Stochastic Mechanics and Stochastic Processes: Proceedings of a Conference held in Swansea, UK, August 4-8, 1986 (Lecture Notes in Mathematics, 1325) 9783540500155, 3540500154

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Lecture Notes in Mathematics Edited by A Dold and B. Eckma nn

1325 A. Truman I. M . Davies (Eds.)

Stochastic Mechanics and Stochastic Processes Proceedings of a Con ference

held in Swansea, U. K., Aug . 4-8, 1986

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Lecture Notes in Mathematics Edited by A Dold and B. Eckma nn

1325 A. Truman I. M . Davies (Eds.)

Stochastic Mechanics and Stochastic Processes Proceedings of a Con ference

held in Swansea, U. K., Aug . 4-8, 1986

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo

Editors

Aubrey Truman Ian M, Davies Department of Mathematics, University College of Singleton Park, Swansea SA2 8PP Wales, U.K.

Mathematics Subject Classification (1980): 60GXX, 60HXX ISBN 3-540-50015-4 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-50015-4 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 1988 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

Preface This volume primarily contains papers presented at the meeting entitled 'Stochastic Mechanics and Stochastic Processes' held in Swansea from 4 August to 8 August 1986. Also included in the volume are some related papers, not presented at the meeting, but in the same subject area. The topics covered herein are quite varied but all related to the central themes of the meeting, including large deviations and statistical mechanics, Nelson's stochastic mechanics and quantum diffusions, simulations of Brownian Motions and stochastic flows. The meeting was most worthwhile both in the quality of the talks given and the level of discussion generated on the subject of stochastic processes and stochastic mechanics. Most of the papers herein are reasonably self-contained and should be readily accessible to researchers in this field. For beginning students in Nelson's stochastic mechanics we recommend that they start by reading the first paper in this volume by Batchelor and Truman, which deals with stochastic mechanics for excited states of a system with finitely many degrees of freedom. Some of the corresponding stochastic flows are discussed in the paper by Chappell and Elworthy, where new results are given for their Lyapunov exponents. There are related papers by Yasue, Durran and Williams and Steele. Zambrini's paper includes new results for Bernstein processes and stochastic mechanics. An interesting new treatment of stochastic mechanics for systems with infinitely many degrees of freedom (quantum fields) is given in the paper by Carlen. Supersymmetry is discussed in the paper by Haba. There is an excellent expository account of large deviations in statistical mechanics by Lewis, and two papers by Lewis and co-workers in statistical mechanics itself. There have been exciting new developments in this area recently. An introduction to the algebraic theory of quantum diffusions is given in the paper by Hudson. New analytical results for stochastic processes are given in the papers by Kifer, McGill, McGregor and Watling. It is a pleasure to thank Brian Davies, of King's College London, and Nick Bingham, of Royal Holloway, for their assistance in helping to organise the meeting and in producing this volume. We are grateful to the SERC for financial support through research grant GR/D/88847. We should like to thank Mrs E. Evans, Mrs M. Prowse and Mrs E. Williams for their patience and application in typing some of the papers. Last but far from least we would like to thank the referee for his important contribution.

A. Truman and I.M. Davies Swansea

CONTENTS A. Batchelor, A. Truman: On first hitting times in stochastic mechanics M. van den Berg, J.T. Lewis: Limit theorems for stochastic processes associated with a Boson gas

16

M. van den Berg, J.T. Lewis, J. Pule: Large deviations and the Boson gas

24

E.A. Carlen: Stochastic mechanics of free scalar fields

40

M.J. Chappell, K.D. Elworthy: Flows of Newtonian diffusions

61

R.M. Durran, A. Truman: Planetesimal diffusions

76

R.M. Durran, A. Truman: Brownian motion on hypersurfaces and computer simulation

89

Z. Haba: Supersymmetry and stochastic processes

101

R.L. Hudson: Algebraic theory of quantum diffusions

113

Y. Kifer: A note on the integrability of Cr-norms of stochastic flows and applications

125

Y. Kifer: A note on stochastic models and expanding transformations

132

J.T. Lewis: The large deviation principle in statistical mechanics: an expository account

141

P. McGill: Excursions from random boundaries

156

M.T. McGregor: A solution of an integral equation in convolution form and a problem in diffusion theory

162

K.D. Watling: Formulae for the heat kernel of an elliptic operator exhibiting small-time asymptotics

167

D. Williams, N. Steele, A. Truman: Stochastic mechanics for a point source

181

K. Yasue: Computer stochastic mechanics

188

J.C. Zambrini: New probabilistic approach to the classical heat equation

205

ON FIRST HITTING TIMES IN STOCHASTIC MECHANICS by Andrew Batchelor and Aubrey Truman Department of Mathematics and Computer Science University College of Swansea Singleton Park, Swansea SA2 8PP 1.

A Resume of Stochastic Mechanics

Q of mass

The Schrodinger equation for a particle

m subject to a force

-2V

in JRd is equivalent to (1 )

*

where Here

h

denotes the complex conjugate and

is Planck's constant divided by

is the quantum mechanical wave-function and the time

=

the positive reals, and Since

E

lR

t

E

lR

2IT.

+

d, d-dimensional Euclidean configuration space.

V is real-valued, equating imaginary parts of the above equation gives

the continuity equation

where

p

=

*

+ div j = 0,

at

is the quantum mechanical particle density and

is the probability current.

t

i

for each Borel set

the probability that

E

*)

-

Q is in

A at

A at time t.}

(3)

d A c lR .

Following Nelson [6J,[7J, introduce the real-valued functions __

,I, 'Y

*

is given by IP{Q

by

h 2mi

The last equation merely expresses the conservation of

particle number in that, for the state time

(2)

e R+ i S,

so that

p

e 2R

=

and

i

(h_ n S ) e 2R. m:'v

Rand

S

defined

We can then deduce from the

continuity equation that

.

1t

i.e. for

v

=

2R )

div(A'Ve 2m -

2R

div(v2p - £p),

h

2m and

£ h. V (R + S).

(4) (5)

2

The last equation is just the forward Kolmogorov equation for the density for the diffusion

satisfying ;

p;

where the forward drift

,-

J[{B. (t)B. (s)} ; 6 .. min(s, t :L

J

p

:LJ

) ,

ill

rv

for

+ S)

m

(6)

-

in cartesians, with

,

i,j

1,2, ... ,d,

being a

Moreover, equating the real parts of Eq. (1) gives for

as ; -/I( IVR 1 -at 2m 2

I VS I 2 + fiR)

lji ;

e

BM(lR

d)

process.

R+iS

,1

V - T'

(7)

Nelson's remarkable discovery was that the last equation embodies a dynamical principle for the diffusion To see this Nelson defines the mean forward and backward time derivatives

D±

by ; lim hi O Then it follows from Eq.

(8)

±h

(6) that ;

,t) ;

and from Its's formula that for sufficiently regular D f(X(t),t) ; +

ot

(9 )

+ R) ()S(t) ,t) f

+b.Vf +21:t lIf)(X(t),t). m

(IO)

-

Nelson went on to show that for sufficiently regular functions

g

and

h

ddt J[ (g()S(t»h()S(t») ; E(g()S(t)D_h()S(t») +E(D+g()i(t»h()S(t») (see e.g.

p. 98 Ref (6».

We can now establish Nelson's result. f

d

00

E

Co (lR ),

where

)i

=

for

(Xl ,X

Firstly, from above, for any

i ; 1,2, ... ,d,

2,···

,X d)

in cartesians.

Since

f

E(g()S(t),t»;

lRd

integrating by parts, using the identity, for

i ; 1,2, ... ,d,

in cartesians, we obtain for each

f

00

E

d

Co (lR )

(11)

3

H: {f ()S (r ) D_Xi (t ) } +

H: {Xi (t ) (2S. 2f) ()S (r ) , t

) }

+

J[ {

(Ili R) f ()S (t ) , t ) } • (14 )

But Ehrenfest's theorem for the quantum mechanical state sufficiently regular

e

R+iS

gives for

g (15)

Hence, setting

gelS)

f()S)X

we obtain

i,

!:H:(Il.S(X,t)f(X» m 1 for

i

=

(14) above, we see that necessarily

D X. (t) -

i

(16)

1,2, ... ,d.

Comparing this with Eq.

for

+ m

1,2, ... ,d.

1..

=

ill

1..

- Il.R)(X(t),t) L

Hence, the backward drift D_)S(t)

(17)

r..>

is given by (18)

- 2R) ()S(t), t )

=

and from ItS's formula DJ(Q,(t),t)

+

=

-2R).17f - fmM)()S(t),t).

(19)

Nelson's amazing discovery now follows from Eqs. (9), (10), (18) and (19).

After

a tedious calculation we obtain (20)

or from Eq. (7) (21)

This is the Nelson-Newton law i.e. a stochastic version of Newton's second law of motion Force

=

(22)

Mass x Acceleration.

Therefore, we have seen that the net content of the Schrodinger equation for the state drift

m

ljJ = e

R+iS

+ IlS)

is just the Kolmogorov equation for the diffusion

and the dynamical principle for

suggests that the sample paths of the diffusion

)S

,-

contained in Eq.

)S

with

(21).

This

have some physical significance.

We investigate this for the stationary states of the Coulomb problem below presenting some new results for first hitting times.

We do not give all the details of the

proofs here and we refer the reader to the original references for details.

(See

4

Refs. (2) and 0)). 2.

First Hitting Times for Ground States of Spherically Symmetric Potentials We consider Nelson diffusions corresponding to ground state solutions of the a self-

Schrodinger equation fbr a quantum mechanical Hamiltonian 2

adjoint linear operator on some appropriate domain in the case

d

symmetric, Let

3

and further we assume that the potential V( 1,lS I),

V

L ( JRd ) .

'!'E(,lS,t)

=

JS

for

3

JR,

E

2

'!'E(x) EL ( JR3 ) t

I·

I

V

We specialize to

is spherically

being the Euclidean norm.

be such a ground state with

-

(23) E

being inf spec

i.e. setting

x

(H).

Then, as is well-known,

f

E

satisfies

d2

'/:12

'/:12

H

d2

- - - + Vex) 2m dx2

r

IS

the radial Hamiltonian.

f

E

on

> 0

and

(0,00)

fE(O)

=

We assume that

V is

is and,

o.

A straight-forward application of corresponding to the state

(24)

0,

piecewise continuous with finite discontinuities on (0,00) so that of course,

0,

IlSl,

=

(-- + Vex) - Elf (x) 2m dx2 E

where

(H - E)f r E

formula yields for the Nelson diffusion

'!'E (25)

B being a

BM(JR)

process.

This is a one-dimensional time-homogeneous diffusion

with generator L

By virtue of Eq.

(26)

x

(24) the radial diffusion D +- + D_D+)

2

Moreover, since for any

C2

I (t) 1

function

satisfies the Nelson-Newton law = -

(27)

d Idx 1 V( 1X1 (t ) .

f,

2

f;I(Hr-E)(fEf) f

E

=

+ (V-E)fEf}

-'/:1Lf,

(28)

being positive, we obtain formally at least (29)

5

Pt

heat kernel.

t

,

being the transition density for For convenience now set

exp {-K(H - E)}(x, y) r

j1 = m = I.

We now set about finding the distribution of 'x(a) = i.nf {s > 0 : the first hitting time of the level

being the appropriate

T

x

(a):

I = x},

= a,

a

(30)

for the process starting at

x.

The key

result here is f

where

(H

r

+ A - E)

-1

(x,a)

-]

E

(H (x) (H

r r

+ A - E) + A - E)

-I

_J

(x,a) (a,a)

is the resolvent kernel.

The last identity follows because

is a one-dimensional time-homogeneous diffusion. gone from

x

to

y

must be less than

time t.

and any intermediate

t

Since the process starts afresh from

=

a,

for each fixed

a x,y

a, ftg(x,a;u)p

=

o g(x,a;u)du

For such a diffusion to have

the first hitting time of any intermediate point

Pt(x,y)

where

(31 )

(A > 0),

fECal,

TI'hx(a)

E

(u,u+du)}.

Laplace transforms and letting y -+ a.

t-u

(32)

(a,y)du,

The desired identity follows by taking

This leads to:

PROPOSITION I Let

E«O)

be inf spec (H),

where

-2-

H

J

/:; x

+V(liSl).

For each

A>O,

let

the equation -2

2

--1 +

_J d

(V(x) +A -E)y

dx

have two linearly independent (0)

gE-A (0)

=],

c2

solutions

(0)

=],

=

0.

Vex) -+0

as

x-+"',

for

'( (a) x

radial process, for the Hamiltonian

(33)

0

and

gE-A

defined by

(This will be the case e.g. if

analytic in a neighbourhood of the origin.) with

fE-A

=

Then, if

V

fE-A (0) 2

x Vex)

=

0,

is

is piecewise continuous,

the first hitting time for the ground state H,

for

x < a, (34 )

whilst for

x >a (35)

h

being the unique solution of Eq.

(33), which is exponentially decreasing at infinity.

6

The above result yields: COROLLARY 1 Let

x> a

and further let

V have compact support with

for the radial ground state process with energy

supp V c [O,a).

Then

E«O)

E{exp(-ATX(a»} = exp(-/(-2(E-'\»(x-a»exp(+I(-2E)(x-a».

(36)

Thus, ]P(T

x

(a)

E

ds)

(21T)

In this case, therefore, for

_11 a-xis -3/2 exp{Es 2

x »a ,

with

>

(x_a)2 I 2s + (-2E)(x-a)}ds.

supp V c [O,a),

the distribution of

is the same as that for Brownian motion with a constant (10».

(37)

Tx(a)

(see e.g. Ref.

The last example is atypical in that one has an explicit formula for the

distribution of

Tx(a).

More typical is the next example.

Example 1 (Spherical Square Well Ground State)
0)

and

be such that

vex)

L

x

E(TX(a»

2

-I d

2

-

dx2

d + b(x)-

lim f

2 E

(x ) v ' (x)

o

Then for the radial ground state

dx

=

-]

2

together with the boundary conditions

xt oo

(0,00),

satisfies LxV(x)

and

is Lipschitz continuous on

being entrance and natural boundaries for the radial gound-state diffusion

with generator process

b(x)

for

v"(x) v(a)

b(x)v'(x)

+ =

0

and

=

(4] )

-1, 2

lim f (x iv ' (x ) E x+O

o

for

x a.

The proof of this relies heavily on the methods of Mandl (see Refs. (5) and (3». Here we content ourselves with a formal explanation of what is going on. differentiate with respect to

A the equation

First we

8

yielding

= -(Hr+A-E)-lf If we assume

v(x) = [(Tx(a»

the desired conditions on

lim HO

d J\[

(42)

E_ A

{exp(-hx(a»)

the last identity leads to

v.

To see this refer back to the results of the last section for

-lim HO where

(H

r

_E)-l

Recalling that

using

-1 (H -E) fE(a) r fE(a)

[{exp(-h (a»} A

x < a,

X

(43)

is an integral operator whose boundary conditions are to be found. L

-f

-J (H -E)f r E E,

we obtain v(a) = 0,

for

v(x)

(44)

The remaining boundary condition comes about because df E _ J\

v(x)

Differentiating, using the facts that formally at least, for

A I 1,=0 (x )

fE(x) -TO

(45)

and

-TJ

as

X-TO,

gives

x --4 mne

1 2 -2x (x + 1 + 2x) e dx

10

-18

secs.

(51 )

Experiment gives: Expected Decay Time for which is consistent.

+ -12 P rr < 10

sees,

This begs the question as to whether or not one can determine

the expected first hitting times for excited states. Denote by

al,l

the first zero of the radial wave function for the first excited

state for the Coulomb problem. i.e. the first zero of

the Laguerre polynomial.

10

We

let

-I

lE+(T(a))

denote

lE(Tr(a)lrc (a,al,l)

IS

distributed according to the

probability distribution of the first excited state) and

the corresponding

expression for a diffusion initially distributed according to probability distribution of first excited state on

(O,a).

Then the methods above yield:

PROPOSITION 4 For the first excited state for the Coulomb potential

2 8a o

----C-_-o-

(I -7e

2

f

)

I 1

a/2a

a

(I -p)

2

-Ze / I JS I

1 -p 7e -(2-p) 2 -} dp 2 {( P 3 + P + 1 +-)e 2p 2p

(52)

and

(53) (I - 7e

Further details of the above calculation are given in Batchelor's Ph.D. thesis. It is clearly going to be very difficult to compute analogues of the above for the nth excited state.

We therefore resort to asymptotic methods.

(8»

.

4.

Asymptotic Results The

n

th

(See Refs.

(5) and

excited state, with zero angular momentum, for the Coulomb problem

has radial wave-function -x I 2x x exp ((n+ I)a )Ln((n+ l)a

fE(x)

being the Bohr radius,

o 1 L n

a

n

0,1,2,...

(54)

a Laguerre polynomial, the energy level

a. l,n

The zeros of the Laguerre polynomial, labelled by 2(n+I)2 increasing order, are unattainable points so internodal regions (a.

1,n

non-communicating (see e.g. Ref. (I». to find first hitting time of

a - 0

We work with wave-function for each energy level

En

f

E

,a.

I

l+,TI

on

)

in

are

(O,a] ,n)

for zero angular

momentum. We follow the methods of Mandl and Newell (see Refs. yea) t=

as

a .. 0,

with

(5) and (8)

and seek a

11 AT (a)

E{exp(for each fixed

A > O.

That such a

(I + A)

)} y

-1

(55)

,

exists can be seen from the results of the

last section. From Eq , (35), since

lim hE-A (x )

=

HO

f (x ) , E

for

AT (a)

7

>

a,

we obtain

fECal

1 im

1 im E{exp (- y a) )} aW

x

(56)

h __A_ (a) a+O E

yea)

and from de l'Hopital's rule

lim a+O

r;: __A_(a)

-

yea)

(57)

-

Hence, we obtain (J + A)

if

y

-1

(58)

,

is defined to satisfy dh E _ d>J

ll

\ _ (a) y , (a) ll-O /(a)

The following (5) and

(59)

-1.

f

a minor extension of a result of Mandl and Newell (see Refs.

(8».

PROPOSITION 5

For the radial Nelson diffusion corresponding to the eigenfunction

fE'

restricted to

(O,aI,n)'

yea)

2

n

and for each fixed

aI,n f 2 fo E

(x ) dx

fa f

n

x AT (a)

lim E{exp(a+O

Let us see why above

nth eigenvalue

y

)}

(J

is consistent with our Eq.

+ A)

-1

(59).

•

Firstly, if

E

n

2

E

n

and

(y) dy

12

(I+A)-I,

for small

A,

it is necessary that, as

I

dh I A=O(a) liE-A

dh -liE-A A=O (x) f

+

(x )

E

a-O,

(60)

fECal

Therefore, our equation reduces to y , (a)

-2-

I

for a function

as

a

0,

(6 J)

Evidently this equation is incapable of determining the

c(x).

multiplicative constant

2

gives f

y'(a)

-2-(

f

a

a J, n 2 fE(x) dx.

However, observe that

r )0

-2 (a) E

2[

a

y (a)

as it should, since

fECal (y(a) + c(x)) ,

dh dAE- A A=O(a)

y (a)

-2 E

fECal

(x)dx)

O.

2

2

f

r )

f

-2 E

(x) d x

-3 (62)

a f

yea)

f

-2 E

(x) dx f

-2 E

(a)

The multiplicative constant

Formally then for the radial diffusion corresponding to the

lS

determined by Eq.

nth eigenfunction

f

(60).

E

n

(63)

2

f

a I,n 2 (x) d x E

fo

n

fa f; 2

The following proposition was

(y) dy ,

n

discovered by Andrew Batchelor on the computer. PROPOSITION 6 lim l'a.m -0-yn(a) nt oo a-l-O y (a)

j

J

16

1,1 3

f

0

j"l being first zero of first Bessel function

for non-zero angular momentum states:

dx,

(64)

Jj •

Further details are given in Refs. (2) and (3). tabulation is given at the end of this paper.

2

x (Jj (x )

Batchelor's computer

There is a corresponding proposition

13

(65)

Reinstating

and

m,

we obtain:

PROPOSITION 7

For the radial Nelson diffusion corresponding to above radial wavefunction 2 8mao [.£. + I) U+4 lao) u+! Y o(a) - - - - (2.£.)! 0,"2 a

(66)

Moreover,

I (2.£. + I)! 8 (2.£. + 2) (.£. + I) U+3

jj,2.£.+1

being first zero of

J

Jj

I,U+1

3

2

x (J 2.t+ I (x ) ) dx ,

(67)

0

2.£.+1 Bessel function.

possible Experimental Tests Revisited Calculations show that for the first few excited states the above asymptotic results give reasonably good agreement with exact values as calculated by Eq. for

+ P n ,

for instance. for

unity.

P

+ IT

This gives, as

Also, one can calculate the numerical value of

system with lowest possible energy and with angular momentum

h

a - mlTc '

['£'=1 (T(a))

4.2 x 10-

which when compared with actual decay rates of accessible to experiment. as the

P

S

(52)

12

10-

se c s ,

12

secs looks as if it might be

Here though one has to guard against being too optimistic

transition rate may be too great (see Ref. (4)).

Nevertheless

stochastic mechanics has an important role to play here in suggesting corresponding results in quantum mechanics.

14

Batchelor's Computer Tabulation Zero Angular Momentum Case n

n 0 lim y (a)/y (a) aW

I

0

1.00000

1

.42122

2

.38735

3

.37686

4

.37221

5

.36973

6

.36826

7

.36731

8

.36666

9

.36620

10

.36586

11

.36560

12

.36540

13

.36524

14

.36511

15

.36500

16

.36492

17

.36484

18

.36478

19

.36473

Note that: I

j

f

T6 0

I, I 3

2

u (J 1 (u)) du

.36424

(5 d.p.)

ACKNOWLEDGEMENT It is a pleasure to thank Professors John Taylor and David Williams at Cambridge for helpful conversations. REFERENCES [1]

Albeverio, S., Blanchard, Ph., Hoegh-Krohn, R. (1984): Newtonian Diffusions and Planets with a remark on non-standard Dirichlet forms and Polymers. In 'Stochastic Analysis and Applications', Proceedings, Swansea 1983, editors

15

A. Truman and D. Williams, 1 - 25. Verlag.

Lecture Notes in Maths. 1095, Springer-

[2J

Batchelor, A., Truman, A., First Hitting Times for Excited States, in preparation.

[3J

Batchelor, A., Truman, A., First Hitting Times in Stochastic Mechanics, in preparation.

[4J

Bethe, H.A., Leon, M. (1962): Negative Meson Absorption in Liquid Hydrogen. Phys. Rev. 636 - 647.

[5J

Mandl, P. (1968): Analytical Treatment of One-Dimensional Markov Processes. Springer-Verlag, New York.

[6J

Nelson, E. (1967): Dynamical Theories of Brownian Motion. Princeton: Princeton University Press.

[7J

Nelson, E. (1985): Quantum Fluctuations. Princeton University Press.

[8J

Newell, G.F. (1962): Asymptotic Extreme Value Distribution for One-Dimensional Diffusion Processes. J. of Math. and Mech. 481 - 496.

[9J

Truman, A. (1986): An Introduction to the stochastic mechanics of stationary states with applications. In 'From local times to global geometry, control and physics', editor K.D. Elworthy, 329 - 344. Pitman Research Notes in Maths. Series 150. Longman Scientific and Technical.

Mathematical Notes.

Princeton Series in Physics.

Princeton:

[10J Williams, D., Rogers, L.C.G. (1987): Diffusions, Markov Processes and Martingales, Volume 2 It$ Calculus. Chichester: John Wiley.

LIMIT THEOREMS FOR STOCHASTIC PROCESSES ASSOCIATED WITH A BOSON GAS M. van den Berg Department of Mathematics, Heriot-Watt University Riccarton, Edinburgh EH14 4AS, United Kingdom J.T. Lewis Dublin Institute for Advanced Studies 10 Burlington Road, Dublin 4, Republic of Ireland

§1 INTRODUCTION In thIs lecture, we discuss the density of particles having energy less than The notation is that of in a boson system as a stochastic process indexed by [1] in this volume. Recall that the hamiltonian for the free boson gas is given by (1.1 ) R (2) where 0 = R (1) grand canonical pressure

p

R

(M) = -

1

Rn

I3V R

{

L

WEn

e

... PR(M)

For a system in a region of volume is defIned for M < 0 by

S(MN(W)-HR(W»}

the

(1. 2)

•

In [2] in this volume, we recalled results (proved in [3]) on the existence of the pressure in the thermodynamic limit: (1. 3)

In order to discuss the phenomenon of boson condensation, we introduced in [3] the family of random variables {XR(' 0) defined by X( R

)

L

1

= V

R

(1. 4)

o·(w).

(j :A(j )o) J R

For the free boson gas, we have the following result:

THEOREM 1 Suppose that

(51) and (52) hold;

then, for

Pc:

finite,

[Conditions (51) and (52) and the critical density Pc are defined in §2 and §3 of Here [ .] denotes the expectat i on taken with respect to [2] in th i s vo 1ume. the grand canonical probability measure with M = MR(P), defined in §3 of

17

[2)1

p.)

it is the expectation at fixed mean density

Proof: From the definition of

=

XR(A),

J P'(ILR(P)IA)dFR(A)

we have

=

[O,A) But, for

< A,

It

(1.6)

P - J p'(ILR(P)\A)dFR(A). [A,OO)

the sequence

{J

p ' (ILI A)dF (A ):R

(1. 7)

= 1, 2, ... }

lA," ) converges uniformly in

IL

on compacts to

Jp' (1LIA)dF(A).

(1.8)

[A. (0) Hence. by Proposition 2 of [2), we have for Hm..

A

>

0:

P - Jp(IL(A)IAldF(A).

=

(1.9)

[A," ) But, by hypothesis,

Pc

is finite so that we may invoke the dominated convergence

principle to conclude that P

< Pc'

P

Pc'

Thus we have

In the free boson gas there Is a second effect, discovered by M. Kac in 1971. We saw in §3 of [pc''');

[2) that the free-energy has a first-order phase-transition segment

it follows that for

P>Pc

there is no guarantee that the weak law of

large numbers will hold for the distribution

=

XR

that

=

N/V R • =

In

1.2 •••• }

fact,

there

is

no

guarantee

of the number density for.

o > Pc-

the

subsequence will converge.

but the 1 i mit d i str i but i on will depend on the deta i 1ed

behaviour of the corresponding subsequence of the sequence other

words, =

it

is

1,2, ... l ,

density of states

sequence

will converge 1 nevertheless. by the Helly Selection Principle, a

possible

to

have

two

sequences.

{A R(' ):R

= 1,2 •... }.

In

1,2, ... }

and

=

each satisfying (51) and (52) and having the same integrated

F(')

but having limit distributions

lKP and

JRp

which

are

18

distinct for P > Pc' (For p < Pc' they must both be equal to 5p ' the degenerate distribution concentrated at p, by Theorem 1 of [1].) For example, Kac showed that in the standard example (described in §3 of [2]) the limit distribution is the exponential distribution supported on [pc'''') with mean p, for p > Pcl other examples are investigated in detail in [3J. We shall see in the next section that, in the mean-field model, this phenomenon disappears: there is no first-order phase-transition segment, the grand canonical pressure exists for all values of and is a differentiable function; the weak law of large numbers holds for XR for all values of the mean density PI nevertheless, condensation persists. In these circumstances it becomes interesting to regard XR{' as a stochastic process and to enquire about the convergence in distribution of a re-scaled, centred version of it. This we do in §3.

§2 THE MEAN FIELD-MODEL To describe =

1,2, ... ) =

with

the

mean-field

model,

we

define

a

sequence

of

hamiltonians

by

HR{w} +

2VR

a > O. The term

N2(w)

2VR

{2.1l

'

which provides a crude caricature of the interaction,

can be understood classically, it arises in an "index of refraction" approximation in which we imagine each particle to move through the system as if it were moving in a uniform optical medium and so receiving an increment of energy proportional to the density XR = N/V R 1 since a is positive the interaction is repulsive. First, we compute the pressure PR (JL), as explained in §4 of [1], writing - %x2 ,

u tx ) =

a straight-forward manipulation gives (2.2 )

for each a < 0, where = But x u Ix ) is continuous and = 1,2, ... ) satisfies the Large Deviation Principle with bounded above and rate-function ra(x} = p{a}+f(x)-etx, by Theorem 1 of [2]. Hence, by Varadhan's First Theorem, =

=

RoO ee

pIa} + SUp{u(x) - Ia(x»

x

where the mean-field free-energy

fIx}

=

exists and is given by

fIx} +

2

=

f(')

SUp{JLX - f(x}},

x

(2.3 )

is given by (2.4 )

19

Thus we have proved:

THEOREM 2 Suppose that and is given by

(51) and (52)

hold; the II the mean-field pressure exists for all real

(2.5)

P(IL) - SUp{ILX - f(x)}, x where x ---+ f(x) is the mean-field free ellergy, given by

f(x) - f(x) +

Next, we introduce the mean-fieid expectation functional -

J

E1['e

=

13M

13M

R

IL

2

defined by

R1,

(2.6)

and the associated probability measure

J, where (2.7)

COROLLARY The mean-field pressure of distribution 6p

functions

concentrated at

{VR}

---+ P(IL)

p' (IL)

0

and

is di f ferentiable for all values of converges weakly to

satisfies the

x

u. The sequence

the degenerate distribution

Large Deviation Principle with constants

IIL(x) = p(lL)tf(x)-lLx.

---+ f Ix ) is strictly convex for x < '" and x ---+ x2 is strictly convex for 2

Since Pc

p -

and rate-function

IL

o 0

x < Pc

and

constant

x < '" ,

the

function

for

x2 is strictly convex for o x < (0; hence there is no 2 first-order phase-transition segment; equivalently, IL ---+ P(IL), the Legendre transform of x ---+ f(X), is differentiable for IL < "'. It follows from Theorem 1 6p ' where p - P' (IL), and from Theorem 4 of [L] that of l I l that - 1,2, ... } satisfies the Large Deviation Principle with constants {V R} and rate-function IIL(.). Although the first-order phase-transition segment, which was present In the free energy function of the free-gas, has disappeared, the phenomenon of condensation persists: x

---+ f(x)

[(x) +

20

THEOREM 3 Suppose that (51) and (52) hold;

lim lim i./l-I[X I (>, ) ] 1-"· where

[.]

-

then. for

Pc finite. we have

(2.8)

(p-pc)+ ,

is the mean-field expectation functional and

P

=

p' (/I-).

Proof. First, we remark that an elementary exercise yields the following alternative formula for the mean-field pressure p(/l): (2.9)

where p{a) is the free-gas pressure. The idea of the proof of (2.8) is that we compute the cumulant generating function of since (2 .10)

we get tXt 0

-

where

- {s

lE R.'

,,[ 13sV RXR] ,

,,... e

is the mean-field expectation functionai for which the free-gas

hamiltonian has been modified by the addition of the term

I

considerations yield the formula

aas

sa·. J

These

(2.11 ) SaO

where

and p{als,Al

+

=

[0,

Jp{al S + "")

A standard argument, using Griffith's Lemma, yields the result . •

§3 FLUCTUATIONS IN THE MEAN-FIELD MODEL Fluctuations in

Xl = N/V I

in the mean-fieid model in the thermodynamic limit

21

were studied for the standard exampie, described in §3 of [2] in this volume, by Davies [4], Wreszinski [5], Fannes and Verbeure [6] and Buffet and Puie [7]. The mean-field model in the generai situation, where the oniy assumptions about the single-particle spectrum are that (51) and (52) hold, was investigated in [8]; we have summarized the results of [8] in §2 and now go on to investigate the fluctuations in X.Q' In fact, we do rather more, we regard ). -7 X.Q{)') as a stochastic process and prove a central limit theorem:

THEOREM 4

Let

Z.Q{).)

VV

=

2

{ X i { ). ) -

-

JEt[xQ{).)]}; then. for

r{).1').2)

is gaussian with mean zero and covariance

Z{).)

J/l

).111).2 - 1 +

where

1

IJ.

< apc'

Z.Q().)

(d)

Z{).),

2

(3.1 )

f p"{a{/l)I).)dF{).)

Jf =

where

given by

(3.2)

[0,). )

and

a{/l)

is the value of

a at which

inf

a aPe' we introduce (3.4)

and prove in analagous fashion:

THEOREM 6

W(>,)

r (>'1' >'2)

covariance

is a gaussian process with mean zero and

given by

(3.5) where =

fp"(OI>.)dF(>.>.

(3.6)

[>., In this case.

W(>.)

(3.7)

The method by which we discovered the representations may be of some interest. The stochastic differentiai equation satisfied by a process filtration (Ft) is discussed by Nelson [9], see also McGill [10].

Suppose that a process

Xt - Xs + then

(Xt,F t)

satisfies the stochastic differential equation

f o(u,Xu)dB{u) + f T(u,Xu)du, t

s

t

s

with

(3.8 )

23 (3.9)

and (3.10 )

Assuming that the processes Z(.),W(') satisfy stochastic differential equations, the corresponding coefficients 0 and T can be computed using (3.9) and (3.10), this is a routine exercise starting from the expressions (3.1) and (3.5) for the covariances since the processes are gauss ian. Obv ious time-changes then give the stochastic differential equatIons for a brownIan bridge and a brownian motion respectively.

REFERENCES [1)

J.T.Lewis: The Large Deviation Pr i ncip le Expository Account (in this volume).

in

Statistical

[2]

M. van den Berg, J.T. Lewis, J.V. Pule: Large Deviations and the Boson Gas, (in th l s voIume ) .

[3)

M. van den Berg, J. T. Lewis and J. V. Pule: A General Theory of Bose-Einstein Condensation, Helv. Phys. Acta, 59, 1271-1288 (1986).

[4)

E.B. Davies: The Thermodynamic Limit for an Imperfect Boson Gas, Commun. Math. Phys. 28, 69-86 (1972).

[5)

W.F. Wreszinski: Normal Fluctuations in some Mean-Field Models in Quantum Statistical Mechanics, Helv. Phys. Acta, 1£, 844-868 (1974).

[6]

M. Fannes, A. Verbeure: The Imperfect Boson Gas, J. Math. Phys. £1, 1809-1818 (1980 i,

[7]

E. Buffet, J.V. Pule: Fluctuation Properties of the Imperfect Boson Gas, J. Math. Phys. 24, 1608-1616 (1983).

[8)

M. van den Berg, J.T. Lewis, P. de Smedt: Condensation in the Imperfect Boson Gas, J. Stat. Phys., 37, 697-707 (1984).

[9)

E. Nelson: Dynamical Theories of University Press, 1967.

Brownian Motion,

New

Mechanics:

Jersey:

an

Princeton

[10] P. McGill: Seminaires de Probabiiites, XX, LNM 1204, Springer: Heidelberg 1986.

LARGE DEVIATIONS AND THE BOSON GAS

M. van den Berg

Heriot- Watt University, Edinburgh J.T. Lewis

Dublin Institute for Advanced Studies, Dublin 4

J.V. Pule University College, Dublin 4 and Dublin Institue for Advanced Studies, Dublin 4

§1 Introduction

In this lecture we review some large deviation results for probability distributions associated with the free boson gas and discuss briefly their application to models of an interacting boson gas. In §2 we describe the probabilistic setting; in §3 we review results on the free boson gas which we shall require; in §4, §5, §6 and §7 we summarize large deviation results in increasing order of sophistication; in §8 we sketch some applications.

§2 The Probabilistic Setting Our ultimate aim is to compute thermodynamic functions for certain models of an interacting boson gas. The physical relevance of these calculations will not be discussed here; we shall concentrate on the probabilistic aspects of the investigation.

n on which the models are defined is the space of terminating sequences of non-negative integers: an element w of n is a sequence The probability space

{wU) E N:j = 1,2,oo.} satisfying

n

wU)
..I(j) : j

=

1,2, ... lis an ordered sequence of real numbers associated with a

region Al of some Euclidean space R d : (2.3) The total number of particles N(w) is defined by

L CTj(W).

N(w) =

(2.4)

j2: 1

As in §2 of [1], we are in a position to define for /l < 0, the grand canonical measure

Pi[ . ] on

n and the grand canonical pressure PI(/l): Pi[w] =

e!3(p.N(w)-H/(w))

r;);---

--e-;;!3C;-;v,:-/ P-;/

(2.5)

where

PI(/l)

= (,BVi)-lln(I: e!3(p.N(w)-H/(w))).

(2.6)

WEO

Because of (2.3), both (2.5) and (2.6) hold for all /l < O. The mean particle number density Ei[XI], where XI = N/Vi and the probability measure

En .], denotes the expectation with respect to

Pi[ . ] is given by

(2.7) Using an identity known to Euler, we have exp ,BViPl(/l) =

II (1-

e!3(P.-A/(j)))-l,

(2.8)

j2: 1

so that we write

PI(/l)

= VI- 1 LP(/lIAI(j)).

(2.9)

j2: 1

where the partial pressure p(I-tIA) is given by (2.10)

26 Lemma 1. For each P


1((3) = J[O,oo) e-{JAdFI(A), we formulate conditions:

(Sl) exists for all (3 in (0,00)

(S2) 4>((3) is non-zero for at least one value of (3 E (0,00). These conditions are weak restrictions on the sequences; their verification in a particular instance can involve some hard analysis.

27

§3 Results Concerning the Free Boson Gas

In this section we review some results on the general theory of the free boson gas; the proofs can be found in [2]. Proposition 1. Suppose tbat (Sl) and (S2) bold; tben ihe following limits exist.

J-l < 0,

(1)

F(A) = lim FI(A).

(2)

1-'00

Tbey are related by p(J-l) =

f

1[0,(0)

p(J-lIA)dF(A).

Moreover, we bave p'(J-l)

= f

1[0,(0)

p'(J-lIA)dF(A).

The standard example is the following one: let hi = in AI with Dirichlet conditions on 8AI where {AI: 1 = 1,2... } is a sequence of dilations of a convex region in

R d which eventually fills out the whole of R d ; let c/(l) = c/(2) ::; ... be the eigenvalues of lu and put A/(j) = c/(j) - c/(l); then (51) and (S2) hold and F(A) = CdAd/2. Next we define the critical density Pc: if A -+ p' (0 IA)is integrable on [0,00) witb respect to F, put

pc =

f

1[0,(0)

(3.1)

p'(O IA)dF(A);

put pc = 00 otherwise. It follows from the dominated convergence principle that if pc is finite then

pc = lim

f

/LiO 1[0,(0)

p'(J-lIA)dF(A) = lim

f

dO 1[ - - - - -1- ml

+1

ml

)

o

m,+ m,+1

(4.8)

+1

and

P[Nz:::; no];::: P[Nz:::; ml +mz];:::

ml ml

+

(4.9)

mz

by Markov's Inequality. Hence

(4.10) D Returning to the proof of Theorem 1, it follows from Lemma 1 that rically distributed; applying Lemma 2 with N 1 = e,B(/L+t,j and ml + mz = V/y; thus ml + 1

=

0"1

and N z

= N -

0"1

is geomet-

0"1,

we have

(4.11)

31

for Vi

t. It follows that vI

1-+00

since, for y

Pc, j.t + t,

-t

(4.12)

= 0

liminf (3:,r

O. Thus we have, in both cases,

-p(!1) - fey) +!1y

liminf _(3l InKf[G]

Vi

1-+00

=

-IIl.(y)

(4.13)

for all y in G, since 8 was arbitrary. Hence liminf _(31 In Kf[G] 1-+00

sup( -IIl.(y))

Vi

G

(4.14)

= -infIll.(y) G

o §5 The Large Deviations of a Vector-valued Random Variable. The Large Deviation result established in §4 enables us to apply Varadhan's Theorems to suitable functions of X, = N /Vij to deal with functions of the m + 1 variables uI/Vi, ... , Urn/Vi, N/Vi we prove a Large Deviation result for the sequence of probability distributions of a vector-valued random variable. Define the vector-valued random variable X, : n - t Rm+l by

X I( m + 1) CW ) --

v.-I 1 "L.J i>m

In order to prove a Large Deviation result for Ki

u J. ( W) •

= proX/-1,

it is necessary to make

a further hypothesis about the single-particle spectrum. First, we define the cumulant generating function Gn . ] by

GIl.[t] = _I_In Ell.[e,BVi] I

(3 Vi

I

.

(5.1)

Lemma 3. Suppose that (S 1) and (S 2) hold and that liml--+oo AI(j) = A(j) exists for j = 1···, m + 1; then the cumulant generating function

GIl.[tJ = lim Gnt] 1-+00

32

exists for all t in R m +1 and is given by CI'[tJ = {p(J-l

+ t m +1 -

A(m

+ 1) -

00,

p(J-l) , t E VI'; otherwise.

.(5.2)

where VI' = {t: tj +A(j)

< -a,j

= 1···,m+

I}

(5.3)

Proof: Put (5.4) for 1 ::; j ::; m and put

(5.5) Since A/(j) -+ A(j) as 1 -+ 00, p;j)(J-l + tk) is defined, for all 1 sufficiently large, for J-l < A(j) - tj. On the set VI" we have, by Proposition 1,

(5.6) while for 1::; j ::; m,

(5.7) It follows that lim C/1'[t J = P( P, + t m +1 1 -+ 00

>,(m + 1)) - pep,)

put C I' [t] = 00 for t in the complement of V 1" Then t function on Rm+! with dom CI' = VI'; put

f-t

(5.8)

C I' [tJ is a closed proper convex

11'[x] = sup {< x, t > -CI'[t]}.

(5.9)

tER"'+'

Theorem 2. Suppose that (51) and (52) hold and that lim/-o oo A/(j) = AU) exists for 1 ::; j ::; m + 1; then, for p, < 0, the sequence

{Kt=P(oX/- 1 :1=1,2,···} satisfies the Large Deviation Principle with constants {Vi : 1 = 1,2, ... } and rate-function

]1'[ . J.

Proof. The proof that (LDl) and (LD2) holds follows, as in §9 of [1], by the fact that 11'[ . J is the Legendre transform of CI' [ . J. to prove that (LD3) holds, we follow Ellis [3J and adapt to our situation Gartners's Lemma:

33

Let K be a non-empty closed subset of Rm+! define II'[K] = inf K II'[x]. If 0 < JI'[K] < 00 then there exists a finite set -r(1), . . . , T(r) of non-zero vectors in Rm+! such that, for c = II' [K] - f, f > 0,

c

K

J=l HI'+ (T(j)·"c)

(5.10)

Ur

where Ht(T;c) = {x:< X,T > -GI'[t] c} if II'[K] = +00 then, for each R > 0, there exists a finite set T(j), . . . ,T(r) of non-zero vectors in R such that

(5.11) First suppose that K is such that 0 < JI'[K] < 00; then r

Kj[K] :::;

L Kj[Ht(T(j); c)] j=l

L Kj[{x :< x, T(j) > r

=

>

CI'[T(j)]

j=l

+ c}].

(5.12)

But by Markov's Inequality,

Kj[{x :< x, T(j)

GI'[T(j)]

+ c}] s

e-/H'i{c"[r(i)l+c} (

i-:

e,BVi00

VI

In KnC]

G

G

o

§6 A Large Deviation Result for a Banach Space-valued Random Variable Let XI :

n --t

be defined by - l', J. >

X I- 1 is a probability measure on = {Xj as the positive cone of the real Banach space [1

thenKt = Pi regard

0

;

0 : I:j>o Xj < oo}. We equipped with the norm

topology, [1 is a complete separable metric space (a Polish space). However, for our purposes, the weak--topology on [1 (the a( [1, co) topology induced by the space Co of

35

real sequences converging to zero) is the appropriate one for our purposes. The space [I equipped with the a(ll, co) topology is not metrizable; nevertheless, the theory of large deviations is still applicable since the a-field of Borel subsets of [1 is the same in both the norm topology and in the a(ll, co)topology (see Azencott [4] for a full discussion of this point and Yamasaki [5] for the measure theory). Notice also that each of the measures Kj is supported on the convex set {x E

Xo =

Xj} since N(w) =

[+ :

aj(w).

The proofs of the results in this section are more technical and we will not give them here. Lemma 4. Suppose that (Sl) and (S2) hold and that lim/->oo A/(j)

then, for Jl

< 0, we have for each t in CI![t] =

l

= 1,2, ...

Co

lim C"'[t] = {p(Jl

-+ 00

= 0 for j

I

00,

+ to) -

p(Jl),

t E V,..,

(6.1)

otherwise,

where

and

VI! = {t E Co : to + Jl < 0, to + sup tj Let J!'[x] =

SUPtfc

o

+ a: < OJ.

(6.2)

{< x, t > -CI![t]}; then a straightforward caculation yields

V;,

x E otherwise,

where

= {x

E

[+ :

Xo

I>j}.

Theorem 3. Suppose that (Sl) and (S2) hold and that lim/->oo A/(j)

1,2, ... ; then, for Jl

(6.3)

(6.4)

o for

j =

< 0, the sequence {Kj = Pi

of probability measures on and rate-function J!' [ . ].

o

X/- 1 : t-: 1,2, ... }

[+ satisfies the Large Deviation Principle with constants {'Vi}

36

§7. A Large Deviation Result for the Occupation Measure We introduce a measure-valued random variable

, where 8>.[B]

=

1 if >. is in B and is zero otherwise. Then L/ maps

n into the space

E = Mt (R+) of positive bounded measures on the positive real line. Let Kj = PiaLIl be the induced probability measure on E; in terms of this we can express the expectation of a functional of L/ as an integral over E. For example,

where G(m) =

and

Ilmil

=

fro,oo) m(d>').

But even in this simplest of examples

there is a difficulty in applying Varadhan's Theorem (supposing that we have established a Large Deviation result for {Kj}. It is this: in order to prove a Large Deviation result, we have to make use of the weak-e-topology on E determined by Co(R+), the continuous functions vanishing at infinity; but the function

m

f-+

IImll

is not continuous in this

topology and Varadhan's Theorem does not apply. We get around this difficulty as follows: we introduce a cut-offT and prove a Large Deviation result for Kj = Pi aLII where now

L

L/(w; B) = V;-l

O"j(w)8>,,(j) [B];

(7.1)

{j:>'j (j) ::S;T}

then we prove an estimate for Pi [xT

8] where

Xnw) = V;-l

L

O"j(w).

(7.2)

{j:>', (j»T}

We state these results without proof:

Theorem 4. Suppose that (Sl) and (S2) hold; then, for 11 < 0, the sequence {Kr =

Pi aLII} of probabili ty measures on M: ([0, T]) satisfies the Large Deviation Principle with constants {Vi} and rate-function 11'[ . ], given by [I'[m] =

sup {« m, t > -CI'[tl} C([O,Tj)

(7.3)

where

CI'[t] = {J[O,Tj{p(1l 00,

+ t(>.)I>.) -

p(lll x )} dF (>' ), SUP[O,Tj{t(>.) - A} < -11, otherwise.

(7.4)

37

Lemma 5. For 8 > 0 and T such tbat

1

T,oo

we beve

p'(JlIA)dF(A)
O}.

Remarks: (1) The results is independent of m for m ;::: 1, so that it is reasonable to conjecture that the same result holds for the pressure PI in the H-Y-L model with hamiltonian Hfoo); we hoped to prove this using Theorem 3 of §6, but, so far, technical difficulties have prevented us.

38

(2) No explicit assumption is made concerning the existence of liml_ex> AlU), while

= liml_ex> AlU), j = 1, ... , m + 1, occur explicitly in the statement of Theorem The reason is that inf {A : F( A) > O} = 0 implies that AU) = 0 for j = 1,2, ....

the AU)

2.

are equal. The H-Y-L model is a special case of the diagonal model [9] for which the hamiltonian is

(8.3) The last two terms in this hamiltonian have different asymptotic behaviour for large 1. To understand the effect of each of these two terms we study them separately. Therefore we consider the regularized hamiltonian:

HlR(w) = Hl(w) If we assume that v : R

ff

+

1 ;=1 j=l

v(.X'l(i), Al(j))O';(w)O'j(w).

(8.4)

R is continuous, bounded and positive then we can use

--+

Theorem 4 and Lemma 5 of §7 to obtain the following result which is proved in [10]. Theorem 6. Suppose that (Sl)and (52) hold; then the pressure pR(;.;.) corresponding

to the sequence of hamiltonians {HI:} is given by

pR(;.;.) =

{;.;.llmll-

sup

fR[m]}

(8.5)

mEMt(R+)

where

fR[m]

=

1

[0,00)

Am(dA) +

_/3-1

1

1

[0,00)

m(dA)

1

[0,00)

s(p(A))dF(A)

[0,00)

m(dA')v(A, A')

(8.6)

and

s( x)

= (1 + x) In(l + x) -

x In Xj

(8.7)

here

medAl

=

+ p(A)dF(A)

(8.8)

39

is the Lebesgue decomposition oEm with respect to dF(>").

References

[1] J.T. Lewis: The Large Deviation Principle in Statistical Mechanics: An Expository Account, (this volume). [2] M. van den Berg, J.T. Lewis and J.V. Pule: A General Theory of Bose­Einstein Condensation, He1v. Pby«. Acta, 59, 1271 ­1288 (1986). [3] R. Ellis: Entropy, Large Deviations and Statistical Mechanics, New York: Springer 1985. [4] R. Azencott: Grandes deviations et applications, Ecole d' Ete de Probabilites de Saint­Flour VIII ­ 1978, 1 ­ 176, LNM 774, Berlin :Springer 1980. [5] Y. Yamasaki: Measures on Infinite Dimensional Spaces, Singapore: World Scientific 1985. [6] M. van den Berg and J.T. Lewis: Limit Theorems for Stochastic Processes Associated with a Boson Gas, (this volume), [7] M. van den Berg, J.T. Lewis, J.V. Pule: The Large Deviation Principle and some models of an interacting Boson Gas, to appear in Commun. Math. Phys. [8] K. Huang, C.N. Yang, J.M. Luttinger: Imperfect Bose gas with hard­sphere interactions, Phys. Rev., 105, 776 ­ 784 (1957). [9] D.J. Thouless: The Quantum Mechanics of Many­ Body Systems, New York: Academic Press 1966. [10] M. van den Berg, J.T. Lewis, J.V. Pule: (in preparation).

STOCHASTIC MECHANICS OF FREE SCALAR FIELDS

by Eric A. Carlen" Princeton University Princeton NJ 08544 U.S.A.

INTRODUCTION Here we discuss and develop stochastic mechanics in the treatment of quantum field theory. The subject is in its infancy, so we restrict our attention to free fields, where already difficult and interesting questions arise. The paper is based on the fact that, just as in the stochastic mechanics of finite particle systems, there is a direct relation between the stochastic mechanical description of a field theory and the Schrodinger representation for the dynamics of the corresponding quantum field theory. Either can be recovered from the other, and this suggests two kinds of investigation. First, if we have a nice Schrodinger representation for the dynamics of a quantum field theory, we can enquire after the sample path properties of the corresponding diffusions in stochastic mechanics. Of course this is the case with the free scalar fields. A very interesting question arising in this direction is how the diffusions corresponding to the single particle states in quantum field theory tend in the non relativistic limit to the familiar free particle diffusions in stochastic mechanics. We will return to this question, and others, later. The questions arising in this kind of investigation seem to be either very easy or very hard, and for the most part we will only answer easy questions here. Second, and perhaps more interesting from a physical point of view, we can undertake to directly construct the stochastic mechanics of an interacting field, and then to use the correspondence with the Schrodinger equation to construct a self adjoint Hamiltonian for the corresponding quantum field theory. One may object that in the most interesting case of four space­time dimensions, one expects difficulties with the type of Schrodinger representation that works in lower dimensions. The simple connection between stochastic mechanics and the Schrodinger equation used here may then require modification, or it may just plain break down. On the other hand, stochastic mechanics provides a probabilistic approach to constructing quantum fields that proceeds in real time­there is no analytic continuation from Minkowski space to Euclidean space involved. This may prove useful on curved space­times where such analytic continuation is impossible.

*

Research partially supported by an N.S.F. postdoctoral fellowship grant

41

Unfortunately, the interesting problems ansmg in this kind of investigation seem very difficult at this time. We therefore stick to the first kind of investigation here. Nelson has discussed some of his ideas regarding the second kind of investigation in his Ascona lectures [2]. Some familiarity with the Nelson's book [1] may be useful in reading what follows, though an aquaintance with diffusion theory and elementary quantum mechanics from any source will probably suffice. In the first section of the paper we develop the connection between stochastic mechanics and the Schrodinger equation on a Gauss space. This permits us to make the transition to infinitely many degrees of freedom. The second section is devoted to developing the Schrodinger representation for the scalar free field of mass m in considerable detail. Some of the results here may be of intrinsic interest, though cartainly they are not deep. Finally the third section presents some results on the stochastic mechanical diffusions corresponding to single particle excitations of the field. The results we discuss all follow fairly easily from the results of the second section. Several harder and more interesting questions are raised but not answered. At the present state of the subject, I must hope that the paper will be interesting not only for the new devlopments within, but also for the exposition of some beautiful ideas of Nelson which still await their full development.

STOCHASTIC MECHANICS ON A GAUSSIAN SPACE Let us begin by recalling some results about diffusions in R N making the modifications that follow upon everywhere replacing Lebesgue measure on R N_ which doesn't have an infinte dimensional analog­with a Gauss measure­which does. Let C be a positive definite operator on R N . The Gauss measure with covariance C is We put 1 1 ) A c = "2(\7-Cx.\7

and call this operator the Ornstein- Uhlenbeck generator with covariance C. The unique process generated by A c with pc(x)dx as its invariant measure is called the ground state process with covariance C. Now consider any diffusion t f-+ in R N generated by

9

= Ac

+ b(x,t).\7

where b is a smooth bounded time dependent gradient vector field on R N . Then of course is smoothly distributed with respect to pc(x)dx. Let p(x, t) denote the corresponding density; that is p(x, t) is defined by

42

Pr{ t E S} = for any Borel subset S of R N . To say that 9 is the generator of t function f on R N X R,

1

p(x,t)pc(x)dx

t means that for any smooth, bounded

f-*

The limit taken with the increment in the other time direction also exists and defines the backward generator 9*:

There is a simple expression giving analogous result in [1], just write

9=

1

9* in

terms of

9

and p. To read this off the

1

+ (b(x, t) - 2C - I X ).\7 .

Then it follows from the formulae on page 32 of [1] that

9* =

1

+ (b(x,t)

1

- 2"C- Ix).\7 - \7log(p(x,t)pc(x)).\7

But \7log(p(x, t)pc(x)) = \7logp(x, t) - C-Ix, so

9*

= -A c

+ (b(x, t)

- \7logp(x, t)).\7

The change of reference measure hardly changes the form of the corresponding formula in [1]. Again with hardly any change in form from the analogous definitions in [1] we define the time dependent vector fields u, v and b; by 1

u = 2\7logp,

b= u

+ v,

b;

=v -

u

= b-

\7logp.

Note that with these definitions,

and

u

1

= 2(b* -

b).

As usual, we call v the current velocity and u the osmotic velocity. The operations D and D* are stochastic time differentiations, and combining them in the following way, we have Nelson's definition of the stochastic acceleration a( t) :

43

Clearly one computes +DD*) by evaluating + at This is easy to do. 9x = + b and so 9*9x = %tb - iO-2x + Acb + b*.\7b. Averaging this with the analogous computation for 99*x yields: a ()

1 4

+ -0

a = -v at

-2

1 2

Acu + -0

-1

u + v.\7v - u.\7u

(We have eliminated band b; in favor of u and v, and the right hand side is evaluated at t).) If we specify by equating it to an external force, as in Newtonian mechanics, we have an evolution equation for v in terms of u and the given force. We can get an equation for u from the evolution equation for the density. Adapting formula (5.24) of [1] to our Gaussian reference measure we see that the density p satisfies the continuity equation:

a P = -\7.(vp) at

+ (0- 1 x.V ) P

· a _l'7(-la) Smce at u - 2" v P at P ,

a

-;::lu

ut

= -Acv

1 -1 + -0 v2

v.\7u - u.\7v

Once a force is specified, say by setting = we have a coupled pair of equations for u and v. Once we have solved them (for some chosen initial conditions), we have the drift field b = u+v of a diffusion satisfying HD*D+DD*) = - \7vtot . Of course it still remains to solve a stochastic differential equation for the diffusion itself. Nelson discovered that with the force specified as above, it is easy to solve this non-linear coupled system of equations because a change of dependent variable transforms it into the Schrodinger equation. Suppose V is smooth bounded function on R N and V; satisfies the Schrodinger equation i :t VJ(x, t) = (-A c

+ V(x))VJ(x, t)

I

Suppose also !VJ(x,tWPc(x)dx = 1 for some, and hence all, t. functions Rand S by VJ(x,t) = eR(x,t)+iS(x,t)

Define the

and, temporarily abusing notation, define the vector fields u and v by u = \7R and v = \7 S. Then the real and imaginary parts of the Schrodinger equation yield the following pair of equations for u and v:

a

-;::lV

ut

1 -1 = Acu + u.\7u - v.\7v - -0 u - \7V

2

44

a

-;-U = -Acv - U.\JV - V.\JU ut

1 -1 + -C v

2

This equation for this u is the same as our previous equation for our previous u. And if we identify

the two equations for the two vector fields v coincide. This is the same as putting a(Et) = -V'Vtot(Ed where Vtot(x) = V(x) + ix.C-1x. Working in the other direction, we define R(x, t) = t)) so that u = V'R. Since by assumption b, and hence v, is a gradient, we define S(x, t) + a(t) by v = V'S. The notation explicitly takes into account the fact that for each t, 5 is only defined up to a constant. One then easily sees that if u and v satisfy the equations (1.5) and (1.6), and if a(Ed is defined by (1.9), then a proper choice of 5 can be made so that 'Ij; defined in terms of Rand 5 by (1.8) satisfies the Schrodinger equation (1.7). This proves the following result:

Theorem 1.1

A smooth diffusion generated by

9

=

a at + A c + b.\J

with density p satisfies Nelson's equation

exactly when there is a solution of the Schrodinger equation i :t 'Ij;(x, t) = so that

(-A c + V(x))'Ij;(x, t)Et

V''Ij; \J'Ij; b = Re- + Im'Ij;

'Ij;

and

p(x, t)

= 1'Ij;(x, t) 12

Now let us consider the infinite dimensional case, first on an informal level. We substitiute L 2 (R d - 1, dx) for R N and take C = + m 2)- 1/ 2. Here d - 1 2: 2 and m > O. Then when V = 0, the free case, we have an equivalence between solutions of

45

and diffusions satisfying

If the paths of t I-t were smooth, our stochastic time derivatives would coincide with the usual time derivatives, and we would have

and so t

I-t

would then satisfy the Klein-Gordon equation

Of course the paths are not smooth. There is all the same a very interesting connection with the Klein-Gordon equation [2]. First consider the ground state process for this covariance. This does exist as a nice diffusion, although it only has continuous sample paths in the completion of L 2 (R d-l, dx) in a weaker norm. It is thoroughly investigated at the sample path level in Carmona's paper [3] in a context we will develop more carefully later in the paper. The results of Albeverio and Hcegh-Krohn [4] also provide a nice realization of the ground state process. For now we simply state that it follows from these authors' results and the computations sketched above that if we choose an appropriate norm on the state space, the ground state process does actually solve the stochastic mechanical equations of motion. Now let ¢(x, t) be a smooth solution of the Klein-Gordon equation:

Consider t I-t ¢(", t) as a path in the state space, and define a new diffusion t by the Cameron-Martin shift

I-t

= + ¢(',t) Of course the transformed diffusion has a generator of the form we are discussing, and since the equation (*) is linear in and since the deteministic process t I-t ¢(",t) satisfies this equation, so does t I-t So we can produce a large family of solutions to our stochastic mechanical equation of motion just by adding solutions of the Klein-Gordon equation to the ground state process. The corresponding solutions of the Schrodinger equation are the coherent state solutions of the Schrodinger equation for the free quantum scalar field of mass m. We will write down these wave functions in the next section after we have carefully formulated the Schrodinger representation for the quantum dynamics of the free scalar field.

d.

46

Guerra and Ruggerio [9] were the first to investigate the stochastic mechanics of free scalar fields. They put cutoffs on the classical field theory to obtain a system of harmonic oscillators; they then treated these stochastic mechanically. The minimal energy process so obtained is Gaussian with mean zero and an explicitly computable correlation. Removing the cutoffs at the level of the correlation, they identified the minimal energy process as the ground state process of the corresponding Euclidean quantum field theory. Our approach to the subject avoids cutoffs ­ which are not needed in the free case. While it provides a slightly more direct route to the result of Guerra and Ruggerio, its real advantage is in treating the processes corresponding to N­particle excitations of the field. It is when looking at these N­particle diffusions that we should expect to see interesting sample path properties. While the sample path properties of the coherent state diffusions are related to the sample path properties of the ground state diffusion in a transparent way, much more interesting­but rather difficult­questions arise in the consideration of the diffusions associated to N­particle excitations of the field. Before we can discuss these, we need to produce the corresponding solutions of the Schrodinger equation; we do this in the next section. Before leaving this section, it is useful to rewrite some of our results in the form they assume upon changing variables so that our Gauss measure becomes a unit Gauss measure. This is easy to do. Let new variables y be defined by y

= C- 1 / 2 X

so that

pc(x)dx

= PI(y)dy

Under this change of variables, A c becomes

A c = 2"1 ( '\7 y A

-

Ac

where

)-1 '\7 y y.C

Continuing to use a hat to denote the change of coordinates, we have b = C- 1 / 2b and i, = C- 1 / 2b*. Defining u and v in terms of band as before, we get yet another power of C- 1 / 2 when we express u in terms of p since u is a gradient:

t.

A

_

U ­

lC­ 1r7 VyP 2 A

­

Similarly, the continuity equation becomes

a

at p = ­ (('\7 y.v)

+ 2(Cu.v) -

(y.v)) p

47

and the formula relating stochasic mechanical diffusions to solutions 'ljJ of the Schrodinger equation becomes

b(y, t) =

c:' (Re \l y'ljJ(y, t) -+- 1m \l y'ljJ(y, t)) 'ljJ(y, t)

'ljJ(y, t)

Unfortunately, when V is not quadratic, there is no simple way to directly produce a large family of solutions to our stochastic mechanical equations of motion. Without linearity in (*), one must work much harder.

THE SCHRODINGER EQUATION FOR THE FREE FIELD The quantization of the classical scalar field obeying the Klein-Gordon equation

is usually carried out in terms of a second quantized occupation number representation. While this is satisfactory for the computational purposes of orthodox quantum field theory, it is less than satisfactory for our purposes here. We must represent each quantum state with a wavefunction I]i on a Riemannian manifold Q, the configuration space of our field system. Only then can we develop the connection between wavefunctions on Q and diffusions in Q as we did in the last section. For this reason, we must construct the Schrodinger representation for the quantum mechanics of the scalar free field of mass m in a little more detail than usual. This is easy to do, and it may also be useful for other purposes. First it is important to emphasize the role of the Klein-Gordon equation in this endeavor. The Klein-Gordon equation is not, despite some early confusion, a relativistic generalization of the non-relativistic free Schrodinger equation. Rather, it is a kinematical equation. The Klein-Gordon equation

stands in the same relation to Schrodinger equation

at = Hw for the quantized field-we will soon define the terms in this equation-as does the Newton equation for a free particle in R 3

48

to the Schrodinger equation .8 dt

1 2m

t) = - -fl'IjJ(x, t)

for a free particle of mass m in R 3 . In all cases, the kinematical equation is used first to define the classical phase space of the system. As a set, this is just the space of initial data for the kinematical equation. Since the particular kinematical equations we are interested in come from Hamiltonian dynamical systems, the initial data space has a natural symplectic structure. With this symplectic structure, it is the classical phase space of the system. The next step is to identify the phase space, as a symplectic manifold, with the cotangent bundle T*(Q) of some Riemannian manifold Q , the configuration space of the system. (Q will be a Hilbert space in the special case we consider here.) The problem with this step­one of the reasons quantization is not an algorithm­is that there may be no unique or even distinguished way to represent the phase space as a cotangent bundle. Even worse, there may be no way to do this consistent with certain symetries of the phase space. This unfortunate circumstance occurs already with the free quantum fields, as we will soon see. One then chooses a measure on Q. The complex Hilbert space L 2 (Q) is the quantum mechanical state space. Regarded as a real Hilbert space, it has a natural symplectic structure given by the imaginary part of its inner product. The quantization is finished by writing down a dictionary relating the symplectic structures of T*(Q) and L 2(Q)-it is here that representations of the canonical comutation relations are of interest­and then translating the dynamics from the first setting to the second. A clear and thorough discussion of quantization at this general level can be found in Segal's book [5]. In our special case, we will be able to translate the dynamics in a fairly transparent way, and so we will not have to worry very much about symplectic structures and representations of the canonical comutation relations. But that is only because we are restricting our attention to free fields here. We now proceed with the actual quantization of the scalar free field. First, define the single particle Hamiltonian H by

H

= (-fl + m 2)1/ 2

Then ((%t)2 ­ fl + m 2) = (i%t + H) (-i %t + H). And so if q\(x, t) is any smooth solution of the Klein­Gordon equation with compact support in x for fixed t, we can naturally decompose q\ as

q\(x, t) = (q\+(x, t)

+ q\_ (x, t))

49

where In fact, Fourier transforming in the spatial variables, we have

where

a: a+.

w(k) = J(P

+m2)

and since rP is real, = Actually, it is easiest to take the Fourier transform in the space and time variables first. Formally one gets a multiple of 8 (w 2 - k 2 - m 2 ) , and then since 8(w 2 - >.2) = 2\. (8(w + >') + 8(w - >')), integrating over w yields the above expression. Clearly then

so that

and the right hand side is clearly Lorentz invariant (since wtk) dk is the invariant measure on {(k, w) : w2

-

k2

= m2} ).

Also note that

and or

rP

= RerP+

and

H- 1 J;

= I nub.;

so we can easily recover the Cauchy data from rP+. This leads to the following result, which is proved in more detail in [5], that if we equip the space of real, smooth, compactly supported (for fixed t) solutions rP of the Klein-Gordon equation with the Hilbertian norm

then the completion is a real Hilbert space ).Ir, and the natural action of the Poincare group on the smooth solutions extends to an orthogonal representation. The representation is irreducible, but we will not use this fact here. It is clear from the above that ).Ir posseses a natural complex structure, even though we are working with real solutions. Namely,complex multiplication acts on rP+ in the obvious way. Note however, that this complex structure depends on the choice of the t = 0 hyperplane. We define ).I to be this complexification of

50 ){r' ){ is called the single particle Hilbert space, and it, equipped with its standard symplectic structure is the phase space of our system. Note that time inversion C : ){ f-+ ){ defined by C¢(x, t) = ¢(x, -t) is a complex conjugation. Let Q be the eigenspace corresponding to the eigenvalue 1. That is, Q consists of solutions of the Klein-Gordon equation with ¢(x,O) = O. We identify Q with the real Sobolev space obtained by completing the space of smooth compactly supported functions ¢ in the norm II (-.6. + m 2)l/ 4¢ 11£2 (Rd-l). Clearly we may regard ){ as T* (Q) in a natural way, and also ){ = Q ffi i Q. Now let Pr be the unit Gauss measure on Q. That is, let B be a Banach space densely containing Q so that the norm of B is measureable over Q in the sense of Gross [6]. Let B be the Borel field of B. Then Pr is the Borel probability measure on B uniquely determined by the finitely additive unit Gauss cylinder measure on

Q.

Select such a B and put

that is, K is complex L2 of B, and loosely speaking, we will say of Q. The fact that B is a Banach space is important because it allows us to do differential as well as integral calculus. Other than that, the choice of B is quite immaterial. In fact, there is a natural isomorphism between the realizations of K arising from any two choices of B. This is a consequence of the fundamental result of Segal [6] that K is naturaly isomorphic to the symmetric tensor algebra over ){. We shall soon use this isomorphism in computations. In order to spell it out, we first need to recall the homogeneous chaos decomposition of K. Ko is defined to be the constant subspace of K. K I is defined to be the closed linear span of all the random variables of the form

e

e

where is in Q. This is only defined pointwise when happens to lie in the dense subspace B* of Q, but it is well defined as an element of LP(B, B, Pr) for all finite p by the obvious approximation procedure. And actually, as we have described it, K I is already closed. Clearly K o .L K I. We define K(n) to be the closure of all polynomials of degree n or less in the elements of KI . We then define Kn to be the orthogonal complement of K(n-I) in K(n)' It is then an easy matter to establish that

As usual, we use Wick dots to denote orthogonal projection from is if

K(n)

to

K n;

that

51

is a generic monomial in

K(n),

we use

to denote its projection onto Kn . The effect of the Wick dots can be explicitly expressed using Hermite polynomials. The nth Hermite polynomial h n is defined by

From the commutation equation [

d (-x - -d )] (-X2 + -), dx 2 dx

= 1

it is immediate that

and that the leading term of h n (x) is z ". Clearly then for any •.

in Q

(.,(

and since Iw(e, t)1 = JPl[t), we need next just find the phase of W. This can be done by extracting v, which is H\lS, from the continuity equation. Differentiating the above expression for P one finds

Since u is the constant (in e) vector field

u(e,t) = H¢;Ct) it follows from the continuity equation that

e

otP(e,t) = -(\l.v)p(e,t)

+ ((e - ¢;(",t)).v)p(e,t)

Simply differentiating our expression for P we find

and the equations agree if we have

v(e,t) = This determines S up to a constant; we have

In turn, this yields

55

which solves i

%t '11 =

Hw provided

This is solved by

This finally gives us w(

e, t) =

e!(U,( ,t)) Q-

tllet>( ,t)

+ t(E,H- 1 1>( ,t)) Q+f(et>( ,t),H- 1 1>( ,t)) Q

It is useful to express this another way. First let us extend the inner product on Q to be bilinear (not 8e8quilinear) over C in the obvious way. Then

and since II¢{, + IIH- 1 ¢{ , is a constant of the motion-namely we can rewrite our expression for '11 as •

III¢III2-

w(e,t) = Using another Hermite polynomial identity

(the Hermite identities above show that e- x 2/2 ,\nhn(x) and e-(x->..)2/ 2 2(R) both are L solutions of the same Sturm-Liouville problem) one sees that

which makes it transparent that '11 satisfies i %t '11 = Hw. Because of the last formula, one usually writes : e(E,et>+(,t))Q : for '11; this defines the Wick exponential. For some purposes, our first expression for '11, though less familiar, is more useful. For any pair ("l, IT) of elements of Q, define the coherent state vector '11 '1,"- (e) by w'1,,.-(c;) = Let {C;) : j E N} be an orthonormal basis for Q. For any N and any q and p in R.", let q and p also denote the elements

56

The following small result may be marginally new; however, it is closely related to a well known result about the Weyl transform [10]. Theorem 2.2 Let TN be the sigma-algebra u{(",6)Q, ... ,(",CN)Q} Then for any \II in K, the conditional expectation of \II given TN may be computed in terms of the coherent states according to the formula

where (q,p) = (,

\IIq,p) K

with the inner product being conjugate linear on the right. Proof For Gaussian spaces orthogonality implies independence. Note then that if 1.. L 2 (B, TN, Pr), then both sides of the equation vanish. Therefore, by a simple density argument, it suffices to prove the result when

where that

f

is a smooth bounded function on R N

.

To show this, we just need to show

(4rr)Nf(x) = {

({

JR2N JRN

dy(2rr)-N/2 e- y 2/2 f(y)e 1( y ·q ) 1 2 i i ( e2"1 ( x·q)-;rlql +2"(x.P)+;r q.p )

2 t lq I - t(y·p)- f(q·p)) X

dpdq

But the (q.p) terms cancel out of the exponent, and doing the p integration yields a 8 function in (x - y). Doing the y integration, the right hand side is reduced to

which is of course just f( x).

This result will not be used in this paper. We present its derivation as a simple example of how stochastic mechanics leads directly to interesting objects in the Schrodinger picture for the free field. However, it may have many uses. Coherent states can be used to provide an approximate diagonalization of perturbations of H just as in ordinary quantum mechanics one may use coherent states to produce

57

an approximate diagonalization of + V. Work is underway to determine whether useful spectral bounds may be obtained in this way.

SINGLE PARTICLE DIFFUSIONS By the results of the first section, the stochastic mechanical diffusion corresponding to a single particle excitation of the scalar free field of mass m satisfies a stochastic differential equation of the form = b(

t)dt

+ d.X,

where dXt is an increment of the ground state process and b is defined by

t) = u(6) +

t)

=

+ with \II (C t) a single particle solution of the free field Schrodinger equation. Using the results and notation of the last section, we know that any such \II is of the form = (C­

Ty M

(6)

maps ffiP, for some p, linearly into TyM, thereby injecting the increments of the p­dimensional Brownian motion {Bt:t O} into the tangent spaces to the manifold. There are many choices of A, p, and X which will give Brownian motion. For example when M = ffi a natural choice is p = 1 , A = 0, and X(y) the identity, giving dYt = dB t . However the periodic choice p = 2, A = 0, and X(y) = (cosy,­siny) , giving (7)

is equally suitable. The choice of X, A gives additional structure to the and physically means making some assumptions on how the fluctuations are being introduced into the system: i.e. on the 'background fields'. As observed by Stroock the coefficients A and X can be obtained from the joint distribution of {(xt,B t) : t O} on M x ffiP, by considering it as a Markov process and looking at its generator using Ito's formula, for example. The joint distributions of (xt'Yt) on M x M would do equally well, as would the '2 point motion' on M x M obtained by solving (5) at 2 different points, as used by Baxendale [2J.

63

C. If the coefficients were smooth and M were compact the equation (4) would have a flow t " 0

of diffeomorphisms, for W E 0 the probability space associated to {Bt:t " o} . See [8J, [11 J, [12J. For Xo in M the process {Ft(x O'-) : t "O} is then the solution of (4) starting from Xo (The same is true for Xo an M-valued random variable independent of {B t : t " O}). In the smooth case for M not necessarily compact local flows exist by a result of Kunita, see [12J, [8J, [llJ, and we can apply this to our situation by considering the equation for (Yt,t) on space-time M x [0,00) with the nodal sets removed. In these circumstances open sets may be sent by Ft(-,w) to infinity (or into nodal sets) in finite time even though this almost surely never happens to individual points: the system is complete, but may not be strongly complete (= strictly conservative). We show in §2 that this does indeed happen even for the ground state of the hydrogen atom with the most obvious choice of X , A . An important property of flows is the behaviour of the distances d(Ft(xO'w), Ft(yO'w)) between two solutions for large time. This infinitesimally is measured by Lyapunov exponents as in Ruelle's ergodic theory for ordinary dynamical systems. Some remarks about these for ground state flows of stochastic mechanical systems were made in [7J and these are further developed here. Even when there is not stochastic completenessJderivatives in probability of Ft(xO'w) with respect to Xo E M exist, [llJ, to give a process vt(w) = TxoFt(-,w)(v o) for each tangent vector Vo at xO. There are then the exponents

i log!vt(w) I = lim i log!vt(w)[ = TTm t->

and

(8)

p;;;

For equations like (4) with smooth coefficients on compact M, Carverhill [5J showecr that for almost all points (x,w) of M x 0 there is a filtration of TxM by linear subspaces o = Vr +1 (x,w)

c

r

with real numbers V

E

< .•.


= A

-

j

A dim(V1x,w) I

is independent of

(x,w)

and

j

j

to

r

64

AE =

(9)

log det TxFt(-,w) .

This comes from the multiplicative ergodic theorem, following Ruelle, and continuing with a stochastic version of Ruelle's work Carverhill gave the stable manij fold theorem: if A < 0 then for almost all (x ,») ( M x Q the stable manifold

vJ('

x,w

)

TTm -tl log d(Ft(w,x),Ft(w,y))

= {y EM:

\j}

t->co

is an immersed manifold in

M tangent to

V1x,w)

at

(10)

x.

Here we will show how some of these results go over to stochastic mechanical flows associated to superpositions of bound states of (2). We also briefly contrast them with what can happen for a simple class of scattering states. Finally we consider the ground state flow for the hydrogen atom. This has some special features, for example a flow does not exist on the obvious state space: however we give Chappell 's result that the top exponent is negative. This goes a long way towards explaining the near coalescence observed in the computer simulation, Figure 1, of the article by Durran and Truman in this volume. The picture that seems to emerge is that, for a process obtained from the evolution of a superposition of bound states, if an observer looks at time t at the process it has density However if he then follows the progress of this picture as it moves in the future under the influence of a generic sample path of fluctuations he will notice it concentrate rapidly so that all its mass is around one or more moving points. The exponential rate of concentration (at least measure theoretically) is controlled by the osmotic velocity, and in the case of ground state processes is proportional to the quantum mechanical kinetic energy (at least for the most natural choices of X, A).

§1.

Ergodic theory for superpositions of bound states

Let eigenvalues

A.

for

(- 2m to + V)ljJj Set IjJt(x)

j = 1 to k be eigenfunctions of our Schrodinger operator with to k:

Ej , j = 1

k E

j= 1

E. IjJj

(11 )

J

aj e

iE.t J

IjJj (x)

t

E

IR

where (al"" ,ak) E: [k is fixed. Taking M = IRn , mechanical diffusion corresponding to {ljJt t O} dX t = bt(xt)dt + !

.dBt

represent the stochastic by a solution of the equation ( 12)

where {B t: t O} is a Brownian motion on M, and b is given by (3). We assume that V is also sufficiently regular so that each IjJj is smooth and of finite energy

65

2) . (i .e. E L solutions {(Xt,T t) and admits at least n n TxFt(-,w) : lR +lR

Then (12) coupled with the equation dT t = dt has strong : t :> O} in lRn x [0,00) - N where N = = O} a local flow (X,1) + (Ft(x,w) , t + 1) There is a derivative in the LO sense [llJ which is given by Vo + vt(w) with (13)

We will show that Ruelle's method of dealing with quasi-periodic time dependent dynamical systems [15J, implies the existence of a Lyapunov spectrum for {Tlt(-,w) : t :> O} and in particular the existence, almost surely, of lim t+oo

i log[vtl

.

B. Let Tk denote the

k-torus. t

E

Define lR

t itE itE k 1 by g (ql' ... ,qk) (e l ql,· .. ,e qk) considering each q. E S as a complex t J number. Let S be the closure of the orbit of 1 under {g : t E lR}. This is another torus; let a be its normalized Haar measure, so the restriction of gt to S is measure preserving, and ergodic, and even minimal (e.g. see [19J). t E lR} be the evolution of the wave function For q E S let where k . =

L

j= 1 J J

=

Then

and the corresponding stochastic mechanical process is given by (14) b(q)t defined as in (3).

for

b(q)t(x) = b(9tq)0(x)

Then

and let r be a connected component of Let N(q) be the nodal set of n n (R x S)' "lR x S - {(x,q) : x E N(q)} One way to proceed would be to consider the (degenerate) stochastic dynamical system on r given by (14) rewritten as (14) , and dqt

=

(iEl, .. ·,iEk)·qt dt

(15)

We might then hope using the multiplication of [k and inclusion of S in [k to obtain the results of [5J for this system i.e. Lyapunov spectrum and stable

66

manifold theorem. A slightly different way and one closer Ruelle is to replace the basic probability space (n,FJP) = n x S , = F * Borel S, and W = IP 61 a One then its space-time version with N removed, as an S.D.E. with of randomness. In either case there is a measurable

to that followed by by $) where considers (14), or rather this additional element

t

'V

of the form ¢t(x,q,w) = (Ft(x,q,w),g q,Btw) where Bt: n + n is the shift (taking n as classical Wiener space) and F is a measurable solution flow for (14) " (15) off the nodal sets, see [11 J: a smooth vers i on may perhaps not exi s t. Since the Markov process determined by (14)', (15) has invariant measure 12dx a(dq) the semi-group of transformations ¢t preserves the measure 1.1 given by 2dx 1 @ a(dq) @ lP(dw) . In fact 1.1 is ergodic: as can be seen by observing that for any Dirac measure O, qO) on r the measure 8(X O,qO) @IP gets mapped by ¢t to one whose projection onto M has support of the whole of r n (JRn - N(9tqO)) .

8(X

'V

C. Let TxFt(-,q,w)(v O) denote the partial derivative in measure of Ft in the direction va E ffin. To apply the multiplicative ergodic theorem as in [5J we need the following key lemma: Lemma lC log

+

For each t

a both quantities

>

'V

and log

IIT/t(-,q,w)11

are integrable with respect to

+

'V

-

1

II

1.1

Proof (Chappell [9J). We can take the norms to be operator norms on L(ffin ;IRn) . Define Vt = Vt(q,w) by

and

Then

dV t

=

D(b(qt)O)(xt)(Vt)dt

logllVtl12

for

qt

= Jt 2 IIVsl [-2ds

o

s

s

=

t

g q

By Ito's formula, if Ilvoll

1 ,

Consequently

logIIT/t(-,q,w)11 s

This shows, via the Cauchy-Schwarz

f log+

67

r°

_ :u p {vO·!!vo!!-l}

inequality that

IIT/t(-,q,w) 1111(dx,dq,dw) s

-
0,

O.

Outline proof By inspection

S

being a

BM(R)

process.

Therefore, we obtain

0,

lim

E+O by the above. II

Planetesimal diffusions associated with the correspondence limit Following Albeverio et al [IJ, we consider a protosolar nebula with its mass concentrated near the origin. the

z

We model a situation in which the nebula is rotating about

axis and in which a planetesimal is condensing out of the nebula.

Due to

collisions with the dust particles of the nebula the planetesimal accrues mass and exchanges energy and angular momentum with the nebula. In the above we take the planetesimal to be at take

-I

m

and we set

to be unity. At time

to be the gravitational mass of the nebula,

("A/m-

E

2)

we

m- I

take the total energy per unit mass of the planetesimal to be From the results above it is natural to interpret

t

We

to be the third com-

ponent of the angular momentum per unit mass of the planetesimal. As in Albeverio et al [2J, we assume that when the mass of the planetesimal, energy per unit mass and angular momentum per unit mass of the planetesimal are all constant the Nelson-Newton law is valid in the form: 1 2- (D +D- + D_D+) .-

; -

ne b u 1 a ,X(t)/IX(t) . . ., ...

3 1

.

83

Several authors have tried to derive this law by making various statistical assumptions about the cloud but so far without success.

Nevertheless the above

Nelson-Newton law is derivable from the stochastic variational principle of Guerra and Morato [6J. To obtain a model in which the Nelson-Newton law is valid we use the above model with

m

= met), A = A(t).

2

as

E:

We shall assume that the diffusion coefficient t too.

tends

We show in the next few pages that, if

to zero sufficiently rapidly and if

A(r ) / m(t ) -+ L (> 0) O

as

and

t too,

then after a sufficiently long time the orbit of the planetesimal is in an 2 L - ne i ghbour hood of

K(r ) ' O

the Keplerian circular orbit of radius r

O

(> 0).

K(r ) with O an angular speed consistent with Kepler's laws but it seems likely that the only We have not yet managed to prove that the planetesimal describes

extra requirement needed to ensure this is that gravitational mass of the sun, as

t t

).lnebula (t )

m- I (t ) -+ ).lsun' the

=

The latter condition is necessary for the

00.

deterministic part of our equations to give the correct values of planetary years. The proof is no harder when in what follows that

.l(t) -+ L than it is when 2. mom

and that

m

f(t)

=

=

We shall assume

1.

met) j),2(t) -+ I, f(t)

>

O.

The key to

getting the correct deterministic behaviour is the next proposition. Proposition 5 Let

dR

dt

I

[11-f(U)ldU 0,

as

tt

as

tt

ln such a way that

oo

oo •

Outline proof t

R( t ) - R(O)

to infinity also.

fo

ds R(s)

But we can write our equation of motion in the form ddt (R(t) -I) + (R(t) -1) R(l t)

giving

e

J:

ds R(s) (R(t)-I)

R(O) - 1 +

I> -

=

f Iu) )

(I -f(t»,

e

J:

ds R(s) duo

84 Hence,

I R (r )

so that

A

lim R(t) < 00.

- ]

'IR(O)-'I.(I'-£',,)ld" ••.

I

Since

t

ds

- I

I "

diverges to infinity, de l'HSpital's rule

fo R(s)

tt ro

gives lim tt co

IR(r )

I (1

A 1 im tt oo

-

f (t))

I

O. II

The corresponding stochastic equation which we have to consider is dRE (

t )

(_1_ -

=

E

f (t

)

Jd t

+

R (t )

d

t )

d

S(

t ) .

Proposition 6 then

limlE{(R tioo

E-R)2(t)}

exists.

Proof formula gives

The desired result now follows from Cauchy's generalised convergence criterion. as

t t

co.

We now show that

f

=

O.

We

require a further lemma: Lemma 7 For each

t:> 0

Proof This is a simple bootstrap based on the remark after the first lemma [7J. oo

f

E

2

2

t (R - R) (u) e (uj du}

=

E

4(R - R)

3

Consider

85 S

being a

process.

BM(R)

negative semidefinite.

Observe that, as

the first lemma, the first term is

Then, as before, {(RE:_ R)4(t) + 6 f(RE:_ R)2(u)s2(U)dU} t

is a true supermartingale giving E: 4 E(R - R) (r ) " 6 lE

f

at (RE:

fooa E: 2 (u i du fta E: 2 (u I du

2 2 - R) (u)£ (u j du ,,6

Therefore, by Doob' s martingale convergence theorem, surely.

lim(RE: - R) 4(t)

0

is defined by

the stochastic process is, after a sufficiently long

neighbourhood of

K(r

o) ,

the Keplerian circular orbit of radius

We hope to show in a future publication that, if only

)Jnebula

)Jsun'

rOo

the gravita-

tional mass of the sun, then after infinite time K(r is being described with an O) angular speed consistent with the observed duration of the planetary years. We conclude with a simple result in this direction: Proposition 10 Let

be a function of time with

f:

2(u)du 0,

a

Proof Reinstating

A and

m to emphasise the physical significance of the result, the

proof is a trivial extension of the proof of proposition 4. REFERENCES [IJ

Albeverio, S., Blanchard, Ph., Hoegh-Krohn, R.(1984): A stochastic model for the orbits of planets and satellites: an interpretation of the Titius-Bode law. Expositiones Mathematicae 365-373.

[2J

Albeverio, S., Blanchard, Ph., Hoegh-Krohn, R: Newtonian Diffusions and Planets, with a remark on non-standard Dirichlet forms and Polymers. In 'Stochastic Analysis and Applications', Proceedings, Swansea 1983, editorS-A. Truman and D. Williams, 1-25. Lecture Notes in Maths. 1095, Springer Verlag.

[3J

Carlen, E. (1984):

[4J

Durran, R.M., and Truman, A: in preparation.

[5J

Gihman, 1.1., and Skorohod, A.V. (1972): Stochastic Differential Equations, Ergebnisse der Mathematik. Berlin: Springer Verlag.

[6J

Guerra, F., and Morato, L.M. (1983): Quantization of dynamical systems and stochastic control theory. Phys. Rev. D, 1774-1786.

[7J

McKean, H.P. (1969): Stochastic Integrals. Probability and Mathematical Statistics Monographs. New York: Academic Press.

[8J

Nelson, E. (1967): Dynamical Theories of Brownian Motion. Princeton: Princeton University Press.

[9J

Nelson, E. (1985): Quantum Fluctuations. Princeton: Princeton University Press.

Conservative Diffusions.

Commun. Math. Phys., 94, 273-296.

Planetesimal Diffusions and Stochastic Mechanics,

Mathematical Notes.

Princeton Series in Physics.

[IOJ

Nieto, M.M. (1972): The Titius Bode Law of Planetary Distances, its History and Theory. Oxford: Pergamon Press.

[11J

Truman, A. (1986): An introduction to the stochastic mechanics of stationary states with applications. In 'From local times to global geometry, control and physics', editor K.D. Elworthy, 329-344. Pitman Research Notes in Maths. Series 150. Longman Scientific and Technical.

[12J

Williams, D. (1979): Diffusions, Markov Processes and Martingales. Volume 1. Foundations; and, jointly with Rogers, L.C.G. (1987) Volume 2. Chichester: John Wiley.

[13J

Zheng, W.A. (1985): Tightness results for laws of diffusion processes and applications to stochastic mechanics. Ann. Inst. Henri Poincare 3l, #2, 103-124, and references cited therein.

88

Footnotes added in proof 1.

We are grateful to Professor W.R. Schneider for a helpful remark correcting our

expression for

2

lli{1 } here.

The discrepancies between the stochastic and quantum

mechanical averages disappear altogether if we work with 2.

£+

by itself. (See ref (4».

After this paper was typed we received a very interesting preprint from Professors

Piotr Garbaczewski and Dariusz Prorok, 'Semiclassical quantum mechanics for the Coulomb-Kepler problem', discussing the same problem by very different means.

It

would seem likely that more progress could be made by combining the methods of their paper with ours.

MOTION ON HYPERSURFACES AND COMPUTER SIMULATION by Richard M. Durran and Aubrey Truman Department of Mathematics and Computer Science University College of Swansea Singleton Park, SWANSEA, SAZ 8PP

I. Introduction

One of the motivations for this work was to construct a simple and accurate m3 . To this end, d we shall discuss a characterization of Brownian motion on hypersurfaces in m , in computer simulation of Brownian motion on a surface embedded in terms of local (parametric) coordinate processes.

We shall obtain, via a straight

forward application of Ito's formula and some elementary differential geometry We go on to discuss

expressions for the stochastic differentials of these processes.

the relation between this characterization, and those given by Baxendale [ZJ, Lewis 3

[6J and, for Brownian motion on surfaces in m , by Williams L7J. For Brownian 3 motion on surfaces in m we give an especially simple formula for the stochastic differentials of local coordinates. Brownian motion on surfaces in

3

m

Using this formula we give a few examples of and discuss, in detail, their computer simulation.

Finally, as the emphasis of this work is on stochastic mechanics and stochastic processes, we discuss briefly some simulations of the stochastic processes arising from the stochastic mechanics of the first few excited states of the hydrogen atom. The coloured pictures which one can obtain here could give quite detailed information about the charge distributions for the excited states of hydrogen. is only possible to give black and white reproductions in this paper.

Unfortunately it We have used

a screen dump program with shading to give some impression of colour. Z. Brownian Motion on Hypersurfaces Let n

n I x)

d be a hypersurface embedded in m with orientating normal vector field d sd-l (the unit sphere in m ) at each point x E V • Let P be the

V E

projection onto the tangent plane, summation convention define, for i.

(n],n

Z""

,n

d)

x

to the hyper surface at

m,

(PV)

x .

Using the

by

df (6 .. -n.n')3 (x ) J x j

(Pvt) . (x )

n

T

f:V

... ,d,

being the unit normal in cartesian coordinates.

.,u Let uj,u Z,·· .,u d be a local coordinate system, with u],u Z,·· d_] parametric coordinates of the hypersurface, so that each point x

x(u)

x(u j ,u

(Z. I)

z , · · · ,u d_ 1)

.

The tangent vectors

3oxu

(u), l

3oxU

z

(u), ... ,

the local

90 are linearly independent and span The Laplacian

6

Tx(u)

by assumption.

on the surface is defined (writing

M The all important identity

=

PV

vp )

by

Trace (v V f) .

(2.2)

P P

what follows is contained in the next lemma.

We feel

that this identity has not been given sufficient emphasis in the literature.

Lemma 6x(u)

where

H(x)

i

(d - I)H(x)n(x),

s the mean curvature of

V

at

(2.3)

x.

Proof Let

f

af aUk aUk aX , j

af ax.

since

Then by definition (2.1),

feu) = feu] ,u 2 ' · · · ,u I ) d_

J

vp feu)

(2.4)

Hence using definition (2.2) and the rule for differentiating a product we see that Trace

(v v

p p

f)

Trace

Uk

(u)V V Uk + (v P P

2!

P aUk

(u»V u \ p kJ

M(u)

(2.5)

But, using definitions (2.1) and (2.2) componentwise and the fact that it is easy to check that for

x

E

n.n

1,

V 6x

- (d i v n t x ) )n(x)

(2.6)

and observing that H(x) = where

H(x)

is the mean curvature of

1

- (d-!)

V

at

(2.7)

div n Cx) , x

we see that

6x = (d-I)H(x)n(x) Finally, applying (2.5) componentwise to

x

(2.8)

and using (2.8), identity (2.3) follows

II

easily.

We shall use this identity in the proof of the following proposition, but first we note a simple corollary originally due to Lewis [61. Corollary Denote by

BM(V)

(a Brownian motion on

V) a process on

V

with generator

91

The

BM(V)

process on the hypersurface

V

is a martingale in the ambient

Euclidean space if and only if the mean curvature of

V vanishes identically.

Proof 1 2- 6X

We merely observe that the vanishing of the drift

gives precisely this

II

by virtue of (2.8). Proposition Let X(t)

=

B(t)

Then a Brownian motion,

be a BM(JRd) , i. e. a Brownian motion on JRd.

X(u(t))

X(u

... ,u

on a hypersurface

d_ l(t)), is characterized by the stochastic differentials l(t),u 2(t),

1,2, ... ,(d-l) ,

i

of the

(d-I) parametric coordinate processes,

R(

E

V embedded in JRd

Oed))

being any

(2.9)

non-anticipating

orthogonal transformation. Proof Let

X i = l,2, ... ,(d-])

a set of (d-l) Ito equations. X(t)

=

X(u(t))

X(u j (t),u

=

V .

(2.10)

We try to choose

2(t),

ax

b and dSi(t) so that i is a Brownian motion on the hypersurface

... ,ud-I (t))

Applying Ito's formula to dX(t) =

I

X(u(t))

gives 2

I a x auiau (u) dUi (t) duj(t) j

2

(u) dUi (t) +

Assuming the above set of (d-I) Ito equations admits a unique solution the point clearly remains on the hypersurface

V.

Hence for a Brownian motion on I

require the generator of the diffusion (2. \ 1) to be 26. I

2 6X(u) dt

=

ax

I

i.e. we require

a2x

(u ) bidt + 2 aU aU (u) dS i (r ) dS j (t ) i j

Now from the left hand equality of (2.3) j

2 6X(u) dt

2x

ax 1

I I a (u) 2 lIu.dt + 2 -a-a 1

U

i uj

X(t)

V we merely

(u ) (V u .. V u.) dt P

1

P J

(2.12)

92 Comparing these two equations we see that for a and

BM(V)

we merely require

v u .. V u. d t, i, j

dB. (t)dB.(t) J

=

P J

P

(2.13)

1,2, ... , d-I .

Evidently we can solve the second set of equations in (2.13) by the choice i

d S. (t ) = (RV u.) .dB(t) , P

i.

where

B(t)

BM(lR

is

Thus a process

d)

and on

dX(t)

=

V,

II

R Ie Oed) any orthogonal transformation.

X(t)

is Brownian motion on

(2. 14)

1,2, ... ,(d-l)

=

V

satisfying,

±

lIX(u) dt

+

(2.15)

dX (u ) (RV u.).dB(t) dU i

p

i.

with coordinate processes,

u (t), i

satisfying 1,2, ... , (d-I)

(RV u.) .dB(t) p

The relation between this characterization and others, is borne out by specific choices of the transformation clearly orthogonal,

R be the identity transformation,

Firstly let

R.

so that (2.15) reads, after substitution from the outer equality

of (2.3), dX( t Noting that tangent plane

dX u

i T

(u) X

{d-r l )

H(X)n(X) dt

--2-

)

vp u i... dB(t)

dX (u)

+

dU i

vp u i... dB(t) dB(t)

F(X) dB(t) , the projection of

=

onto the

x , this reads,

at

(d-I) H(X)n(X) dt + F(X) dB(t) ,

dX(t)

(2. ] 6)

2

which is the characterization given by Baxendale [2J, used also by Lewis [6J, for Brownian motion on a hypersurface. We now focus our attention on Brownian motion on surfaces embedded Let the orientating vector field

n

=

n(x)

t

n

3. lR

be given by (2.17)

n Cx)

It is not difficult to show that n(x) Thus, taking

R

3x

dB(t)

i\

u

i

(u)(v u. p i.

to be a rotation of 90° about

from (2.6) and (2.18), dX(t)

=

-

i\

n(X».dB(t) .

n ,

±(d v n(X»n(X) dt i

(2. 18)

(2.15) reads, after substitution + n(X)

i\

dB(t) ,

(2. 19)

3 which is the characterization for Brownian motion on surfaces in lR given by Williams [7J and van den Berg and Lewis [3J. equations have a particularly simple form.

In local coordinates the corresponding Let

E,

F

and

G be the fundamental

93 magnitudes of the first order defined by

I ClX

E

Clu

(u)

I

2

Clx Clx (u) '""3 Clu u (u) l 2

F

I

I Clx (u) I

G

Clu

z

2

then j

dU (r )

"2L'lu I dt -

I Clx (u ) . dB (r ) (EG-F2) 1/2 dU2

(2.20)

dU (r ) 2

1 "2L'1u Z dt +

1 Clx (u) . dB( t (EG-F2) j /2 CluJ

(2.21 )

j

)

We feel that these equations should be given more prominence as they are so useful for computer simulations. coordinates

u

We should emphasise that these equations are valid for

which are not necessarily orthogonal. Using these equations 2 in the next section we consider some examples of Brownian motion on parametrized l

and

u

surfaces embedded in 3.

.

Examples

(i) Surface of Revolution Let

V: I

-+

]R3 be the parametrized surface of revolution obtained by rotating

the parametrized curve, V(s,e)

= {!S

E

3

c Cs) :

(a(s) ,8(s»

=

!S(s,e)

=

8(s)

a

a Vs E I, about the a-axis.

(a(s) ,S(s)cos e, S(s)sin e) s

For a surface of this kind, parametrized 1n suppressing the dependence of

>

and

13

on

sand

e,

E

I, e

the operator

Thus (3. I)

L'I

is given by,

s (3.2)

Using equations (2.20) and (2.21) of the previous section and the above, a Brownian motion on

V,

starting at

!So =

differentials of the parametric coordinates ds(t)

1S characterized by the following stochastic sand

e:(0,,2+

I

13,2)

1/2

2 (dB sine- dB3 cos e) t t (3.3)

de(t)

where

J?,(t)

I

13(0,,2+8,2) 1/2 (B I B2 B3) t' t' t

2 3 {a'dB l + 13' (dB cos e + dB sin e ) } , t t t

is the driving Brownian motion in

(3.4)

94 (ii)

Pseudo sphere If we take c(s)

=

(r

(I - e- 2u lc) 1/2 du, c e -sic ) ,

S E [0,00),

0.5)

c >0 ,

o

In the above example, the surface i.e.

V

.

the parametrized pseudosphere in

lS

has constant negative Gaussian curvature

V

K =

- I Ic

2

(Here

.

is the

s

arc length along the surface.) The stochastic differentials for V

sand

8

characterizing a Brownian motion on

are, by (3.3) and (3.4),

_J..- d t + dB2 sin 8 - dB3 cos 8

ds(t)

2c

t

2 3 dB cos 8 - dB sin e}

d8(t)

(iii)

(3.6)

t

t

0.7)

t

Torus Let

be the parametrized surface of revolution obtained by rotating

the parametrized circle

c(¢)

plane, about the z-axis.

(a+b(l+cos¢),bsin¢)

(a,b>O),

in the

(x,z)

Thus 2

v(e,¢) = {O) of injective homomorphisms from

and a family

u into IT.

In this paper we shall consider processes of this type, in which U is an algebra of bounded operators on a Hilbert space

G is =

the algebra

and

of all bounded operators on the Hilbert space is the Boson Fock space over L2 (R>O)' jo will

where

always be the ampliation

The state

Wwill

be the tensor prod-

uct of a state Wo of G with the vacuum state on Following [43, we define the creation and annihilation processes in

as the families of operators defined on the algebraic

tensor product of

with the span of the exponential vectors (co-

herent states) W(f), fEL 2

in

f\(tlu ® W(f) A +(t)u :9 ,(f)

A(tlu;) *(f)

i f ;)

by the actions

.(exp EX[o,tJf)IE=o W(f+EX[0,tJ) IE=o

S[O,tJf u ® ,(f)

These should not be envisaged as quantum stochastic processes in the sense of [1J but, in the present context, regarded as mathematical devices with the aid of which quantum stochastic processes are con­

114

structed, just as classical Brownian motion is used to construct more physically realistic models, even of physical Brownian motion. The mechanism of the construction is the quantum stochastic calculus of[4J which gives meaning to stochastic integrals of certain operator­valued processes against the integrators d A, dAtand dA. Thus we assume that, for each xfG, x

: = jt(X o ) constitutes an adapted prot cess in the sense of [4J, and that these processes satisfy quantum

stochastic differential equations of the form

a, a + and T are maps frm Q to itself; we call them the struct­

Here

maps of the quantum diffusion j on Q. We write (1.1) succiactly as dx

=

+ a(x)dA

t

+

oj

a (x)dA

+ T(x)dt,

but emphasise that the structure maps are from Q to itself and that for instance A(X) is a notational abbreviation for the adapted process jt(X o

».

Thus quantum diffusions are governed by systems of auto-

nomous quantum stochastic differential equations. The fact that each jt is a homomorphism enforces properties of the structure maps. They inherit linearity from the linearity of jt' Since jt is a *­map, A

= At,

T

= r+

and at is indeed the adjoint map

of a, defined by

Since jt(I)

=I

the structure maps vanish on I.

To exploit the multiplicativity of jt we require the technical condition that jt is contractive for the uniform operator norm. This

*

is automatic if Q is a C ­algebra or if it is closed

under formation

of holomorphic functions of self­adjoint elements (for instance if Q is the algebra Cffi(V) of smooth functions on a compact Riemann manifold V acting on L2 (V) by multiplication). When each jt is contractive, for arbitrary x.and y. in Q the stochastic differentials dx and dy satisfy the condition ensuring validity of the quantum Ito product formula of [4J. Comparing coefficients of the basic differ ntials in dx Y + x dy + dx dy

115

further identities to be satisfied by the structure maps are obtained; these are called the restraint equations. This calculation is carried out in §2. In 93 it is shown

the

restraint equations are cohomological in character and the classification of their solutions is considered. In 94 the effect of replacing the structure maps by cohomologically eqmvalent maps is examined , and it is shown that the new diffusion can be constructed from the old by a perturbation procedure. In

examples are considered; in particular

when Ci = B( '\';>0) every quantum diffusion on G. is shown to be completely described by a unitary process of the type constructed in[4J. In contrast, when G. is commutative,

as well as classical diffusions, a

class of quantum diffusions related to generalised Poisson processes is constructed which are not governed by such unitary processes. I wish to thank several colleagues and students, particularly David Applebaum, Frank Ball, Mark Evans and Michael Schdrmann for valuable

and comments on a first draft of this work.

92. Restraint equations Recall from [4J, Theorem 4.5, that if = E d A + F dA+ + GkdA + H dt, k K

k=1,2

are stochastic differentials satisfying

for each t>O, then d(M 1M2' is also a stochastic differential, given by

where the basic differentials d r l

,

dAt, dA and dt commute with adapted

processes and the correction dM1dM is evaluated by combining this 2 with extension by bilinearity of the basic multiplication rules dAtdA dt

+

dA

d/\ dA 0 0

0

dA

dA dt 0

a

dt

0

0

+ dA 0

0

0

0

0

Now let j be a quantum diffusion on Q. Provided each jtis contrac­

116

tive, for arbitrary Xo ,yoEG, the stochastic differentials dx

t A(x)dA + a(x)dA + aT(x)dA + T(x)dt

dy

A(y)dA + S(y)dA t+ at(Y)dA + T(y)dt

satisfy the condition (2.1), so that we may write

=x

d(xy)

dy + dx Y + dx dye

On the other hand

( 2.4) Using (?2) to compare coefficients of the basic differentials between

(2.3) and (2.4) we obtain the restraint equations A(XY)

XA(y)

a(xy)

xa(y) + a(x)y + A(x)a(y) t x r Cy ) + r(x)y + a (x)a(y).

r(xy)

+ A(X)y + A(X)A(y)

( 2. 'i)

( 2.6) ( 2.7)

These may be regarded as identities to be satisfied by the structure maps for arbitrary x,yEG.

93.

Solutions of the restraint equations By adding xy to both sides of (2.5), we see that A + id is multi-

plicative on G, where id is the identity map. Since A is a *­map and A(I) = 0 we may therefore set 0' ­

where

id

is a unital *­endomorphism of G. Conversely, given any such 0, A defined by (3. 1) is a *­map vanishing on I which satisfies (2.,). 0'

Substituting (3.1) into (2.6) gives a(xy)

= o(x)a(y)

+ a(x)y

for all x,yEG, that is a must be a

(see Lemma 3.2 in [FJ where

Since

K; Ilidll1" + 1.

where a >0

there exis ts

t (a)

1

depends only on

bt

i nf Is :

T

giEU

foraH

l;

Define

t

continuous in

and

bo

id

then for any

> 0 such that

PhI ,;t(a)} ,; where

1".

bs i ov,

Let

a

0.2)

un; {b EDifr(M): b; gn ogn_l

0

•••

ogl

and

i ; l , ... ,n}

then by 0.1),

any Put

Tn

inHs:

bs i Un}

then repeating the argument of Lemma 3.1 from [Bl J we have

Ph

for some

s

satisfying

n

< z } ; P{b iU s

n

for

some

s ,;t}

Tn-I'; e ,; t}

using the strong Markov property of the process Tn_I'

(3.3)

bt

in

Difr(M)

at the Markov time

Thus (3.4) Taking

t

t(a)

and using (3.2) we get

Ph ,;t(a)} ,; an.

0.5)

n

Now by (3.3) and (3.5),

E(max(o,;ss,;i(6) Ilb SII1"'o,;ss,;ui(a) IlbS-llll"»k

,; Ic

kn (K1" + l ) kn l,;t(a)0 3

a random variable

Then for any

independent of n; 0,1, .... D

K

K

>0

there exists

such that with the probability one (2.11)

for all

k; 0,1, ••. ,

Proof.

where

1

is given by (2.2) according to Theorem 1.

Set T- l 8 n-l n

(2.12)

then by (2.1) T- l 8 )

n-l n

(2.13)

From (2.10) and the Chebyshev's inequality it follows that (2.14) Since variable

-(l+K)

t. m

m;l

MK

then by the Borel-Cantelli lemma there exists a random




dv m+n

(re m> 0, ren > 0)

1 +s

and

u

n;

- s,

I2Bel+s,l-s).

we deduce that

'" e2st dt f_",[cosh (x + l 3/2 t j

and with

s

replaced by

-s,

we get on adding the two formulae

'" cosh 2st dt ; f_",[COSh(x+t)]32 With

x

replaced by

!TTX,

t

= !TTT,

12 cosh 2sx Bel + s , 1- s ) ,

s ;

'IT

we only need some well-known properties

of the beta and gamma functions to complete the proof of (8). 3.

A Wiener-Hopf identity Let

IT(x,8); £[IT(x,y)](8),

with

IT(x,y)

as given by (4) and (5).

Our

integral equation (3) then takes the form 'IT

'IT"""

sin(t; + 8)IT(x,8) + sin(4 - 8)IT(x,-8); f'V

I 2 cosh 8x.

(IO)

We proceed to show that this Wiener-Hopf identity is satisfied by -

2 -

8

(II)

IT(x,8) ; -r,;-G(x,8) BO +-;,1).

Substituting into the left-hand side of (10) for we have

IT(x,8)

and

IT(x,-8)

from (11),

165

With

8

! + 11 and

z

1_.Q. in 4 'Il

z

nisin n z

f(z)f(1 -z)

the left-hand side of

(10) may be reduced to 2n(G(x,8) + G(x,-8»

f(14 since

f(j)

/n.

is satisfied by

.Q.)f(1 - .Q.) 1T 4 'IT

+

Using (8) this becomes IT(x,8)

/2 cosh 8x

as given in (I I).

and we have shown that (10)

To establish (11) we apply the Laplace

transform to the convolution (4) to get _1

t'V

f"-'

IT(x,8)

/(2n)G(x,8)£[ (sinh jnv) 'J(8)

and it only remains to prove that I

£[ (sinh jnv)

To obtain (12) we simply set

e

2

-nv

/2

( 12)

](8) 1n the Laplace integral to get

u

I

8

3

11-'

TI 0 u

f

(I - u )

which is the required beta function integral.

_1

'du

It is now clear that the convolution

(4) satisfies our integral equation (3). 4.

Evaluation of the convolution To evaluate the convolution (4) we set

fo

y -

V

-T

to get

y

Jl(X,y)

and if we substitute in for determine.

Let

X

jnx,

-/(2n)

G(X,T) Y

I(X,Y)

jny

G(X,T) dT " [sinh jn(y + T) J 2

from (5) then we have two integrals to and consider

• •_ _ o [ s i. nh rY + t )Jl [cosh ( X + t )J3/ 2

(13)

166

with

e

2t

and (6) with

Replacing

this becomes

v

=

a

X by

=

Y

e ,

-x I

2

and setting

T =

2t 1T

-e

b

-Y

p

e

X

and

q

e

-x

gives

and adding gives (I (X Y)

I (-X Y)) =

,+,

-2 cosh Ycosh X (sinh YJ . 2 2 cosh X (s1nh Y + cosh X)

in our integral (13) gives

I1(x,y)

-I { I (I,1TX,,1Ty I ) 472

+

which is (7).

REFERENCES [lJ

R.R. LONDON, H.P. McKEAN, L.C.G. ROGERS and DAVID WILLIAMS, A martingale approach to some Wiener-Hopf problems, II, Semina ire de Probabilites XVI, Springer Lecture Notes in Math. 920, 68-90.

[2J

R.R. LONDON, H.P. McKEAN, L.C.G. ROGERS and DAVID WILLIAMS, A martingale approach to some Wiener-Hopf problems, I, Seminaire de Probabilites XVI, Springer Lecture Notes in Math. 41-67.

[3J

N. BAKER, Some integral equalities in Wiener-Hopf theory, Stochastic analysis and applications, Springer Lecture Notes in Math. 1095, 169-186.

FORMULJE FOR THE HEAT KERNEL OF AN ELLIPTIC OPERATOR EXHIBITING SMALL-TIME ASYMPTOTICS by Keith D. Watling Department of Mathematics and Computer Science, University College of Swansea, Singleton Park, Swansea. SA2 8PP Great Britain

§O Introduction

Let L be a smooth scalar second-order elliptic differential operator on a smooth connected n-dimensional manifold M. The inverse of the symbol of this operator is the natural choice of smooth Riemannian structure on M. The 'LeviCivita' connection of this natural Riemannian structure decomposes the operator as L = !.6.. + b + c, where .6.. is the 'Laplace-Beltrami' operator for this natural Riemannian structure, b is a smooth vector field and c is a smooth real-valued function on M. Let p(t, x, y) be the fundamental solution of the heat equation,

with respect to the natural Riemannian measure. We wish to examine the small-time behaviour of p(t, x, y). This has been extensively studied by numerous authors, see for example Molchanov [1], Kifer [1], Azencott et al [1], Azencott [1], Bismut [1], Ikeda and Watanabe [1] and references contained in these. We will give an extension of the simple ideas of Elworthy and Truman [1] and Elworthy [1] (Chapter IX §12), where the emphasis (motivated by consideration of the Schrodinger equation) is on obtaining exact formulas for p(t,x,y) which do not lose any information: for example involve all the geodesics not just any minimal one, which is all that is relevant in the small-time asymptotic expansion approximations to p(t, x, V). See also Arede [1], Elworthy, Ndumu and Truman [1], Ndumu [1], [2] and Watling [1] for other extensions of these ideas. Under the simplifying assumption that y is a pole of the natural Riemannian manifold we obtain exact expressions for p(t, X, y) which clearly exhibit the smalltime asymptotic behaviour to any order. These may be extended to the case that the exponential map is a covering map in a similar manner to Elworthy [1] (Chapter

168

IX §12 Remark 12D(i)) and Arede [1]: so in particular we can obtain such formulze for 'Cartan-Hadamard ' manifolds and connected nilpotent Lie groups. Moreover the terms in the exact expressions will still give the correct terms in the smalltime asymptotic expansion approximations to p(t, x, y) on more general complete manifolds: where there is a unique minimal geodesic between x and y along which they are not conjugate (in other words x is not on the cut locus of y), because by Varadhan's estimate (see Azencott et al [1]) the asymptotic behaviour only depends on the behaviour in a neigh bourhood of this geodesic.

§1 Some Riemannian Geometry (1.1) Definitions. We assume that y is a pole of the natural Riemannian manifold i.e., the exponential map based at y is a diffeomorphism, so there is a unique geodesic parameterised to take unit time between y and any point of the manifold. So we may define the following functions on M motivated by the discussion in §6 of Watling [1]: E>y(x) = the square root of the Jacobian determinant with respect to the natural Riemannian structure of the inverse of the exponential map based at y, at the point x, i.e., the square root of the inverse of Ruse's invariant, see Besse [1] and Elworthy and Truman [1]. (1.1.1)

By(x)

= exp

{il (l(u),b({(u))) dU},

(1.1.2)

where "I is the unique geodesic from x to y parameterised to take unit time and "I' is its velocity, (the integrand is the work done by b in moving along the geodesic

"I),

Cy(x)

= E>y(x)By(x), 1

2

Ey(x) = 2d(x,y) ,

(1.1.3) (1.1.4)

where d(x, y) is the natural Riemannian distance between x and y, (Ey(x) is the energy of the geodesic "I). Finally we may define the following functions on R + x M:

ry(t,x) =

1 8 y(x) exp { -Ey(x) } , J(21ft)n t qy(t,x) = By(x)ry(t,x).

(1.1.5)

(1.1.6)

(1.2) Remark. We need the important fact from Elworthy [1] (Chapter IX §12 Example 12D) that if j(x) is a real valued function on M which depends only on r = d(x, y) (i.e., it is invariant under rotations about y) then: 2

d j {n- 1 0 } dj t::.j(x) = dr 2 (r) + -r- - 2 or 10g8 y(x) dr (r).

(1.2.1)

169

(1.3) Lemma.

e

!:}ry(t,x) at

1

=2

-

Proof: Using the fact that functions f on M, we see that:

x) =

{ b. (log 6 y -

= -1 { 2

+ II\7fI1 2 } exp(J) , for real valued

=

E y) (x) +

log 8 y(x) - 1 t

8 ( ) ry(t,x). 2 y x

IIV' log 6 y(x) -

V'Ey(x)

II'} 'y(t, x),

- 2 (\7 log 8 y(x), \7 Ey(x))

+ II\7log8 y(x)11 2+ t12 11\7Ey(x)11 2} ry(t,x). So from (1.2.1) applied to E y and using the facts that

+ II\7logfIl 2 }

=

for real valued functions

I,

f on M, and

we deduce that: 1 2

= =

n

2

8 ( ) - y x 2t

28 y(x) ry(t, x)

+ 2t1 E y(x )}

ry(t,x)

or y (. ) t, x

+ 7ft

(1.4) Proposition. Ifpy(t,x,z) is the fundamental solution of the heat equation:

with respect to the natural Riemannian measure. Then Py(t,x,y) = qy(t,x). Proof: As in Elworthy and Truman [1], Elworthy [1] (Chapter IX §12B), we see that qy(t, x) (considered as a distribution with respect to the natural Riemannian measure) tends to the delta function at y as t tends to 0, by a simple change of variable argument.

170

By (1.3) and the fact that .6.(1 g) functions 1 and g:

3qy Bi(t, x) = By(x)

= l.6.g + g.6.1 + 2 ('V I, 'Vg), for

real valued

y(X)} {I2".6.r y(t,x) - .6.8 28 ry(t,x) , y(x)

.6.Cy(x) } - 2C qy(t, x), y(x) LCy(X)} I = Lqy(t,x) + { 2" (b(x) + 'VlogBy(x), 'VEy(x)) - Cy(x) qy(t,x). =

1

2.6.qy(t, x) +

{ 1

t ('V log By(x), 'VEy(x))

Observe that if b is a gradient then b + 'V log By = 0, so in this case we have the result required. (1.4.1) In general we will prove that

(b(x) + 'VlogBy(x), 'VEy(x))

= o.

(1.4.2)

i.e., b + 'V log By only rotates about y. Firstly note that we have 'VEy(x) = '1'(0), where I is the unique geodesic from x to y parameterised to take unit time, so:

But:

10gBy(t(r))

=

1 1 1

(a'(u)'b(a(u))) du,

where a is the unique geodesic from I(r) to y parameterised to take unit time, 1

=

((t'(s),b({(s)))ds,

where I is as above, so:

whence the result.

§2 Semi-Classical Semigroups and Bridge Processes (2.1) Definition. Let P(t) be the semigroup associated with p(t, x, y). Define the operators Qy(t, s) by:

{Qy(t, s)J} (x) for t

= qy(t, X)-1 {P(t -

s)(qy(s, .)/(.))} (x),

s> 0, on smooth functions of compact support.

(2.1.1)

171

(2.2) Remark. Formally these operators form a two parameter semigroup i.e.,

Qy(t,s)Qy(s,r) = Qy(t,r). This is like a 'Doob h-transform' of the semigroup P(t) but with respect to a function qy on space-time that is only 'approximately harmonic' for the operator: L, with 'error' given by (1.4).

gt -

(2.3) Lemma. For f smooth of compact support we have:

a

as {Qy(t,t - s)J} (x) = {Qy(t,t - s)(Iy(t - s)f)} (x),for t > s and

a

0,(2.3.1)

s > 0,

at {Qy(t, s)J} (x) = {Iy(t)(Qy(t, s)f)} (x), for t

(2.3.2)

where

LC (z) {Iy(t)J} (z) = {(L - c)J} (z) + ((V'logqy(t,z), V'f(z)) + Cy(z) f(z), = L(Cy{\(z) _ (V'Ey(z), V'f(z)) Cy Z t

=

(2.3.3)

+ (V' log By(z) + (b + V' log By)(z) (z)

+ { 2B;(z) + +

1

log By(z)

log B y(z)11

2

V'Ey(z), V'f(Z))

+ (V' log By(z), (b + V' log By)(z))

+ (b(z), V' logBy(z)) + c(z) } f(z) (2.3.4)

In particular when b is a gradient we have:

{Iy(t)J} (z) =

1

+ (V' log ry(t, z), V'f(z)) +

z

-

2

divb(z) -

2

+ c(z)} f(z) (2.3.5)

Proof: By (1.4) and definition of P we have:

as

t - s)f}(x) = Qy(t, t _ s) {L(qy(t - s, .)f(.)) _ Lqy(t - s, .) f(.) qy(t - s,.) qy(t - s,.)

+

LCy(.) () } ( ) C y (.)

f.

x,

172

whence (2.3.1) follows from the fact that !:lUg) real valued functions f and g. Similarly:

at

= f!:lg+g!:lf+2('Vf,'Vg),

for

s)f}(x) = L(qy(t, .)(Qy(t, s)f)(.)) (x) _ Lqy(t,x) {Qy(t, s)f}(x) qy{t,x) qy(t,x) LCy(x) + Cy(x) {Qy(t,s)f}(x),

whence (2.3.2) follows as above. Finally (2.3.3) follows from the same identity, (2.3.4) from the definitions of Land C y , and (2.3.5) from (1.4.1). (2.4) Definition. Let L o denote the smooth scalar second-order elliptic differential operator L - c. Then the qy-transformed Lo-diffusion xt(u) starting from z is the time dependent diffusion with generator

Lo(z)

+ 'V log qy(t -

1

u, z) = Z!:l(z)

+ b(z) + 'V log qy(t - u, z), for t > u

0,

which may be represented as a strong solution of a stochastic differential equation. As a consequence of the isometric embedding theorem of Nash, see Nash [1]' we can obtain a smooth section X of the bundle L(R m; T M), for sufficiently large m, so that if X* denotes the dual section of L(T* M; R m ) then A = XX* is the symbol of L. Then xt(u) is the solution of the following Ito differential equation with respect to the Levi-Civita connection:

dxt(u) xt(o)

= X(xt(u)) dB(u) + {b(xt(u)) + 'Vlogqy(t = x,

u,xt(u))}du,

where B(u) is m-dimensional Brownian motion. (2.5) Remark. If b is a gradient vector field then from (2.3.5) we see that xt(u) is actually Brownian motion on M transformed with respect to r y, i.e., it is the 'Brownian Riemannian Bridge Process' between x and y in time t, of Elworthy and Truman [1] and Elworthy [1] (Chapter IX §12D). So xt(u) tends to y almost surely as u increases to t and consequently the process does not explode. For more general b we get a different process, but as we will see in (2.6), (1.4.2) means that the same argument as in Elworthy [1] (Chapter IX §12D) will show that it is a bridge process which is radially the same as the Euclidean Brownian bridge in R n : so in particular xt(u) tends to y almost surely as u increases to t and consequently this process does not explode either.

173

(2.6) Lemma. The qy-transformed La-diffusion xt(u) is a bridge process, i.e., xt(u) tends to y almost surely as u increases to t, whose radial component has the same law as the Euclidean Brownian bridge in TM y == R" between exp;l x and 0 in time t. We will call this the 'Semi-Classical Bridge Process' between x and y in time t, associated with L as in Watling (1J. Proof: Consider the function Ry : M --+ R defined by Ry(x) = d(x,y). Then we see that R y is 0 2 on M \ {y} while for n 2 and x #. y we have that almost surely x t (u) avoids y for 0 :::; u < t. So we may apply Ito's formula for 0 :::; u < t to deduce that:

Ry(xt(u)) = Ry(x)

+

[U

l

u

(\7R y(x t(s)),X(x t(s))dB(s))

+ i (\7 Ry(xt(s)),b(xt(s)) + \7 logqy(t -

s, xt(s))) ds

o

[U1

+i

o

"2tJ.Ry(x t(s)) ds

From (1.2.1) applied to R y we see that: n -1

0

tJ.Ry(z) = - ( - ) - 2-log 8 y(z ). Ry Z or From (1.4.1) and the observation that \7Ey(z) = R y(z)\7Ry(z) we see that:

(\7 Ry(z), b(z)

+ V' log By(z)

= O.

2

As IIV'Ry(z)11 = 1 we see that:

(V'R y(z), V'e;(z) = If we define:

W(u) =

l

u

n; (z ).

(\7R y(x t (s )), X (x t (s ))dB (s ))) ,

then we see that it is a I-dimensional Brownian motion from Elworthy [1] (Chapter V Corollary 5C), since if we define H : M --+ L(R m , R) by H ( Z ) ( v) then

= (V' n,(z), X (Z ) v)

H* (z) = X* ( z) (en,(z) )

so we have

2

H H* = IIV'R y(z)1I = 1. Thus denoting Ry(xt(s)) by rt(s) we see that rt(s) satisfies:

rt(u)

= rt(O) + W(u) + £(n _ 2

1) [U :(s) _ [U rt(s) ds. ia r s i o t - s

Consequently it is just the radial component of the Euclidean bridge in the statement of the Lemma. It then follows that xt(u) is a bridge process.

174

Lg

y (2.7) Proposition. Assume is bounded above on M, then for f smooth of v compact support and t > s :2: 0:

{Qy(t, t - s)f}(x)

r

LCy(xt(u)) )} t ( { f(x (s)) exp io Cy(xt(u)) du

= Ex

where xt(u) is the semi-classical bridge process between x and y in time t.

(2.8) Remark. This enables us to extend the domain of the Qy(t, s) to say bounded measurable functions f as the right-hand side of the above equality make sense for such functions. They then form a semigroup on this function space. Proof: (Of (2.7)) Define h : [t - s, t] X M x R -+ R by

h(r,x,v)

=

v{Qy(r,t - s)f}(x).

Notice this is smooth. Then:

a

ar h(r, x, v)

= {Iy(r)h(r,., v)}(x), by

(2.3.2).

Consider the process yt(r) = (rt(r),xt(r),vt(r)) on the domain of h given by:

drt(r) = -dr dxt(r) = X(xt(r)) dB(r) + {b(xt(r)) LC (xt(r)) dvt(r) = y vt(r) dr Cy(xt(r)) with xt(O) So:

=

X,

+ \7logqy(r t(r),xt(r))} dr

rt(O) = t, and vt(O) = 1. rt(r) and

=t -

r,

r

t { LCy(xt(u)) } v (r) = exp io Cy(xt(u)) du So the result follows by applying Ito's formula to h(yt(s)) observing the cancellation that occurs to get:

where M(s) is a martingale, with M(O) = o. Then take expectations, observing that the martingale part must be bounded as the other non-constant term is by assumption, to deduce the result.

175

(2.9) Theorem. Recalling the definitions in (1.1) we assume above on M, then for t > 0:

IS

bounded

where xt(u) is the semi-classical bridge process between z and y in time t.

Proof: Recalling (1.4), (2.1.1) and (2.6) simply let s tend to t in (2.7) and use dominated convergence for J a smooth function of compact support taking the constant value one in a neighbourhood of the geodesic segment between z and y.

(2.10) Remark. This is just the elementary formula of Elworthy and Truman [1] and Elworthy [1] (Chapter IX §12 Theorem 12D) in the case that b = O.

§3 Exact Formulre Exhibiting Small­Time Asymptotic Behaviour

(3.1) Definition. For 0

r

s let F(s, r) be the operator {F(s,r)J}(z)

=

J(-y(s ­ r)),

where I is the unique geodesic from z to y parameterised to take time s. These form a two parameter semigroup on, for example, bounded measurable functions. (3.2) Lemma. Assume compact support:

L;lJ y

is bounded above on M, then for

J

smooth and of

8 -8s {Qy(t,t-s)F(t-s,t-r)f}(x) = {Qy(t,t-s)(Lc lJ (t-s)F(t-s,t-r)J)}(x), for t > r 2': s 2': 0, where L c y is the operator defined by:

L

() _ L(Cy(.)g(.))(z) cyg z ­ Cy(z) .

Proof: Follows from (2.3.3) and definition of F(t­s,t­r).

176

L;y

(3.3) Proposition. Assume is bounded above on M, then for f a smooth y function of compact support taking the constant value one in a neighbourhood of the geodesic segment between x and y, we have for any N 0 and 0 :S S < t:

{Qy(t,t - s)J}(x)

= 1 + al(s,x,y)

+ ... + aN(s,x,y) + FN+l(S,X,y),

where for 1 :S n:S N:

and for 1 :S n :S N

+ 1:

where x t (u) is the semi-classical bridge process between x and y in time t.

Proof: By (3.2) for any smooth h of compact support, and any 0 :S r :S s:

{Qy(t,t - r)h}(x)

= {Qy(t,t - r)F(t - r,t - r)h}(x) = {Qy(t,t)F(t,t - r)h}(x) +

I

T

{Qy(t,t - v)LcyF(t - v,t - r)h}(x)dv

= {F(t, t - r)h )(x)

+

J:

1':" { l-o; [F(t - v, t - r)h](x'(v))

exp

(Jro

LCy(xt(u)) )} Cy(xt(u)) du dv,

by (2.6). So taking h = I, r = 8 and v = 81 we see the theorem is true for N = O. Then we proceed by induction taking:

which again is smooth of compact support, r = see that:

Fn(s, x,y) and whence the result.

Sn

and v =

= an(8, x,y) + Fn+ 1(8, x,y)

8 n+1

from which we

177

(3.4) Definition. For 0

r

1 let G(s,r) be the operator defined as: (G(s,r)f)(z)

e

=

f( 0'.( s - r)), where 0'. is the unique geodesic from z to y parameterised to take time (1 - r). These form a two parameter semigroup on, for example, bounded measurable functions.

(3.5) Theorem. Recalling the definitions in (1.1) then we have for N

for 0

rn

rn-l

we assume

is bounded

0, provided that

...

< rl

1, is bounded for 2

n

< N + 1:

where

and for n

2:

and where

and for n

2:

where xt(u) is the semi-classical bridge process between x and y in time t. Observe that Rn(t,x,y)t n+ 1 is o(t n) as t tends to 0, so the above formula gives the asymptotic expansion of p( t, x, y) as t tends to o. Proof: Recalling (1.4) and (2.1.1) simply let s tend to t in (3.3) using dominated convergence for remainder term and finally change variables.

178

§4 Examples (4.1) Example. (Minakshish undaram-Pleijel Expansion) We calculate the first term from Theorem (3.5) of p(t,y,y) in the case c = 0:

=

al(y,y)

i

l

{G(r,o)

(y) dr.; ... drl'

_ LCy(y) Cy(y) , as G(r,O)

== Identity,

as (b+ V'logBy)(y)

=

°

=

1

1 .

1

2

- Z-dlvb(Y) - Z-lI b(y)jj ,

and 8 y(y) = 1, 1 1 . 1 2 = -S(y) - - dlvb(y) - -llb(y)11 12 2 2 '

where S is the scalar curvature, see Besse [1].

(4.2) Example. (Hyperbolic n-space) For hyperbolic n-space with constant sectional curvatures - R- 2 we have: n-1

8 y(x) =

-2

= for r = d(x, y), and 1jJ(r) = sinh( So by (1.2.1) applied to 8 y we see that:

= -(n -

l)1jJ"(r) 41jJ(r)

2E>y(x)

= -(n - 1) 4R2

+ (n -

=-(n-1)2

8R2

In particular when n

+ (n - l)(n - 3)

= 3 it

+

r2

8 l)(n - 3) 8

r2

8

is constant and so:

_

r2

1jJ'(r)2} 1jJ(r)2

1

}

R2

R2

1

}

179

which is just the kth term in the power series expansion of exp (2R12 ), which is what you would expect from the exact formula in Theorem (2.7). In general the scalar curvature S is given by, So observe that in this

-nY;2- 1).

special case of hyperbolic n-space we have

=

as pointed out in (4.1).

References.

Arede M.T. [1] (1983): Geomeiri« du noyau de la chaleur sur les verietes. These de Doctorat de 3eme Cycle Physique Tbeorique, Universite D'aix Marseille, Faculte des Sciences de Luminy. Azencott R. et al [1] (1981): Geodesiques et diffusions en temps petit. Semuieue de Ptobebilites, Universite de Paris VII. Asierique 84-85 Societe Mathematique de France. Azencott R. [2] (1984): Densite des diffusions en temps petit, developpemeiits asymptotiques, Semituure de Probebilite XVIII 1982/83. Lecture Note in Math. 1059, pp. 402-498: Springer Verlag. Besse A.L. [1] (1978): Manifolds all of whose geodesics are closed. Ergebnisse der Mathematik 93: Springer Verlag. Bismut J.M. [1] (1984): Large deviations and the Malliavin calculus. Progress in Mathematics 45, Birkhaiiser Verlag. Elworthy K.D. [1] (1982): Stochastic differential equations on manifolds. London Math. Soc. Lecture Notes series no. 70: Cambridge University Press. Elworthy K.D. and Truman A. [1] (1982): The diffusion equation and classical mechanics: an elementary formula. In 'Stochastic Processes in Quantum Physics' pp. 136-146. ed. S. Albeverio et el, Lecture Notes in Physics no. 173: SpringerVerlag. Elworthy K.D., Ndumu M. and Truman A. [1] (1986): An elementary inequality for the heat kernel on a Riemannian manifold and the classical limit of quantum partition function. In 'From Local Times to Global Geometry, Control and Physics' pp. 84-89, ed. K.D. Elworthy. Pitman Research Notes in Math. No. 150, Longman. Ikeda N. and Watanabe S. [1] (1986): Malliavin calculus of Wiener functionals and applications. In 'From Local Times to Global Geometry, Control and Physics' pp. 132-178, ed. K. D. Elworthy. Pitman Research Notes in Math. No. 150, Longman. Kifer Y.I. [1] (1976): On the asymptotics of the transition density of processes with small diffusion. Theory of Probability and its Applications XII no. 3. Molchanov S.A. [1] (1975): Diffusion processes and Riemannian geometry. Russian Math. Surveys 30, 1-53. Nash [1] (1956): The Imbedding Problem for Riemannian Manifolds. Ann. of Math. 63, 20-63.

180

Ndumu M. [1] (1986): An elementary formula for the Dirichlet heat kernel on Riemannian manifolds. In 'From Local Times to Global Geometry, Control and Physics' pp. 320-328, ed. K. D. Elworthy. Pitman Research Notes in Math. No. 150, Longman. Ndumu M. [2] (1987): Ph.D. Thesis, University of Warwick, in preparation. Watling K.D. [1] (1986): Foriuulee for solutions to (possibly degenerate) diffusion equations exhibiting semi-classical and small-time asymptotics. Ph.D. Thesis, University of Warwick.

STOCHASTIC MECHANICS FOR A POINT SOURCE by David Williams, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge, CB2 ISB Nick Steele and Aubrey Truman, Department of Mathematics and Computer Science, University College of Swansea, Singleton Park, Swansea, SA2 8Pp I. Introduction

The basic motivation comes from the following result. LEMMA.

Let

0,

X(jM),

(5)

••••

This time series can be computed by the iteration with the forward difference equation (6)

X«j+l)M) = X(j!'1t) + b(X(jM),jM)M + W«j+l)M) ­ W(jM),

for

j

0,1,2,....

The increment of the forward Wiener process in the right­hand

side is a Gaussian random variable with mean

°

and covariance

/Zt.

Samples of

such a Gaussian random variable can be generated as a sum of many uniformly distributed pseudo random numbers.

In this way, we can assign by computer a definite

sample value to the right­hand side of Eq. (6) in each step of iterations.

The time

series (5) thus obtained approximates a sample path of the stochastic process

X by

linear interpolation.[5] It is difficult to understand the dynamical property of a stochastic process by regarding only one sample path. sample paths as possible.

For the better understanding, we need to see as many

Then, the totality of those computer generated sample

paths of the stochastic process X should be taken into account as a qualitative representative of the unseen quantum trajectory of a particle. to the case study of quantum dynamics.

Now we will proceed

191 3.

Case Study 1 (Superposition) One of the most strange concepts in quantum mechanics is the superposition.

Each wavefunction subject to the Schrodinger equation (1) carries each dynamics of a quantum mechanical particle.

Given two wavefunctions

and

say, we can

always get another one by a linear combination of them. +

=

and a Z l superposition of

where

a

are complex numbers chosen such that and

= 1.

Since the new wavefunction

a

We call

also satisfies the

Schrodinger equation (1), it certainly carries quantum dynamics of the particle. However, dynamical meaning of the superposition

is obscure.

In the orthodox

interpretation of quantum mechanics, it is used to be understood as follows: suppose

and

operator

P

representing a certain dynamical variable.

proper values. value

P

z

are mutually orthogonal proper wavefunctions of a self-adjoint

PI

Then, we will find the value

with probability

the

lazlz,

Let

PI

and

with probability

if the dynamical variable

As long as the dynamical variable Z with probability lall

P

Pz

be their

lallZ

and the

is measured in

P

is concerned with, the Z a with . zl This is not the case, however, for other dynamical variables incompatible with P. superposition

looks like

In the superposition

l

and

there exists a kind of intereference between

and

This can be seen, for example, by computing the probability c1ens:ety, p

Z

+

JaIl

Z

Z

(8)

+ lazl

Z

Z

+ a{/Z

+ ala Z

The last two terms in the third right-hand side of Eq. (8) manifest the interference of dynamcics..

It seems difficult to extract some dynamical meaning of superposition

from the complex expression (8).

Therefore, it will be of interest to compare the

sample paths of the stochastic process

X associated with the superposition

those of stochastic processes

X

and

Z Dynamical aspect of the superposition

assume a and a l Z decompositions of

with and

associated respectively with then, will be less obscure.

to be real, for simplicity.

We

Let us write down the polar

and

(9) (10) = exp (R +

Then, it is immediate to see

Rand

8

(11)

i 8) •

in terms of

R

I,

R 8 Z' 1

and

8Z '

obtaining

192

R

(12)

arctan

S

The stochastic processes and

Xl' X

(13)

and

2,

associated with the wavefunctions

X

respectively, satisfy the forward stochastic differential equations b (Xl (t),t) dt + dW(t)

(14)

l

(15)

dX(t) where the drift vectors

b

l,

b

=

2,

b(X(t) ,t) dt + dW(t) , and

(16)

are given respectively by

b

(17) (18) b

(19)

VCR + S),

=

By Eqs. (12), (13), and (19), we may understand how complex the dynamics in the superposition

W is.

The drift vector

shows highly nonlinear and complicated

b

b and b 2. l In order to extract some dynamical and probabilistic aspects of superposition,

dependences on the drift vectors

we consider here the simplest case of a free particle in one dimension.

Let

exp{+ipx-i(p2/2)t}

be an outgoing wave, and = exp{-ipx-i(p2/2)t} be an = Ir and a = 11 _r 2 for a positive constant r, and 2 l introduce the superposition = alw + a w. For the equal weight superposition =

incoming wave.

with

r =

We put

a

2

1 2'

becomes a standing wave proportional to

interpretation of quantum mechanics,

p

moving freely with constant momenta r

=

21

and incoming waves density

and

-p,

and

cos(px).

In the orthodox

represent each particle dynamics and the superposition

W with

represents a particle dynamics standing in a local spatial region.

see the sample paths of P 2 "1,

WI

Xl' X and

2,

o:

with

Let us

In figures 1, notice that the outgoing

carry uniform probability densities

whereas the standing wave P

X.

and

r

=

cos 2 (px).

PI" 1

and

21 carries a periodic probability

The totality of those computer generated sample paths well manifests the qualitative dynamical aspects of particle trajectories associated respectively with outgoing wave

wI'

incoming wave

W2'

and standing wave

What about the dynamics in the general superposition show the sample paths of

Xl' X

2,

and

X,

for

r

=

W. with

r #

1 2? Figures 2

0.1, 0.2, 0.3, 0.4, 0.5, 0.6,

193

Fig.1 b

Fig.1 a

Fig.lc Sample paths of a quantum mechanical free particle associated with the outgoing wave the standing wave p

= 1.0,

t-range

=

= 5.0,

(Fig. la), the incoming wave +

(Fig. lc).

and

x-range

(Fig. lb), and

Numerical factors are chosen as

= 7.5.

194

Fig.2a

Fig.2c

Fig.2d

Fig.2e

Fig.2f

195

Fig .2g

Fig .2h

Fig .2i Fig.

Sample paths of a quantum mechanical free particle associated Z with the superposition 1jJ = li1jJl + IO-r )1jJZ' for r = 0.1 (Fig. Za), O.Z (Fig. Zb), 0.3 (Fig. Zc), 0.4 (Fig. Zd), 0.5 (Fig. Ze), 0.6 (Fig. Zf), 0.7 (Fig. Zg), 0.8 (Fig. Zh), 0.9 (Fig. Zi). same as Fig. 1.

Numerical factors are the

196

0.7, 0.8, 0.9.

There, you will see that the more you have

in the superposition,

the more you find sample paths going away from left to right. 4.

Case Study 2 (Two Slit Interference) Let us recall the physical setting of the famous two-slit interference thought

experiment.

A quantum particle is emitted from a certain source and it reaches a

certain point of a detecting film later on.

Between the source and the detecting film

we place an infinite plate with two parallel slits separated by a small distance.

The

probability distribution of the particle on the detecting film, that can be obtained by successive emissions of the particles, shows the interference pattern when the two slits are open and one does not observe through which one the particle goes.

Once one

observes that the particle goes through one of the slits, the interference pattern disappears. In the orthodox interpretation of quantum mechanics, we cannot say anything about the particle trajectory except the probability distribution of position, though it is indeed a particle that hits the detecting film. through which slit passed the particle.

This prohibited us from asking

It will be quite interesting, therefore, to

revisit the problem of two-slit interference from the point of view of stochastic mechanics.

Since then we can illustrate the unseen trajectories of a particle.

Nelson investigated it analytically in his recent book,[3] and we will use the same notation as he. We wish to claim that the wavefunction does go through both of the two slits but not the particle itself, though it is impossible to know which is crossed.

Indeed,

we cannot observe a particle passing through a slit without giving fatal damage to the wavefunction.

As the wavefunction guides the particle where it has to go, the

dynamics of the particle after the measurement does differ from the original one so that the interference disappears.

It seems worthwhile to see the conceptual

difference between the physical conditioning by measurements and the mathematical conditioning by constraining sample paths. whereas the latter does not.

The former does touch the wavefunction,

As Zambrini and Nelson have shown,[3,7,S] the probability

distribution of the particle with interference is precisely the sum of the conditioned probability distributions, one conditioned to pass through the upper slit and the other the lower one.

The concept of probability and the Bayes rule in quantum

mechanics do not differ from the conventional ones. It has been believed by many physicists that the concept of probability is completely different from the conventional one only because the probability distribution with interference cannot be expressed as a sum of the probability distributions conditioned physically to pass through each of the two slits.

Physical

conditioning such as to close the lower slit modifies the wavefunction and so the underlying stochastic process.

Bayes rule should be satisfied only when we assume

the mathematical conditioning that does not modify the wavefunction.

It is not

proper to modify the good old concept of probability in order to avoid the real

197

Fio.3a

Fig.3b

Fig.3c

Fig.3d

198

Fig.3e

Fig.3f

Fig.3g Fig.

Sample paths of a quantum mechanical free particle passing

through the two slits.

Separation of the two slits is 1.0,

of each slit is

Time scale is chosen to be 0.5 (Fig. 3a), 5.0 (Fig. 3b),

1.25.

and the width

20.0 (Fig. 3c), 90.0 (Fig. 3d), 120.0 (Fig. 3e), 200.0 (Fig. 3f), 500.0 (Fig. 3g).

199

Fig.4a

Fig.4b

Fig.4c

Fig.4d

200

Fig .41

Fig.4e

Fig.4g

Fig.

Sample paths of a quantum mechanical free particle conditioned

to pass through the upper slit.

Numerical factors are the same as Fig. 3.

201

difficulties. In figures 3 and figures 4 we see the totality of sample paths of a stochastic process associated

with the superposition of two Gaussian wavefunctions passing

through the two slits.

It may illustrate the unseen quantum trajectories of the

particle in the problem of two-slit interference. scales.

We show it for various time

We also show the totality of sample paths conditions mathematically to pass

through the upper slit.

The complicated dynamics of quantum interference will be

found there. 5.

Case Study 3 (Harmonic Oscillators) One of the most familiar dynamical systems in physics is a one-dimensional

harmonic oscillator.

Since the potential energy is of quadratic form

v

(20)

the Schrodinger equation (1) can be integrated. proper wavefunctions

such that

Especially, we have a series of (x) exp {-1'(n +})t}

and

"n

solves

the proper value problem (21)

for

n; 0,1,2,....

We call

the ground state,

the second excited state, and so on.

the first excited state,

Those proper wavefunctions

can be

obtained explicitly in terms of Hermite polynomials with a common Gaussian weight. For example, we have (22)

(23)

(24)

They are also called stationary states, since the probability densities are timeindependent. From the point of view of stochastic mechanics, the dynamics of a quantum is given by the

mechanical harmonic oscillator in the stationary state stochastic process

X

with invariant probability density

I

n

1

2

•

For the better

understanding of the dynamical aspect of stationary states of the harmonic oscillator, figures 5 exhibit the totality of computer generated sample paths of the stochastic processes excited state

XIS

associated with the ground state

and the second excited state

respectively.

there the segregation and mutual balancing of sample paths.

the first We will find

202

Fig.50

Fig.5b

Fig.5c Fig.

Sample paths of a quantum mechanical harmonic oscillator in

the ground state

(Fig. Sa),

and in the second excited state 10.0.

in the first excited state (Fig. Sc).

(Fig. Sb),

Time scale is chosen to be

203

Fig.60

Fig. 6a,6b.

Fig.6b

Sample paths of a quantum mechanical harmonic oscillator in

the superpositions

=

+

(Fig. 6a)

and

=

+

+

(Fig. 6b).

204

It has been a physical conjecture that the more you superpose the higher excited states, the more the quantum trajectories approach the classical ones.

To

see whether this conjecture seems reasonable or not, we will compute the sample paths of the stochastic processes associated with the superpositions I)J

r =

I)J" =

Wo

W + I)Jl'

(25)

+ I)Jl + 1)J2'

(26)

o

respectively, and exhibit them in figures 6. seem to claim the validity of this conjecture.

The computer generated sample paths Most of the sample paths gather into

the vicinity of a classical oscillatory trajectory. 6.

Conclusions A numerical analysis with computer of the quantum mechanical trajectory of a

particle has been proposed within the realm of stochastic mechanics.

It has been

shown that the present approach with computer graphics illustrates the typical samples of the unseen trajectory of a particle in quantum mechanics.

Intuitive

meanings of fundamental concepts in quantum mechanics such as the superposition and the interference have been clarified. Acknowledgements The author would like to take the opportunity to thank Professor Aubrey Truman for a very kind invitation to participate in the conference.

He also thanks

Professor Jean-Claude Zambrini for helpful advices and correspondences, Professor Tetsuya Misawa for intensive discussions, and Miss Mihoko Monobe for skillful assistance in preparing the manuscript.

Computation was performed at the Research

Institute for Informatics and Science (RIIS) of Notre Dame Seishin University with TOSBAC UX-700.

The hospitality and support of the Research Center BiBoS of the

University of Bielefeld are gratefully acknowledged. References 1.

1. Fenyes, Z. Phys , 132, 81 (1952).

2.

E. Nelson, Dynamical Theories of Brownian Motion, (Princeton University Press,

3.

E. Nelson, Quantum Fluctuations, (Princeton University Press, Princeton, 1985).

4.

A. Truman and J.T. Lewis, "The Stochastic Mechanics of the Ground-State of the

5.

K. Yasue and J.-C. Zambrini, Ann. Phys. (N.Y.) 159,99 (1985).

6.

E. Nelson, "Field Theory and the Future of Stochastic Mechanics", BiBoS preprint

Princeton, 1967).

Hydrogen Atom", BiBoS preprint 37/84, to appear in Proceedings BiBoS I.

173/85, to appear in Proc. Ascona Conference "Stochastic Processes in Classical and Quantum Systems". 7.

J.-C. Zambrini, Phys. Rev. A33, 1532 (1986).

8.

J.-C. Zambrini, "Variational processes and stochastic versions of mechanics", to appear in J. Math. Phys.

NEW PROBABILISTIC APPROACH TO THE CLASSICAL HEAT EQUATION J.C. Zambrini Mathematics Institute University of Warwick Coventry CV4 7AL 1. Historical survey To look at the history of the theories of Brownian motion is a fascinating exercise in which the indispensable interaction between theoretical physics and mathematics is illustrated in a particularly dramatic way. It also suggests that a requirement to discover today something new in this area might well be the modestyneeded to admit the equal contri butions of both concerned communiti es. After the pioneering time of the botanist Brown and the mathematician Bachelier. the concept of Brownian motion became familiar to the physicists thanks to the works of Einstein. Smoluchowski. Langevin, Ornstein and Uhlenbeck among others. Then Levy. Wiener. Doob and gave mathematical substance to these somewhat formal physical theories. Notice that all of them were motivated by the need of classical physics. After the quantum revolution, but with at least a major exception. to be specified later. physicists lost interest in the subject. till 1949 when Feynman published his work on the path integral formulation of nonrelativistic quantum mechanics. His (formal) result was that it is possible to avoid the conventional description of the quantum dynamics in terms of the evolution group. i.e. of the one­parameter group of unitary operators on the Hilbert space 2 H = L {lR ) (with scalar product