write t _ W(f) :::-- eltP {a (f) - a ( f) } . Quantum Brownian motion [1,5,6J is the family of operators (At' At t

...

t ? 0)

= a t( X[O,tJ')

The duality transformation [8J is a Hilbert space isomorphism, which we may use to identify the two spaces from H onto the Hilbert space L2(w), where w is Wiener measure, under conjugation by which the self-adjoint operators t

0

become multiplications by the canonical realisation X t ? 0 of Brownian motion. t, In [5] (see also [3,4,7 ) a stochastic calculus is developed for quantum Brownian

46

motion generalising the classical It6 calculus in which integration against dX is replaced by integration against the noncommuting independent stochastic differentials t dA and dA , and in which the integrands are operator valued processes (F(t): t 0) which, in the bounded case which concerns us here, are adapted, in the sense that F(t) EB(H )exI for t

O.

H = f(L 2 [ 0 , t ] ) ,

t

t

Here we follow the notation of [5J, setting

H = f(L 2 [ t , 00 ) ) ,

t

and making the canonical identification H

r

operator, the stochastic integral

=

M( t )

H t

rjI

t

H.

When it exists as a bounded

(dA t F + GdA + Hds )

o

r

is determined by the formula (f\s)F(s) +G(s)g(s)

ds .

(1.3)

o

Now let f:[O,oo)

( be locally square integrable and, for t

0, write

W (f) = W( f ). t t

Combining the relation ( h l > = exp< g , h >

< ( g),

with (1.2), we have t

112-+}, t

t

whence

Comparing with (1.3) we see that the Weyl operators Wt(f), t

0 satisfy the stochastic

differential equation Wo(f)

=

Let t

B(H).

(1. 4)

I,

H(t) be a strongly continuous self-adjoint valued map from [0,00) into

The Dyson expansion [7, Theorem X.59] permits the construction of a family of

unitary operators (Wt(H), t

0) in

H satisfying the (strong sense) ordinary differen-

tial equation WO(H) = I,

t t

(H) = iH(t)W (H). t

If H(.) is adapted, so too is (Wt(H): t Given two such maps Hj and H2, the map t

t H2(t) = W t(Hi)H 2(t)W t(H j)

is also strongly continuous, and we have

(1. 5)

0).

Wt(H) is strongly continuous in t.

47

(1. 6)

Our purpose can now be stated; we shall combine and generalise the constructions of the families Wt(f), Wt(H), establishing the existence, for non-anticipating operator valued functions F and H, with H self-adjoint valued, of an operator valued process (Wt(F,H), t

0) satisfying the generalisation of (1.4) and -of (1.6)

WO(F ,H)

(1. 7)

I,

together with the generalisation of (1.2) and (1.6) -

-

1

1--

-1-

W :: Wt(Fj tF 2, Hj tH 2 - 2i(F jF 2 t(Fj,Hj)Wt(F 2,H2) §2.

F2F j

».

(l.8)

ConstX'uction of Wt(F,H) Let h be a Hilbert space and let FO,HO

L

B(h)

Ho with the operators L0511 and HO!5"1 in B(llo 0)

in B( h (X H) satisfying

)

(2.1 )

dU:: U(-dA FotFodA-(lHot;;LOLo)dt ,

I,

so that the adjoint pX'ocess satisfies

t

t ,;;LOL t )d) ,t dUt = (dA t F O FodAt(iHO O r u.

I,

U (0)

(2.2)

We say that the B(H)-valued adapted process F is simple if there exists an increasing sequence 0

::

to < t)
O.

x

in R-D

70

Hence

N(v) = V (

Log (f v ))

(4Log

=V

(f) )

1 ( "Z Log (fv ) n

n-xx> K.

The second assertion of the corollary is easy and left to the reader. o RemaY'k

a) Consider the two following kinds of stochastic equations Xt = Bt + Jt g(Xs)ds

(3.1 )

o

(3.2)

Xt = Bt +

f veda)

(X)

(v is in M(IR)).

IR

COY'ollaY'y 3.2 implies the following results. Any solution of (3.2) is the

strong limit of a sequence of solutions of equations of the form (3.1). Conversely let (gn) be a bounded sequence in L1 (1R ) and let Xn , n=1,2, ... be the solutions of the corresponding equations (3.1). Then there exist a measure v and n

n

a subsequence (X k) such that X k converges strongly to the process X solution of (3.2). b) As shown in the following simple example, there is no analogue of COY'ollaY'y 3.2 for measures v which only satisfy the weaker assumption

for all

x,

Set:

for all

Let Xn be defined by : = Bt + J vn(da)

(X

n)

n

71

Xn is well-defined

see the remarks after theorem 2.3. We can easily

prove that Xn

where X = UB t, t

X

U being a random variable independent of

B and such

that P(U=I) But

X

is not even a

1

P(U=-I) = "2 .

Narkov

process and thus it

cannot

be solution of an

equation of the form (3.2). We now use theorem 3.1 to obtain a new proof and also a slight general ization of a result due to Rosenkrantz (I 7 ] ). Corollary 3.3 :

and X be a continuous semimartingale such that

Let v be in

t = X0 + Bt +

X

f v(da) IR

La (X) • t

Assume that Xo is in LI(rl) and set Xnt

1

- X 2

n n t

Then Xn converges weakly towards the skew brownian motion of parameter a given by : I-a _ _ c l+a - exp( 2v (IR))

(l-V{{y}) ) l+v({yJ)

(recall that the skew brownian motion with parameter a

is the process

uniquely defined by

Remark :

Rosenkrantz treated the case v(da) = g(a)da. In this case we have J

I-a l+a

exp (-2

J g(x)dx) IR

72

Proof : Define, for each integer n and for any Borel set A

Then

so that Xn has the same law as the process

=

*

Xo + Bt +

J vn(da)

Yn defined by

(yn)

IR

Theorem 3.1 together with the relations

f v (x)

1

n

for all

f v (x)---t exp(-2v n

c(lR))

x< 0 for all

IT

y

x >0

complete the proof of the corollary.

o Remark Corollary 3.3 provides a result of convergence towards the skew brownian

moti on with parameter a, where a of a (a=l

or a=-l)

is in the i nterva 1 ] -1; 1 [ . The extreme va1ues

which correspond to reflecting brownian motion, cannot be

obtained through these methods. However it is possible to state a similar result of convergence towards a reflecting brownian motion. Let (f n) be a sequence in f

n

L1(R) S.t

;;;, 0

with

E

n

73

Let Xn be defined by :

x''t = Bt

+

r

ds.

fn

0

Then it is easy to prove that

xn

(weakly) ) n-> 0,

> 1

for any

r.v.

1 > 0,

(1.c) does not hold for

k

= 1,

even when

to be a stopping time ; again, this may be done by taking

L

1S assumed

'\,

1 = T and letting c"" "". c'

These two results clearly show the importance of the stopping times

'\,

{T 1n c} connection with the study of reflecting Brownian motion; in this Note, we take up

the next natural step, that is the study of = inf{t:

for

(p

t)

P t

a Bessel process, with dimension

in this set-up.

d

2, and we extend 1. Shepp's formula

101

Moreover, with the help of the mutual (local) absolute continuity of the Bessel laws, for dimensions

2 , when the processes start at

form of the total winding of complex (see

a

>

0, the Fourier trans-

'\,

BM around 0, up to

T is obtained c'

formula (2.b.2) below).

This formula (2.b.2) is very similar to D. Kendall's formula (32) in gives the Fourier transform of the total winding around complex

BM

[5J,

which

for the pole­seeking

0

stopped when it first hits a circle centered at

O.

In the third paragraph below, a probabilistic explanation is given for this Slmllarity, using the time SUbstitution method, as advocated by D. Williams in the discussion following

D.

Kendall's paper

[5J

([5J,

p.414)

1.

2. An extension of Shepp's formula (l.h) (2.1 )

We consider, on the space

(pt(w)

­

w(t) ; t

" C(IR+,1R), the process of coordinates " o{p s ; s < t.}

0), and its natural filtration

To any couple of numbers

\)

and the distribution

>

on

0, and

a

>

0, we associate the dimension

(il,+1 !:L \> ­ 2 dx2 2x dx'

a.

a >

0, 11, v

stopping time

>

O.

Acoording to

[13J,

and

[7J,

one has, for any

T

dpA (2.a)

a

i,

on

T+

()(T < 00)

where

(2.2) Confluent hypergeometric functions appear repeatedly ln the main formulae below. We have conformed with the notations and definitions used in Abramowitz ­ Stegun ([lJ, p , 504 and (2.3) The main result in this paper is the following

102

Theorem

For any

[t

(2.b)

where

U > 0, v

0, a

a,c > 0, one has :

0, and

1 +

c

2)1/ 2 2 A = (U + v ,

and

A

M, if

A-U

2

1 A(a +1cIe.· 2 ' A+ ;

)

A(a +1cIe.· A+ 1 , 2 2 ' U, if

a < c

a

2

,

c.

>

In particular, 2

A(a ; u+1 ;

(2.b.1 )

) 2

A(a; u+1 ;

r

EO exp ( -

(2.b.2)

aL'

)

a

Before entering properly into the proof of the theorem, we remark that, if 2

denotes the distribution of the lli - va l ue d r.v.

[log(

+

1+Tc ) ; CT] ' c

then : (2.c) for any

'IT

,b

between

a and c, proving at once the infinite divisibility of

a,c

b

*

under

'IT

a,c

pU a'

'IT

b,c

This is a probabilistic proof (and improvement) of Hartman's result

1T

([4J,]).

a,c 271-2),

asserting that the right-hand side of (2.b.1), r e ap , (2.b.2), is the Laplace transform In

a,

resp.

v

2

of an infinitely divisible distribution on lli+.

We also note that identity (2.c) is probabilistically easier understood after time-changing the Bessel process

(p

t),

It is well-known (cf. D. Williams [12J under

with the inverse [7J)

(T ) of (C

t

t).

that:

pU a'

a real-valued BM U loga'

where

stands here for

starting at

(loga). Using (a.a) , one obtains:

BM, with constant drift

\1,

103

Formula (2.c) now appears as a consequence of the strong Markov property taken at time o(b), for BMll(l ). . oga We now proceed to the proof of the theorem, via two steps. We first prove formula (2.b.1). The following notation will be helpful : (20) =

I

Recall that, for any

e

>

II

'"u (20)

(20) ;

K

0, the processes

'"

2

(I (ep ) exp(-.L t ) t IJ 2 are two law of

t .:':. 0), and

plJ-local martingales, an assertion from which the Laplace transform of the a

T

inf{t : P

c

=

t

c}

under

is easily deduced (c r , J. Kent

a < c, and deduce from (2.e) that

We now suppose that

2

e T", )l Ell ['" I (ec 11rr:« + T ) exp(- -a II c 2 cJ Then, following Shepp's method equality with respect to

e'

de e

l'

IJ

(ea).

[8J, we integrate both 2 -e I 2 . P

sides of the previous .

.e , and obtaln, after the change of varlables

c

(2.g)

where

[t£]).

)-

[(1 +

u (c) p

=

f:

de e

We now use the expansion

J..::.p, 21

u (c) p

J

u (a), p

2 .eP.r (ee ). II

r (20) II

l:

n l r( ll+n+1)'

2

n=O 2

2c)

(2.h)

2

a-1

This proves (2.b.1), as a consequence of (2.g) same method, with

'"

KIJ

, where in the case

to obtain

2 .

a > c, we use the

now replacing

Step 2. The complete formula (2.b) now follows from (2.b.1), using the explicit Radon-Nikodym density formula (2.a) for

T

= T'"c

104

Remark: In fact, formula (2.h) has a long history; it is due to Hankel (c r , Watson

[llJ,

p , 384-394) and is a generalisation of formulae due to Lipschitz,

Weber, and Gegenbauer ; at the beginning of the century, formula frequently used by some physicists (again, see Watson [1 I] 3. Another interpretation

0

(3.1) For any

Opv a

from

>

,

(2.h) has been

p. 385).

total winding for pole-seeking

BM :

0, we introduce a new family of distributions

is the distribution of the

d _ 2(v+1)

dimensional Bessel process, starting

a , with "naive drift" 0, that is the distribution of the JR+-valued diffusion

with infinitesimal generator d

A

- 6) dx'

v

'I'he introduction of a terminology such as "naive drift" seems necessary, in

order to avoid confusion with the diffusion obtained by taking the radial part of a

-valued

BM, started at the origin, with

usually called Bessel process with drift (c r , Shiga - Watanabe

[9J ;

Watanabe

[lOJ

t

(E

; this latter diffusion is

Itt

0

[IJ) .

(3.2) In the course of his mathematical study of Bird Navigation, D. Kendall obtained some remarkable formulae (see formula (32) which the following

1S

easily deduced

for

c'




b.

0, one gets : A(I,-v; 1+2A ; 20a) A(A-V; 1+2A; 20b)·

The comparison of this formula with (2.b.2) implies the following extension of (3.c): (3.e)

c

where a

2

. p2v) (d) (C , a T

40a'

c'

c

2

0pv ) a'

40c' .

The proof given In (3.3) for the identity (3.c) is still valid for (3.e), provided the process started at

(Y in (3.d) now stands for t) 0, with constant drift v.

a real-valued Brownian motion,

107

M. ABRAMOVITZ, I. STEGUN

[2J

M.T. BARLOW, S.D. JACKA, M. YOR

Handbook of Mathematical Functions. New-York - Dover - 1970. Inequalities for a couple of processes stopped at an arbitrary random time. To appear (1983). On the norms of stochastic integrals and other martingales. Duke Math. Journal, vol. 43, nO 4, 697-704

(1976) .

[4J

[5J

[8J

[lOJ

Uniqueness of principal values, complete monotonicity of logarithmic derivatives of principal solutions and oscillation theorems. Math. Ann. 241,257-281 (1979). D. KENDALL

Pole-seeking Brownian Motion and Bird Navigation. Journal of the Royal Statistical Society. Series B, 36, n? 3)P. 365-417, 1974.

J. KENT

Some probabilistic properties of Bessel functions. Ann. Prob. £, 760-770 (1978).

J. PITMAN

Bessel processes and Infinitely divisible laws. In : "Stochastic Integrals". Lecture Notes in Maths 851. Springer (1981) (ed. D. Williams).

L. SHEPP

A first passage problem for the Wiener process. Ann. Math. Stat. 38 (1967), p. 1912-1914.

T. SHIGA, S. WATANABE

Bessel diffusions as a one-parameter family of diffusion processes. £.f.W, 27 (1973), 37-46. On Time Inversion of One-Dimensional Diffusion processes. £eitschrift fUr Wahr. 2l (1975), 115-124.

A treatise on the theory of Bessel functions. Second edition. Cambridge University Press

(1966) .

Path-decomposition and continuity of local time for one-dimensional diffusions, I Proc. London Math. Soc., Ser. 3, 28, 738-768

(1974).

[13J

-

Loi de l'indice du lacet Brownien, et distribution de Hartman-Watson. £.f.W, 53, 71-95 (1980).

DISTRIBUTIONAL APPROXIMATIONS FOR NETWORKS OF QUASIREVERSIBLE QUEUES

P.K. Pollett Department of Mathematical Statistics and Operational Research University College Cardiff CFl lXL Great Britain

ABSTRACT. This paper is concerned with establishing Poisson approximations to flows in general queueing networks.

Bounds

are provided to assess the departure of a given flow from Poisson and these lead to simple criteria for good Poisson approximations.

The class of networks considered here are

those with a countable collection of customer classes and where the service requirement of a customer at a given queue has a general distribution which may depend upon the class of the customer.

KEYWORDS.

Queueing networks, Poisson Approximations.

109 1.

INTRODUCTION.

In a recent paper, Brown and Pollett (1982) exhibited a method for approximating customer flow processes in single class queueing networks with exponential service requirements and servers with state-dependent rates. The distance of customer flows from Poisson processes was estimated using formulas derived by Brown (1982) for general point processes.

It is the purpose of the current exposition to extend their results to a class of quasireversible networks with customers of different classes and associated general service requirements.

Bounds are provided to assess

the degree of deviation of arrival processes from suitably chosen Poisson processes.

Although the arithmetic values of these bounds are of doubtful

practical significance, they are of some theoretical interest and give rise to simple criteria for good Poisson approximations. to fall into three categories: buted customer routing. requirements are exponential

These criteria tend

light traffic, heavy traffic and evenly distri-

However, in contrast to the situation where service (Brown and Pollett (1982», the heavy traffic

approximation seems only to be possible if service effort is distributed evenly among all customers present in a given queue (the server sharing discipline).

In section 2 a standard notation is defined and various preliminary results on queueing networks are collected.

Sections 3 and 4 are devoted

to discussing Poisson approximations to arrival processes in both open and closed networks of symmetric queues.

2.

NOTATION AND PRELIMINARY RESULTS.

Let

N denote

a multiclass network consisting of J queues {1,2, ..•• ,J}

(with J possibly infinite) and a countable set of customer classes,

C.

customers are allowed to enter or leave the network it is said to be open;

If

110

otherwise, there is a fixed number of customers of each class and the network In the open case we suppose that arrivals from

is said to be

outside the network occur as independent Poisson streams, the class c arrival Define for each c in

stream at queue j having a bounded rate of v.(c). J

a

A(c)

=

(Ajk(C»

C

to be the collection of probabilities that

govern internal transitions from queues j to k for customers of class c, and let AJ,O(C)

J

=

1-\ A. (c)

be the probability that after completion of service at

queue j a class c customer leaves the network.

If

N is

closed, AjO(C) is

taken to be zero for each j and c.

In the open case define g(c)

=

(a (c) ,a , .•.. ,aj(c» 2(c) l

to be a vector

with non-negative entries that satisfies

g(c)

(1)

':!(c) + g(c)A(c).

In order that this vector be unique, we assume that it is possible for any class c customer to eventually leave the network either directly or indirectly via some sequence of queues.

The quantity a.(c) may be interpreted as the J

equilibrium arrival rate for class c customers at queue j and will be positive if it is possible for such customers to visit the queue.

In the closed case we suppose that A(c) is irreducible and non-null persistent.

This ensures that there exists a unique (up to a constant multiple)

vector with positive entries that satisfies (2)

-

o Ic) = a(c)A(c)

-

J

and it will be of no loss in generality to assume thatL a.(c) By Chang 1. j=lJ and Laverberg (1974) the quantity aj(c)/ak(c) is the ratio of the class c arrival rates at queues j and k.

111

We suppose that each queue in the network is

J.>ymme-tJUc.

(Kelly (1976»),

that is, each queue j in N operates as follows:

(i)

A total service effort is offered at a rate (units per second) when there are n

J

J

customers present;

j

A proportion y.(t,n.) of this effort is directed to

(ii)

J

J

the customer occupying queue position

when this

customer leaves the queue, customers in t+2, ...• ,n. move into positions t, t+l, •... ,n.-l J

J

respectively; When a customer arrives he chooses to occupy position

(iii)

t in the queue with probability y.

J

previously in

J

customers

.... ,n. move into positions J

t+l,t+2, ..•. ,n.+l respectively. J

For each j in {l,2, •.•. ,J} we assume that n

L v• (t,n)

and for n>O,

J

and

The fact that the same function Yj is used in both (ii)

1.

J

=0

J

and (iii) places some restriction upon the types of possible service discipline. However, it allows service requirements to take a quite general form without making equilibrium analysis unmanagable.

We suppose that successive service

requirements for customers of class c at queue j are i.i.d. random variables with distribution function F. (x) and mean JC

J

Thus, when there are n

j

customers present at queue j the rate at which class c customers are served is

J

(c)

. (n

J

i

J

)

Let e(t) the network

(customers per second).

=

N and

(x

l(t),x2(t),

•••• ,x

J(t»

be a Markov process that describes

that contains enough information for one to deduce the number

of customers in each queue and the classes of each of them.

In particular,

112

when queue j is symmetric we let x . = (n.; x , (1), x , (2) , .... ,x. (n.»

J

x.(t) = (c.(t), z.(t), u.(t»

J

J

J

J

J

J

J

J

J

where

describes the customer in queue position t.

Here c.(t) is the class of the customer z. (t) is his service requirement and J

J

u.(t) is the amcunt of service so far received.

In general

J

a continuous state space.

However, if each of the F. ,

C E

JC

(Cox (1955», for example if F.

JC

will have

C, admit a

is Hyperexponential

or a mixture of Gamma distributions, it is sometimes convenient to let z. (t) J

and u. (t) determine, respectively, the total number of (fictitious) stages of J

service and the number of stages reached.

In this case the state space

will be countable.

For each j in {1,2, .••• ,J} and c in C let a.(c) = J

J

J

the

average amount of service required by class c customers arriving in queue j, and let a

\

J.= !..c E

Ca,J (c),

the total average requirement.

For the closed

network let N(c) be the total number of class c customers and define = ( .... ,N(c), .... ) to be the vector which determines the number of customers of each class in the network. m. class c as N -c j

Denote the vector with m fewer customers of

Define n.(c) to be the number of class c customers at queue J

and let J

I

J

n,(c)=N(c), c e C} j=l J

1;N

denote the state space of

•

The following results summarise some of the important equilibrium properties of the network consisting of symmetric queues. consequence of Theorems 3.7(i) and 3.10 of Kelly (1979).

Lemma 1 is a direct

113

Lemma 1.

For the open multiclass network

N consisting and only

equilibrium distribution exists for

I

n n a. /{ IT

n=O J

r=l J

(r)}

of symmetric queues, an


0,

(11)

If

d

2

then

{oJ

E

Kesten ([35J, Theorem 2) gave the first proof of this fact, a proof that was later shortened by Bretagnolle ([aJ).

Kesten and Bretagnolle also

discuss the more difficult problem of when does {OJ E

{oJ

imply

E

r.

We also have the following alternative characterization of when

THEOREM 4.

if and only if

Suppose that X

A u (x), is bounded.

d

=

1

and that

A > O.

{oJ f

We have

has a strong Feller resolvent whose canonical density, In this case there is a positive constant

c

A

such

that O.

(x)

0

is regular u

Ie

,

that

In this case

= U x (-x)ju x (0).

Bretagno11e classifies the circumstances under which this situation occurs. Theorems 4 and 5 are given, with quite different proofs, in Port and Stone ([43J, pp.207-210). one sees that function.

u

Ie

(x)

+

U

Ie

Note that in the circumstances of Theorem 4 (-x)

is almost everywhere equal to a continuous

But this is not sufficient to ensure that the lower

semicontinuous function

u

x (x)

is continuous at the origin.

Consider,

for example, a Poisson process with unit drift.

(b)

The comparison problem. Our main result (Theorem 2) yields an immediate solution to the

comparison problem.

144

THEOREM 6.

processes having exponents 1 Re(A + 1/1 (z ) 2

Proof.

A > O.

Suppose that

and

1/1 1

1/1

Let

The finiteness of

X 2

respectively.

2

0(1) Re(A +

and

Xl

1

(II z

1/Il(Z»

II +

be two Levy If

00)

implies that of

so the result

follows from Theorem 2. The result has been obtained, with progressive weakening of the assumptions, by Orey ([41J), Kanda (r3lJ) and Hawkes ([18J).

The point

here is the very simplicity of the proof that results from our geometric approach. In fact in

then

!

caP

we proved slightly more, namely that if

A A (A) 5 M caP2 (A) 1

consequence that i f

o

< a < 1,

1/I(z) = Izl constants

and

Xl

for all analytic sets X 2

A.

This has the

are two linear stable processes of index

so that the exponents take the form a

{1 - i8sgn(z) t an j-n«}

M 1

and

M 2

-1 5 8 5 1,

with

then there are

such that (13)

M caP (B) 5 ca P (B) 5 M ca P (B) 1 1 2 1 2

for all analytic sets (c)

B.

This answers a quest ion due to Taylor (I 56J) .

The symmetrization problem (Orey). Let

of

a,

X.

X be a Levy process and let Then

Zl

and

Z2

be independent copies

is called the symmetrization of

X.

Orey

conjectured that (14) There are examples of varying degrees of sophistication to show that this

145

inclusion can be strict. Example 1. so that

Let

X(s)

= Pt

X t

= p(s). P(X)

=

t

where

P

is a Poisson process of rate one,

t

One can see that {B:A(B) = O}.

whilst

Thus the one class of sets is smallest possible whilst the other is largest possible. Example 2. y(X)

has

a

Pruitt

=

so that

2 and 3 1 < a < 3

cr 47 J)

has given an example of a subordinator

y(x(s» 2 5'

-

3

= 5'

X

which

I f we choose

y denoting Pruitt's index.

then an intersection argument of the type used in

Hawkes ([17J, the argument deducing Theorem 4 from Theorem 3) can be applied to show that almost all realizations of the range of a linear stable process of index

a

are in

but not in

The inclusion relation (14) follows from Theorem 6 and the observation that 1

(:\ + Re lji) where

§8.

lji

is the exponent of

X

and

ljis

that of

X(s).

ENERGY AND CAPACITY

We now return to the problem that was left unanswered in §4(d). In ([18J) we showed that for Levy processes, and open sets [4 cap(D)J

-1

S I(D) S [cap(D)J

-1

D,

one has

(15)

.

The upper inequality is, as is well known, in fact equality when the process is symmetric. in general.

In ([10J)

The answer is no!

and Rao ask whether this is true This is seen by taking

0 < a < 1

considering the symmetric and increasing stable processes of index

and a

146

and taking varies

D

I(D)

to be the unit interval. can be arbitrarily close to

in (15) cannot be replaced by any number

In 116J we showed that as [2 cap (D)]-I.

e, e

a

Thus the number 4

Thus the Kelvin

< 2.

It would be interesting

principle even fails for linear stable processes. to know the best constants in (15).

For more information on energy see Chung ([8]) and Chung and Rao ([10J).

§9.

WIENER TESTS.

In this section we mention a criterion for a point for a set. theme.

x

to be regular

This criterion is essentially in keeping with our geometric

We also indicate how the comparison results for capacity can be

applied to yield comparison results for regular points. (a)

The classical result.

In 159] Wiener gave a criterion for a point

x

to be a regular boundary point for the Dirichlet problem associated with a given domain. (b)

Brownian motion. is regular for

We shall discuss a probabilistic version of this.

Let K

B t

be the brownian motion in

md .

Then

x

if and only if

I

2 n(d-2)cap [K (x)J n

I

n cap [K (x ) n

n;:o"O

(d

;:0"

3)

(16)

or

n;:o"O where

K (x) n

=

{z

E

K: 2-(n+l) S

(d

Ilz- xii


Io N

Now,

J

0 1 h (N + iy) 1 e -

h(y)e

i8

6

o

ydy

Ydy

I

N i6 -2K8 0 e Ye h(y+2iK)dy

r3

-

r4

- niZres - i

2K 6 0 e- Yh(iy)dy .

J

But h(y+ 2iK) = i

and

where

ex y a i (sin 2K)

h(iy)

g

2 ah(y)

is real-valued.

Therefore,

say,

r

and

3

f

become

4

and Hence, applying Cauchy's Theorem and letting niZres + i

If we now multiply both sides by

(3.2)

Re[(i-

ae K6

i

-a K6 e

_ iae-K6) J:e

N -+

00

,

we get:

a+l {2K -8 0 e yg(y)dy.

and take real parts we obtain: i 6Yh(y)dj 2K sinh K6x 1TX I-a rrx (cOST) sin""2

181 i 1T().

If we bear in mind that

2

e

iCt

cos

1Ta 2

1Ta

+ i sin""2'

then we may

write (3.2) in the form

(3.3)

na

oo

fo(cos ""2 cos 8y sinh K8

+ sin

1TCt

2

sin 8y cosh K8)h(y)dy

=

K sinh K8x 1TX 1-Cl 1TX (cosT) sinT

We can also write (3.3) in the form

(3.4)

f - J_

1Ta 1Ta (cos""2cos8ysinhK8-sinTsin8ycoshK8)(-h(-y»dy

_00

K sinh K8x 1TX 1-a 1TX (cosT) sinT

Hence, to obtain (3.1) we put and add (3.3) and (3.4).

a

= 211

1T '

where

This then gives

11

is defined as above,

1To(X,y),

namely,

zlzl.

1TX a 1TX 1-a sgn(y)sin T (sinh 2k ) (cos 2)

o

1T (x,y)

where

0
czJ = J (cz,oo)/J (cy ;»)

Thus, for some constants

,

E:

and

e.

The fundamental equation (3.9) therefore takes the form:

(4.1)

=

E:

-Rlxl Iyl

e.

The Brownian scaling gives us further information: (4.2)

for

x < O. Y > 0 ,

192

Hence IT(x,y) = cIT(cx,cy), On taking

c

= 1/lxl,

where

u

=

Iy/xl,

l+a

c

p(x,y)

p(cx,cy).

we see that

p(x,Y)

(4.3)

and

Ixl

and

-I-a

=

k(u)

Ixl

p(-I,ly/xl)

-I-a

k(u) ,

p(-l,u) .

Substitution of (4.3) into (4.1) yields: d 2 [(A + U2 +a)k(U)] =

du

2

_ 2ARlxl

Since the left-hand side is a function of 2 d 2+Cl. 2 [(A+u )k(u)] = du and

-3-a-E:

6+a

(A+u

2+a

-2ARu

alone, we must have 6+a

,

o

r

and

00

x= -

°

and

b,

and

p(x,Y) Ixladx