An Introduction to Stochastic Partial Differential Equations 9783540164418

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AN INTRODUCTION TO STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

John B. WALSH

The general p r o b l e m governed

by a p a r t i a l

perturbed

randomly,

perhaps

in time?

u(x,t)

is the p o s i t i o n

sandstorm

Suppose

differential

evolve

Think

in c a l m air u(x,t)

is this.

one is given a p h y s i c a l

equation.

by some

for example

Suppose

of one of the strings

would

satisfy the w a v e

Let W

x and time t.

equation

xt

represent

The number

the i n t e n s i t y

of grains h i t t i n g

time,

so that,

subtracting

a mean

white

noise,

and the

final e q u a t i o n Utt(x,t)

w h e r e W is a white noise two-parameter

white

of o r d i n a r y derivatives

though

higher

of this

= Uxx(X,t)

However,

by a s u c c e s s i o n

if a of

at the p o i n t

at another

point

and and

W m a y be a p p r o x i m a t e d

by a

+ W(x,t),

equation

or,

in other words,

- not s u r p r i s i n g

in it exist.

However,

in this

non-differentiable,

the s o l u t i o n

-

fails:

This

is one of the t e c h n i c a l

involving

.

is

differential

this

most

intensity,

stochastic

dimensions

distribution-valued

hitting

in both time and space,

and then show that

continuous,

xx

a

noise.

One p e c u l i a r i t y

In

= u

tt

If

t, t h e n

the string at a given p o i n t

of the number

after

left outdoors.

of the b o m b a r d m e n t

independent

equation,

u

is then

How does it

at the p o i n t x and time

time will be largely

partial

carelessly

should b l o w up, the string w o u l d be b o m b a r d e d

sand grains.

behavior

that the system

sort of a white noise. of a guitar

system

- is that none of t h e

one may rewrite

form there

is a solution

say,

rather

turns out to be a distribution, barriers

in the subject:

and this has g e n e r a t e d

extensive

it as an integral which

is a

function.

w i t h a drumhead,

solutions,

a fairly

equations

in v i e w of the

use of functional

than a string - even not a function.

one m u s t

deal w i t h

a n u m b e r of a p p r o a c h e s ,

analysis.

267

Our aim is to study a c e r t a i n number of such stochastic p a r t i a l d i f f e r e n t i a l equations, to see how they arise, to see how their solutions behave, and to examine some techniques of solution.

We shall c o n c e n t r a t e

m o r e on p a r a b o l i c equations than on h y p e r b o l i c or elliptic, and on equations in which the p e r t u r b a t i o n comes from something akin to white noise. In particular, one class we shall study in detail arises from systems of b r a n c h i n g diffusions.

These lead to linear p a r a b o l i c equations whose

solutions are g e n e r a l i z e d O r n s t e i n - U h l e n b e c k processes, and include those studied by Ito, Holley and Stoock,

Dawson, and others.

Another r e l a t e d class

of e q u a t i o n s comes from certain n e u r o p h y s i o l o g i c a l models. Our p o i n t of view is more r e a l - v a r i a b l e o r i e n t e d than the usual theory, and, we hope, slightly more intuitive.

We regard white noise W as a measure

on E u c l i d e a n space, W(dx, dr), and c o n s t r u c t stochastic integrals of the form f f ( x , t ) d W directly,

f o l l o w i n g Ito's original construction.

This is a

t w o - p a r a m e t e r integral, but it is a p a r t i c u l a r l y simple one, known in t w o - p a r a m e t e r theory as a " w e a k l y - a d a p t e d integral".

We generalize it to

include integrals with respect to m a r t i n g a l e measures,

and solve the

equations in terms of these integrals. We will need a certain amount of machinery:

n u c l e a r spaces, some

e l e m e n t a r y Sobolev space theory, and weak c o n v e r g e n c e of stochastic processes with values in Schwartz space.

We develop this as we need it.

For instance, we treat SPDE's in one space dimension in Chapter 3, as soon as we have d e v e l o p e d the integral, but solutions in h i g h e r d i m e n s i o n s are generally Schwartz distributions,

so we develop some e l e m e n t a r y

d i s t r i b u t i o n theory in Chapter 4 before t r e a t i n g higher dimensional e q u a t i o n s in C h a p t e r 5.

In the same way, we treat w e a k c o n v e r g e n c e of =S'-valued

processes in Chapter 6 before t r e a t i n g the limits of infinite p a r t i c l e systems and the Brownian density p r o c e s s in Chapter 8. After c o m p a r i n g the small p a r t of the subject we can cover w i t h the m u c h larger mass we can't, we had a m o m e n t a r y desire to re-title our notes:

"An

Introduction to an Introduction to Stochastic Partial Differential Equations";

268

w h i c h means

that the i n t r o d u c t i o n

to the notes,

w o u l d be the i n t r o d u c t i o n

to "An I n t r o d u c t i o n

to begin w i t h an infinite

regression.

an introduction,

not a survey.

on the subject,

w h a t we do cover

knows?

even p h y s i c a l l y

Perhaps

Let's

While we w i l l

which

... ", but no.

It is not g o o d

just keep in m i n d that this

is

forego m u c h of the recent w o r k

is m a t h e m a t i c a l l y useful.

you are now reading,

interesting

and, w h o

CHAPTER ONE WHITE NOISE AND THE BROWNIAN Let random

(E,~,v)

be a ~ - f i n i t e

measure

space.

set f u n c t i o n W on the sets A e =E of finite (i)

W(A)

is a N ( 0 , v ( A ) )

(ii)

if A ~ W(A ~

In m o s t To see that sets of ~ Gaussian

cases,

: {W(A), process

are

exists,

space

and

< ~}.

From

(i) and

C(A,B)

= E{W(A)

W(B)}

t h e o r e m on G a u s s i a n

process

measure•

i n d e x e d by the

(ii) this m u s t be a m e a n - z e r o

C given by

process

and v w i l l be L e b e s g u e

think of it as a G a u s s i a n

function

a Gaussian

such that

independent

covariance

By a general there exists

v-measure

+ W(B).

A £ ~, v(A)

with

and W(B)

E w i l l be a E u c l i d e a n

such a p r o c e s s

A w h i t e n o i s e b a s e d on v is a

randc~a v a r i a b l e ;

B = ~, then W(A) B) = W(A)

SHEET

= v(A ~

processes,

B).

if C is p o s i t i v e

w i t h m e a n zero a n d c o v a r i a n c e

definite,

function

C.

Now

let

A I , • ..,A n be in E= and let el,. -',a n be real numbers. a i a j C(Ai,A j) =

~ aia j fI A (x) I A (x)dx i,j i j

i,j

= f(~ a i IA.( x))2~ h 0. i Thus C is a p o s i t i v e mean

zero G a u s s i a n

definite,

process

so that there

{W(A)}

on (Q,~,P)

exists

a probability

space

such that W s a t i s f i e s

(Q,~,P)

(i) and

and a

(ii)

above• There are other ways of d e f i n i n g w h i t e noise• Lebesgue motion".

measure,

it is often

Such a d e s c r i p t i o n

the B r o w n i a n

sheet

rather

Brownian

is p o s s i b l e

to the case

If t =

is

G a u s s i a n process.

in h i g h e r

as the "derivative dimensions

{Wt,

If s =

too,

of B r o w n i a n

but it involves

motion. E = R n+ = {(tl,...,tn):

(t1,•..,t n) £ R n+ , let

s h e e t o n R~ is the p r o c e s s

a mean-zero function

measure•

informally

than B r o w n i a n

Let us specialize v = Lebesgue

described

In case E = R a n d v =

t £

R~}

(0,t]

defined

(s1,.o°,s n) and t =

=

ti>0 ,_

i=1,...,n}

(0,tl]×...×(0,tn].

by W t = W { ( 0 , t ] } . (tl,...,tn),

and The

This is

its c o v a r i a n c e

270

(1.1)

E{WsWt} If we r e g a r d

Notice W(R)

that

we can

is g i v e n

can

= W

be c o m p u t e d

approximated

recover

b y the

W((u,v),(s,t)]

W(A)

st

the w h i t e

usual

formula

- W

- W

sv

finite

Interestingly, J.

Kitagawa,

in

an idea of w h a t curves

in

R~ I).

Brownian C(t,t')

this

the

uv

Brownian

process

for

R n+ f r o m

Wt,

for

if R is a r e c t a n g l e ,

(if n = 2 a n d 0 < u < s, 0 < v < t, If A is a f i n i t e

).

Borel

of r e c t a n g l e s

in o r d e r

W vanishes

in

function.

A

sheet

was

to do a n a l y s i s

looks

like,

a way

first

consider

n = 2, v = L e b e s g u e

on the axes.

can be

0 . by a statistician,

in c o n t i n u o u s

its b e h a v i o r

time.

along

To get

some

measure.

If s = s

it is a m e a n - z e r o

measure

W(A)

that

introduced

of v a r i a n c e

let's

of r e c t a n g l e s ,

set A of f i n i t e

in s u c h

n

union

- W ( A n ) ) 2} = v ( A - A n) + v ( A n - A) +

, in the c a s e

motion, = s

1951

noise

- W

ut

W t is its d i s t r i b u t i o n

and a general

unions

E{(W(A)

(slAtl)'''(Sn Atn )

as a m e a s u r e ,

by a d d i t i v i t y ,

by

=

Gaussian

> 0 is f i x e d ,

o

process

with

{ W s t" t>0} o

covarianoe

is a

function

(tat').

o

2).

Along

the h y p e r b o l a

st =

I, let

Xt = W t -t" e ,e Then stationary

{Xt,

- ~ < t < ~}

is an O r n s t e i n - U h l e n b e c k

process,

Gaussian

process

with

I, a n d c o v a r i a n c e

C(s,t)

mean

= E{W

zero,

s e

3). process

the

of i n d e p e n d e n t

increments paths

Along

in

are

stationary.

although

The

Just

as

same

in one p a r a m e t e r ,

transformations

which

M t = Wtt

is a m a r t i n g a l e ,

it is not a B r o w n i a n

is t r u e

take

there one

I Ast = ~

Scaling: Cst

Inversion:

= st

W1 1 ; s

~anslation

function

-s W t -t } = e - l S - t l " e ,e

the p r o c e s s

increments,

a strictly

if we c o n s i d e r

and even

motion, W along

by

(So,to):

t

Est = Wso+S,to+t

are

Brownian

scaling, sheet

inversion,

into

increasin@

and

another.

Wa2s,b2 t DSt

= s W1 s

- Wso+S,t ° - Wso,to+t

+ Ws t oo

a

for t h e s e

2 R+ • 4).

translation

not

diagonal

,e

variance

i.e.

271 T h e n A, C, D, and E are B r o w n i a n sheets, and moreover, E is independent of F*s t = ~ {Wuv o o

:

u


t = t + t = %(A,B)+

%(A,C).

Moreover, by the general theory, IQt(A,B)I ! Q t (A,A) I/2 e t (B,B)I/2" A set A x B x (s,t] C

E x E x R+ will be called a rectangle.

Define a

set function Q on rectangles by

e(A x H × (s,t]) = Qt(A,B~ - QsCA,B~, and e x t e n d Q by a d d i t i v i t y to finite disjoint unions of rectangles, A i x B i x (si,ti] are disjoint,

i =

1,...,n, set

i.e. if

290

n

(2.2)

n

Q( U A i x Bi x (si,ti]) = [ IQtl(Ai,Bi) - Qs (Ai'Bi))" i=1 i=1 " i

Exercise 2.3.

Verify that Q is well-defined, i.e. if

n m A = i=IU Ai× B i x (si,ti ] = jU__.=IA"x 3 the same value for Q(A) i n ( 2 . 5 ) . If a 1,...,a

n

e R

(2.3)

B'x 3

(s3't3]' each representation gives

(Hint:

use biadditivity.)

and if A I,...,A 6 E are disjoint, then for any s < t n ~n n [ 7 aia j Q(A i × Aj ×(s,t]) > 0, 9=I 9=I

for the sum is =

[ aiajI t - s) i,j

=
~ 0. i 1

A signed measure K(dx dy ds) on E x E × B is positive definite

if for each bounded measurable function f for which the integral makes sense, (2.4)

f f(x,s)f(y,s)K(dxdyds) ~ 0 EXEXR + For such a positive definite signed measure K, define (f'g)K =

f f(x,s)g(y,s)K(dxdyds). EXExR +

Note that (f'f)K ~ 0 by (2.4).

Exercise 2.4.

Suppose K is symmetric in x and y.

Prove Schwartz' and

Minkowski's inequalities _,I/2, ,I/2 (f'g)K ~ (f'r;K ~g'g;K •

and

,I/2

(f+g' ftg;K

_,I/2

~ (f'z;K

I/2 + (g'g)K

It is not always possible to extend Q to a measure on E x E × B, where B = Borel sets on

R+, as the example at the end of the chapter shows.

are led to the following definition,

We

291

DEFINITION. o-finite

A martingale

measure

K(A,~),

m e a s u r e M is w o r t h y A ~

E x E x B, ~ E Q,

(i)

K is p o s i t i v e

(ii)

for f i x e d A, B,

(iii)

for all n,

(iv)

for any r e c t a n g l e A,

definite

E{K(E

n

x E

simply

replace

special

t > 0} is p r e d i c t a b l e ; < ~;

of M.

We w i l l

cases m e n t i o n e d

and those w i t h n u c l e a r confidence

in x a n d y;

is no restriction.

it by K(dx dy ds) + K(dy dx ds). on M.

a random

IQ(A)I ! K(A).

that K be s y m m e t r i c

is a s t r o n g c o n d i t i o n important

x [0,T])}

n

exists

such t h a t

and s y m m e t r i c

{K(A x B x (0,t]),

We call K the d o m i n a t i n g m e a s u r e The requirement

if there

covariance

Apart

If not,

from this,

show b e l o w that it h o l d s

above:

both orthogonal

are worthy.

that we will have no dealings

In fact,

with unworthy

we

however

it

for the two

martingale

measures

we can state w i t h measures

in these

notes. If M is w o r t h y is a p o s i t i v e

set function.

restrict o u r s e l v e s is f i n i t e l y set f u n c t i o n measure. measure

with c o v a r i a t i o n

for a.e. ~.

dominated

In particular,

on E x E x ~, and the total

Orthogonal = {(x,x):

PROPOSITION supported

PROOF.

measures x ~ E},

2.1.

by A(E)

Q(A x B ×

If M is o r t h o g o n a l IQI[ E x E - A(E)) vanishes

2K,

(2.3),

and white

of E x E x B upon w h i c h

and hence

A worthy martingale

are easily

Q(-,~) additive to a

to a signed

of Q s a t i s f i e s

Q w i l l be p o s i t i v e

be the d i a g o n a l

finitely

can be e x t e n d e d

can be e x t e n d e d

variation

noises

K, then K + Q

so that we can first

T h e n K + Q is a p o s i t i v e

for a.e. ~ Q(o,~)

By

measure

E is separable,

subalgebra

by the m e a s u r e

for all A E E x E x B.

d(E)

The o-field

to a c o u n t a b l e

additive

Q and d o m i n a t i n g

IQI(A) ! K(A)

definite. characterized.

Let

of E.

measure

is o r t h o g o n a l

iff Q is

x R+

[0,t]) = t-

and A ~ B = ~, this v a n i s h e s x R+] = 0, i.e.

supp Q c d(E)

hence x R+

for all d i s j o i n t A and B, M is e v i d e n t l y

.

Conversely,

orthogonal.

if this Q.E.D.

292

S T O C H A S T I C INTEGRALS

We are only going to do the L 2 - t h e o r y here - the bare bones, so to speak.

It is p o s s i b l e to e x t e n d our integrals further, but since w e w o n ' t

n e e d the extensions in this course, we will leave t h e m to our readers. Let M be a worthy m a r t i n g a l e m e a s u r e on the L u s i n space

(E,E), a n d let

QM a n d K M be its c o v a r i a t i o n and d o m i n a t i n g m e a s u r e s respectively.

Our

d e f i n i t i o n of the stochastic integral may look u n f a m i l i a r at first, but we are m e r e l y f o l l o w i n g Ito's c o n s t r u c t i o n in a different setting. In the classical case, one constructs the s t o c h a s t i c integral as a process rather than as a random variable.

That is, one constructs

t {f f dB, t ~ 0} s i m u l t a n e o u s l y for all t; one can then say that the i n t e g r a l 0 is a martingale,

for instance.

is "martingale measure".

The analogue of "martingale" in our s e t t i n g

Accordingly, we will define our stochastic i n t e g r a l

as a m a r t i n g a l e measure. R e c a l l that we are r e s t r i c t i n g ourselves to a finite time interval {0,T] and to one of the En, so that M is finite. the integral for e l e m e n t a r y functions,

As usual, we will first define

then for simple functions,

and then

for all functions in a certain class by a functional c o m p l e t i o n argument.

DEFINITION.

A f u n c t i o n f(x,s,~)

(2.5)

is e l e m e n t a r ~ if it is of the f o r m

f(x,s,~) = X(~)I(a,b](S)

w h e r e 0 _< a < t, X is b o u n d e d and ~a-measurable, it is a finite sum of e l e m e n t a r y functions,

IA(X), and A 6 =E"

f is sim~le ' if

we denote the class of simple

functions by S.

DEFINITION. by S.

The ~ r e d i c t a b l e G - f i e l d P on Q x E × R + is the G - f i e l d generated

A f u n c t i o n is p r e d i c t a b l e if it is P - m e a s u r a b l e . We define a n o r m

II

IIM on the p r e d i c t a b l e functions by

llf IIM = E{~Ifl, Ifl~K} I/2

293

Note that we have used the absolute

value of f to define

Let ~M be the class of all predictable

PROPOSITION

2.2.

I}f IIM, so that

f for which

llf IIM < ~.

Let f E ~M and let A = {(x,s):If(x,s) I > £}.

Then

E × [0,T]>} i T I }{f{{M E{K(E x E x [0,T])}

E{K(A×

PROF.

~ E t + t = ta.s.

Indeed, t = t = t + t + 2 t, and the last t e r m v a n i s h e s since M is o r t h o g o n a l . 2°

A C B => t ~ t S

3°

< t --

=>

< s --

~ is a G - f i n i t e measure:

S

it must be Q - f i n i t e since M T is, a n d

a d d i t i v i t y follows by t a k i n g expectations in I". The i n c r e a s i n g p r o c e s s t is f i n i t e l y a d d i t i v e for each t by I°, but it is better than that.

It is p o s s i b l e to construct a v e r s i o n w h i c h is a

m e a s u r e in A for each t.

T H E O R E M 2.7. measure.

Let {Mt(A) , ~t' 0 < t < T, A E =E} be an o r t h o g o n a l m a r t i n g a l e

Then there exists a family {vt(.), 0 < t < T} of random Q - f i n i t e

m e a s u r e s on (E,E) such that (i)

{vt, 0 < t < T} is predictable;

(ii)

for all A 6 ~, t ÷ vt(A) is r i g h t - c o n t i n u o u s and increasing;

(iii) P{vt(A) = t} =

PROOF.

I all t ~ 0, A ~ ~.

we can reduce this to the case E C R, for E is h o m e o m o r p h i c to a

Borel set F C R.

Let h: E + F be the h o m e o m o r p h i s m ,

Mt(A) = Mt(h-I(A)),

~(A) = ~(h-1(A)).

and define

If we find a ~t s a t i s f y i n g the

c o n c l u s i o n s of the t h e o r e m a n d if ~ t ( R - F) = 0, then v t = vt° h satisfies the theorem.

Thus we may assume E is a Borel subset of

Since M is Q-finite, are compact K n C all n.

R.

there exist E n + E for w h i c h ~ ( E n ) < ~"

E n such that u(E n- K n) < 2 -n.

We may also assume K n C

It is then enough to p r o v e the t h e o r e m for each K . n

assume E is compact in

R a n d ~(E)

Define Ft(x) = t' -~ < x < ~.

Then t h e r e

Thus we may

Kn+ I

300

Then

x t, x', t'~ Q}.

We claim that F

t

is the distribution

This will be the function

of a

measure. c)

t I ~ 1 = = . n

•

C H A P T E R THREE EQUATIONS

IN ONE SPACE D I M E N S I O N

We are going to look at stochastic white noise and similar processes.

x C

R d, d ~ 2.

invariably, continuous.

and x is the space variable.

between t~e case where x is one-dimensional

In the former case the solutions

real-valued

functions.

On the other hand,

are only generalized

equations

The solutions will be functions

x and t, where t is the time variable to be a big difference

partial differential

are typically,

R d, the solutions

of the variables There turns out and the case where

though not

They will be non-differentiable,

in

driven b y

but are usually

are no longer functions,

but

functions.

We will need some knowledge of Schwartz d ~ 2, but we can treat some e x ~ p l e s do that in this chapter,

distributions

in one dimension by hand,

to handle the case so to speak.

and give a somewhat more general treatment

We will

later, when we

treat the case d > 2.

wA~

TH~

EQUATION

Let us return to the wave equation of Chapter one: I

(3.1)

~2V ~t 2

~2V ~x 2

+ W

t > 0, x E

V(x,0) = 0,

x e

R

~(x,0)

x E

R.

= 0,

White noise is so rough that solution will not be differentiable. equation which will be solvable.

(3.1) has no solution: However,

we can rewrite

R ×

any candidate

[0,T], where T > 0 is fixed.

it as an integral

Assume

of compact support and integrate

for the sake of argument

V ~ C (2) . T f f OR

for a

This is called a weak form of the equation.

We first m u l t i p l y by a C~ function ~(x,t) over

R;

"

[Vtt(x,t)-V(x,t)]~(x,t)~dt

T = I I OR

~(x,t)W(x,t)~dt.

that

309

Integrate by p a r t s twice on the left-hand side.

N o w # is of compact support in x,

but it may not vanish at t = 0 and t = T, so we will get some b o u n d a r y terms: T

f f vcx,t)c%t(x,t)-~=cx,t)1~dt + f t~(x,')vt(x,')10-%cx,')vcx,')l~1~ 0

R

R T

= I f ,(x,t)~Cx,t)~dt 0

R

If ~(x,T) = ~t(x,T) = 0, the b o u n d a r y terms will drop out b e c a u s e of the initial conditions.

DEFINITION.

This leads us to the following.

We say that V is a w e a k solution of (3.1) p r o v i d i n g that V(x,t) is

l o c a l l y i n t e g r a b l e and that for all T > 0 and all C~ functions ~(x,t) of c o m p a c t support for w h i c h ~(x,T) = ~t(x,T) = 0, V x , T f ~ V(x,t)[~tt(x,t) 0 R

(3.2)

we have

- ~xx(X,t)]dxdt =

The above a r g u m e n t is a little unsatisfying; satisfies

it indicates that if V

(3.1) in some sense, it should satisfy (3.2), w h i l e it is really the

c o n v e r s e we want.

We leave it as an exercise to verify that if we replace W b y a

smooth function f in (3.1) and (3.2), and if V satisfies it does in fact satisfy

T H E O R E M 3.1. I ^ V(x,t) = ~ W

PROOF.

T ~ f ~dW. 0 R

(3.2) and is in C (2), t h e n

(3.1).

There exists a unique continuous solution to (3.2), n a m e l y (t-x, t+x) ...... , w h e r e W is the m o d i f i e d Brownian sheet of Chapter One.

Uniqueness:

if V I and V 2 are both c o n t i n u o u s and satisfy

(3.2), then t h e i r

d i f f e r e n c e U = V 2 - V I satisfies ffU(x,t)[~tt(x,t) Let f(x,t) be a ~ exists a ~ E ~ C(x,t; Xo,to)

- ~xx(X,t)]dxdt = 0

f u n c t i o n of compact support in

R x (0,T).

Notice that there

w i t h #(x,T) = ~t(x,T) = 0 such that ~tt - ~xx = f" is the indicator f u n c t i o n of the cone

{(x,t): t0} Now V satisfies

(33~ /f

[ ff

(3.2) iff the following vanishes:

½~u,v;u,v>~Cdudv~]~$uv(U',v~du'dv-

ff

{u'+v'>0} { u+v>0}

$Cu,v,~(dudv~

{ u+v>0}

We can interchange the order of integration by the stochastic Fubini's t h e o r e m of Chapter Two :

=

ff {u+v>o}

If f ~ u v v

(u°,v°)du'dv

'

-

~(u,v)

(dudv).

u

But the term in b r a c k e t s v a n i s h e s identically,

for ~ has compact support. QED

The literature of t w o - p a r a m e t e r p r o c e s s e s contains studies of s t o c h a s t i c d i f f e r e n t i a l equations of the form (3.4)

dV(u,v) = f(V)dW(u,v) + g ( V ) d u d v

where V and W are two p a r a m e t e r processes,

and dV and dW represent t w o - d i m e n s i o n a l

increments, w h i c h we w o u l d write V(dudv) and W(dudv).

These equations rotate into

the n o n - l i n e a r wave e q u a t i o n Vtt(x,t) = Vxx (x,t) + f(V)W(x,t) + g(V) in the region {(x,t): t > 0, -t 6).

1 - P3 and

Then

tL ftfL •F (x,t) < c(f0f0Gs2eq(x,y~?p/2q ~{l~(y,s)-vn-1(y,s)IP}G(1-e)e(~,~) ayas --

0

t-s

0

In this case 2eq < 3, so the first factor is bounded;

by (3.13) the expression

is

t < C/

Hn_1(s)

(t-s}ads,

0 1 where a = ~ (1+ep-p)

> -I, and C is a constant.

Thus t (3.14)

Hn(t) < Cf

Hn_1(s)(t-S)ads,

t >__ 0

0 for some a > -I and C > 0.

Notice that if Hn_ I is bounded on an interval

[0,T], so

is H . n t

Ho(t) I and constant CI,

t hn(t) ~ Clf0hn_1(s)(t-s)ads,

n =

1,2, . . . .

Then there is a constant C and an integer k ) I such that for each n ~ t e

such that h 0

I and

[0,T], t (t-s) hn+mk(t) ~ cmf hn(S) ~ l a s , 0

(3.15)

m = 1,2 . . . . .

Let us accept the lemma for the moment. that for each n,

~ m=0

does

I/i)

~ n=0

(Hn(t))

(Hn+mk(t)) 1/p converges

Thus vn(x,t)

converges

It applies to the Hn, and implies

uniformly

on compacts,

and therefore

in L p, and the convergence

so

is u n i f o r m

317

in [0,L] x [0,T] for any T > 0.

In particular,

V n converges

in L 2.

Let V(x,t) = lim

vn(x,t). It remains to show that V satisfies that V satisfies

(3.9).

(3.11) - this follows from (3.12).

show that (3.11) implies

(Note that it is easy to show However, we would still have to

(3.9), so we may as well show (3.9) directly.)

Consider L f (vn(x,t)-V0(x))~(x)dx 0

(3.16)

t L - f f vn(x,s)[~"(x)-~(x)]dx ds 00 t L n - f f f(V -1(y,s))~(y)W(dyds). 00

By (3.12) this is L t L

= f f / f(vn-1(y,s))Gt_s (x,y)W(dyds) % (x)dx 000 L

L

+ I ( ] Gt(x,y)Vo(Y)dY - Vo(X)]4)(x)dx 0 0 t L L I ~ [] Gs(X,y)V0(y)dy

0 0

u L + f I f(vn-1(y,s))Gu_s(X,y)W(dyds)](*"(x)-*(x))dxdu

0

0 0

- ftfLf(vn-1 (y,s))~ (y)W(dyds). 00 Integrate first over x and collect terms: t L

= f f f(vn-1(y,s))[Gt_s(~,y) 00

t - f Gu_s(~"-~,y)ds s

- ~(y)]W(dyds)

L t - f [Gt(4),Y) - *(y) - I Gu(*"-*,y)du]V0(Y)dY. 0 0 But this equals zero since both terms in square brackets vanish by (3.8). (3.16) vanishes

for each n.

We claim it vanishes

Let n ÷ ~ in (3.16).

vn(x,s) +V(x,s)

each T > 0, and, thanks to the Lipschitz ~ifo~ly

Thus

in the limit too.

in L 2, ~ i f o ~ l y

conditions,

f(vn-1(y,s))

in [0,L]x[0,T]

for

also converges

in L 2 to f(V(y,s)). It follows that the first two integrals

does the stochastic

inte~al,

for

tL E{(f f (f(V(y,s)) 00 t L < I< / ~ E{(V(y,s)

0 0

in (3.16) c o n v e r ~

- f(vn-1(y,s))~(y)W(dyds))2}

- vn-1(y,s)) 2} ~(y)dyds

as n ÷ ~.

So

318

which tends to zero. by V.

It follows that

This gives us

(3.16)

still vanishes

if we replace V n a n d V n-1

(3.9). Q.E.D.

We must now prove the lemma.

PROOF

(of Lemma 3.3).

If a > 0 take k = I and C = C I.

If -I < a < 0,

2 t ' t hn(t) < Clf0hn_2(u)(~ ('t-s)a(s-u)ads)du • u If a = -I + E, the inner integral , 2 ~1-e 2[t---~) [

is bounded above by I/2(t-u) 0

dv 1-e

< 4 -- ? (t-u)2E-1

v

so t hn(t ) < C1f0hn_1(s ) _

t ds 4 2 fohn 2(s) (t_s)1_ £ ! ~ C I _

If 2e ~ I we stop and take k = 2 and C = ~4 C 2I.

Otherwise

t

16 4 < - ~ C I f hn_4 (s) E 0

--

until we get

(t-s) to a positive power.

ds (t_s)l_2£

we continue

ds (t-s) I-4£

When this happens,

we have

t hn(t) ~ C f

hn_k(S)ds. 0

But now

(3.151 follows

from this by induction.

Q.E.D.

In many cases the initial value V0(x) V(x,t)

will be bounded in L p for all p.

continuous

COROLLARY

process,

3.4.

(x,t) ÷ V(x,t)

PROOF.

and, even better,

Suppose

that V0(x)

is deterministic,

We can then show that V is actually

estimate

its modulus

is LP-bounded

is a Holder continuous

a

of continuity.

for all p > 0.

Then for a.e. ~,

function with exponent ~ - e, for any E > 0.

A glance at the series expansion Gt(x,y)

in which case

of G t shows that it can be written

= gt(x,y)

+ Ht(x,y) 2 -

where

gt(x,y ) = (4~t)-I/2e

(y-x)

4t

-

t

•

Ht(x,y)

is a smooth function of (t,x,y) on (0,L) x (0,L) x (-~, =), and H vanishes

t < 0.

By (3.11)

if

319

L V(x,t)

=

f 0

t L + ~ f f(V(y,s))Ht_s(X,y)W(dyds) 0 0

Vo(Y)Gt(x,y)dy t

L f f(V(y,s))gt_s(X,y)W(dyds)" 0

+ f 0

The first term on the right hand side is easily seen to be a smooth function of (x,t) on (0,L) x (0,~). function

H.

The second term is basically a convolution

It can also be shown to be smooth;

Denote the third term by U(x,t)° by e s t i m a t i n g the moments E{IU(x+h,

we leave the details to the reader.

We will show that U is ~61der continuous

of its increments

and using Corollary

We will estimate

Replacing

1.4.

Now

t+k) - U(x,t)In} I/n ~ E{IU(x+h , t+k) - U(x, t+k) In} I/n + E{IU(x,t+k)

Burkholder's

of W with a smooth

inequality

the two terms separately.

to bound the moments

- U(x,t)In} I/n. The basic idea is to use

of each of the stochastic

integrals.

t+k by t, we see that t

E{l~Cx+h,t) - ~(x,t~ln} _ 0.

Then there is a n e i g h b o r h o o d G of zero such that

If(x) I < 6.

Jl IJ • n

(We identify H 0 w i t h its dual H 0, but we do

Let us give E the toplogy d e t e r m i n e d by the ]t 11 . n

be

norm

In fact we have: •' ' 3

Let E '

larger

I1 II . n

T h e n H_n is the dual of Hn.

f ~ H

note that

if m < n for, since Jf llm < J1 iln, any linear functional on E w h i c h is

continuous

the

Meanwhile,

f

e E'.

Thus there is

For 6 > 0, if ~xll < £6, then n

This implies that HfII_n ~ I/e, i.e. f e H_n.

Conversely,

if

, it is a linear functional on E, and it is continuous r e l a t i v e to H ~ , a n d n

hence continuous in the t o p o l o g y of E. Note:

The a r g u m e n t above also proves the f o l l o w i n g more general statement:

a linear map of E into a metric space.

Let F be

Then F is continuous iff it is continuous in

one of the norms , II . n W e give E' the strong topology:

a set A G E is b o u n d e d if it is b o u n d e d in

each n o r m il l]n, i.e. if {rlxiln, xeA} is a b o u n d e d set for each n.

Define a s e m i - n o r m

pA(f) = sup{If(x) I : xeA}. The strong t o p o l o g y is g e n e r a t e d by the semi-norms {PA : A ~ E is bounded}. not in g e n e r a l

norr~ble,

but

its

topology

is

compatible

with

the

metric

d(x,y) = ~ 2-n(I + fly-Xlln)-IIy-XXn , n and we can speak of the completeness of E.

If E is complete, then E = ~ n

H . n

N o w E is

332

(Clearly

E C

/'% H n, a n d n

~IX-Xnfl

if x e

< 2 -n =>

n

where H

-n

H

n

~ n

H_n~

is a H i l b e r t

space

is d u a l

to H , a n d n

We m a y of the

Exercise

4.1.

in H

(Hint:

m

...3

Suppose

I]

H

m

the

H0~

following

fl

II n

it is t o t a l l y

d(X,Xn)

H2~

Then

that

a n d we h a v e N n

Hn = E

fl II , -~ < n < ~, n

n > m such

n

x n e E such

< 2 -n+1. )

...D

explicitly H

is

s~ace,

HI~

exists

of t h e s p a c e s

< HS

Thus

a nuclear

to the n o r m

for a l l m t h e r e

properties

show

H_I~

relative

for e a c h n t h e r e

m < n.

it is c a l l e d

not often use

fundamental

then

Ilx-x II < 2 -n n m

If E is c o m p l e t e , E' =

~ Hn, n

that

E is d e n s e

II II m

in t h e

< HS

sequel,

fln

but

n

in Hn,

.

it is o n e

.

the c l o s e d

unit

ball

in H

is c o m p a c t

n

bounded.)

REGULARIZATION

L e t E be a n u c l e a r random

linear

functional

if,

space

as a b o v e .

for e a c h

x,y e E a n d a,b,

X ( a x + by) = aX(x)

THEOREM

4.1.

probability

H

-n

a.s°

Let X be a r a n d o m

linear

in , I~ for s o m e m. m In p a r t i c u l a r ,

Convergence

If

+ bX(y)

functional

11 II m

< HS

X has a v e r s i o n

in p r o b a b i l i t y

A stochastic

with

E

process

xeE}

is a

R,

a.s.

on E w h i c h

is c o n t i n u o u s

H ~i , t h e n X has n values

{X(x),

a version

in

which

is

in

in E'.

is m e t r i z a b l e ,

being

compatible

with

the metric

def

lu x(x)nl If X is c o n t i n u o u s s o m e m by o u r note.

COROLLARY

4.2.

probability

in p r o b a b i l i t y There

Then

on E,

E{JX(x}I^I}. it is c o n t i n u o u s

exists n such that

Let X be a r a n d o m

on E.

=

X has

linear

a version

~ II < m HS

functional with values

in p r o b a b i l i t y

~ H . n

which in E'.

in

T h u s we h a v e

is c o n t i n u o u s

in

II 11 f o r m

333

PROOF

(of Theorem 4.1).

We will first show that

Let (e k) be a CONS in (E,, ITn).

~X(ek)2 < ~. For E > 0 there exists 6 > 0 such that IIIX(x)lll < e whenever

llxU < 6. m

We

claim that Re E{e ix(x)} > I - 2E - 2E6--2flX"2 • --

m

; ~ Indeed, the left-hand side is greater than 1 - ~I EtX2(x)a41,

and if llxllm _< 6,

E{X2(x)A4} ! 4E{IX(x)IAI} ~ 4e, while if nxn

m

> 6,

E{X2(x)A4}

< llxl126-2E{X2(6x/NXltm)~4} < 4 E 6-2nxll 2. --

m

--

m

Let us continue the trickery by letting YI,Y2,...

be iid N(0,O 2) r.v.

N

independent of X, and set x =

~ Ykek . k=l

Then

N Re E{e iX(x)} = E{ReE{exp[i [ YkX(ek )]Ix}}. I But if X is given, [YkX(ek ) is conditionally a mean zero Gaussian r.v. with variance ~2~X2(ek) , and the above expectation

is its characteristic

function:

2N - ~2 { [ X2(ek)} = E{e k=1 }° On the other hand,

E{Re E{e

it also equals

iTYkx~ek)

IY}} > I

= I- 2e - 26-2E

_

2e -

26-2~E{,x, 2}

N [ E{YjY k} < ej,ek> m j ,k=1

= 1 - 2e - 26 -2 e ~

2

N 2 7 "ekllm" k=1

Thus 2 N _ ~_ ~ X2(ek ) 2 E{e k=l } > I - 2e - 26-2£C 2 _

N 7 "ek H2 m ° k=1

Let N ÷ ~ and note that the last stun is bounded since ~ "m < n " n . HS to to see that P{ Let QI = {~: ~ X2(~k '~) < "}" k

~ X2(ek ) < ~} h I - 2e. k=1 Then P{QI} = I.

Define

Next let 2

÷ 0

334

i Y(x,~)

~ nX(ek )

if ~ £ QI"

0

if ~ e ~ - QI"

=

The sum is finite by the Schwartz inequality, Y E H

-n

Moreover,

so Y is well-defined.

with n o r m

,YII_n = [ y2(e k) = ~ X 2 ( e k ) < k k N

Finally, P{Y(x) = X(x)} =

I,

x ~ E.

Indeed,

let ~

X(x N) = Y(x N) on QI' and nX-XNll m ~ HX-XNll n ÷ 0.

= [ nek • k=1

Clearly

Thus

Y(x) = lim Y(x N) = lim X(x N) = X(x). Note:

We have f o l l o w e d some notes of Ito in this proof.

The tricks are due to

S a z a n o v and Yamazaki.

EXAMPLES

Let us see w h a t the spaces E and H n are in some special cases.

EXAMPLE

I.

Let G C R d

be a b o u n d e d domain and let E 0 = ~(G) be the set of C ~

functions of compact s u p p o r t in G.

Let H U 0 be the usual L 2 - n o r m on G a n d set

0 n° w h e r e ~ is a m u l t i - i n d e x of length operator.

d/2, H

n

by Maurin's theorem.

To see why, note

embeds in Cb(G) by the Sobolev e m b e d d i n g t h e o r e m and , Pi < If 11 n 2n HS By T h e o r e m 4.1, M t has a v e r s i o n with values in

particular, M t e H _ d _ 2 , and if d is odd, we h a v e Mte H_d_1. a n a l y s i s h e r e w o u l d show that, locally at least, M t

6

H_2 n.

(A m o r e delicate

H n for any n > d/2.)

In

338

Exercise

4.2.

filtration,

Show that under

etc.)

that the p r o c e s s

has a right c o n t i n u o u s appropriate

the usual

Sobolev

version.

Mt,

hypotheses

considered

(i.e.

right

as a p r o c e s s

S h o w that it is a l s o right

continuous w i t h values

continuous

in ~(G),

in the

space.

Even pathology

has

its degrees.

The m a r t i n g a l e

M t will

certainly

not be a d i f f e r e n t i a b l e

or even c o n t i n u o u s

According

to the above,

it is at worst a d e r i v a t i v e

of order d + 2 of an L 2 f u n c t i o n

or, u s i n g

the e m b e d d i n g

theorem

3 of order ~ d + 3 of a c o n t i n u o u s

function.

Thus a d i s t r i b u t i o n

again, in H

-n

H_n_1 , and the s t a t e m e n t

that M does

regarded

property

as a r e g u l a r i t y

a derivative

relevant

appropriate

space,

until we discuss

it is u s u a l l y process

nuclear

easier

indeed

as an even m o r e p r a c t i c a l

take values

than a d i s t r i b u t i o n

in a certain

most processes

H

in

can be

-n

as h a v i n g values

and put off the task of d e c i d i n g

the regularity

of the process.

for it is often

than to verify

matter,

bad.

of M.

to do it this way;

is d i s t r i b u t i o n - v a l u e d

but it is not infinitely

is "more d i f f e r e n t i a b l e "

In the future we will discuss another

function,

measure

it lives

we shall u s u a l l y

which H

in ~(G)' is

-n

As a p r a c t i c a l

much simpler in a given

matter,

to verify

Sobolov

or

that a

space...and

leave even that to the reader.

CHAPTER FIVE

PA~OLIC ~UATIONS iN£d Let {Mt, ~t' t ~ 0} be a worthy m a r t i n g a l e m e a s u r e on

R d with covariation

Let R(A) = E { K ( A ) } .

m e a s u r e Q ( d x dy ds) = d s and d o m i n a t i n g m e a s u r e K. Assume that for some p > 0 and all T > 0 I

~(dx dy ds) < ~.

(1÷IxlP,~1÷lylP~

RdX[0,T]

~ ( x ) M ( d x ds) exists for each ~ ~ ~(Rd). dx[0,T ]

T h e n Mr(#) =

Let L be a u n i f o r m l y e l l i p t i c s e l f - a d j o i n t s e c o n d order d i f f e r e n t i a l o p e r a t o r with b o u n d e d smooth coefficients.

Let T be a d i f f e r e n t i a l o p e r a t o r on

of finite order with b o u n d e d smooth coefficients. not on t).

d R

(Note that T and L operate on x,

C o n s i d e r the SPDE

(5.1) V(x,0) = 0 We will clearly need to let V and M have d i s t r i b u t i o n values, make sense of the t e r m TM.

if only to

We will suppose they have values in the Schwartz

space ~'(Rd). We want to cover two situations: holds in

Rd .

the first is the case in w h i c h

(5.1)

A l t h o u g h there are no b o u n d a r y conditions as such, the fact that

V t e ~ ' ( R d) implies a boundedness condition at infinity. The second is the case in w h i c h D is a b o u n d e d domain in

R d, a n d

h o m o g e n e o u s b o u n d a r y conditions are imposed on 5D. (There is a t h i r d situation w h i c h is covered - f o r m a l l y at least - by (5.1), and that is the case w h e r e T is an integral o p e r a t o r rather than a d i f f e r e n t i a l operator.

Suppose,

suitable functions g and h. r e a l - v a l u e d martingale,

for instance, that Tf(x) = g(x) f f ( y ) h ( y ) d y for

In that case, TMt(x) = g(x)Mt(h).

so that

Now Mt(h)

is a

(5.1) can be r e w r i t t e n

i dV t = LV dt + gdMt(h) V(x,0) = 0 This differs f r o m (5.1) in that the d r i v i n g term is a o n e - p a r a m e t e r

340

m a r t i n g a l e rather than a m a r t i n g a l e measure.

Its solutions have a r a d i c a l l y

different behavior from those of (5.1) and it deserves to be t r e a t e d separately.) Suppose

(5.1) holds on

integrate by parts.

R d.

Integrate it against ~e ~(Rd), and then

Let T* be the formal adjoint of T.

The weak form of

(5.1) is

then t t Vt( ~ ) = f V s ( L ~ ) d s + f IdT*#(x)M(dxds), 0 0 R

(5.2)

Notice that when we integrate by parts,

~ E s(Rd).

(5.2) follows easily for # of

compact support, but in order to pass to rapidly d e c r e a s i n g #, we must use the fact that V and TM do not grow too q u i c k l y at infinity. In case D is a b o u n d e d region with a smooth boundary,

let B be the o p e r a t o r

B = d(x)D N + e(x), where D N is the normal derivative on ~D, and d and e are in C~(SD).

C o n s i d e r the i n i t i a l - b o u n d a r y - v a l u e p r o b l e m = LV + TM

I

(5.3)

on D x [0,~);

BV = 0

on 5D ×

V(x,0) = 0

on D.

[0,~);

Let C~(D) and C0(D) be r e s p e c t i v e l y the set of smooth functions on D and the set of smooth functions with compact support in D.

Let C~(D) be the set of functions

in C~(D) whose derivatives all extend to continuous functions on D.

Finally,

let

~B = {¢ 6 C~(~): B~ = 0 on ~D}. The weak form of

(5.3) is t t Vt(~) = f V s ( L ~ ) d s + f f T #(x)M(dxds), ~ £ {B" 0 0D

(5.4)

This needs a w o r d of explanation.

To derive

(5.4) from (5.3), m u l t i p l y by

# and integrate f o r m a l l y over D x [0,t] - i.e. treat TM as if it were a d i f f e r e n t i a b l e function - and then use a form of Green's t h e o r e m to throw the d e r i v a t i v e s over on #. b o u n d a r y condition.

This w o r k s on the first integral if both V and ~ satisfy the

Unless T is of zeroth order,

M may not satisfy the b o u n d a r y conditions. D, however.)

Nevertheless,

it may not work for the second,

(It does w o r k if ~ has compact support in

the e q u a t i o n we w i s h to solve is (5.4), not (5.3).

The r e q u i r e m e n t that

for

(5.4) hold for all ~ s a t i s f y i n g the b o u n d a r y

conditions is e s s e n t i a l l y a b o u n d a r y condition on V.

The above situation,

in which we regard the integral, rather than the

differential equation as fundamental,

is analogous to many situations in which

physical reasoning leads one directly to an integral equation, and then mathematics takes over to extract the partial differential equation. derivations of the heat equation, Navier-Stokes

See the physicists'

equation, and Maxwell's equation,

for instance. As in the one-variable case, it is possible to treat test functions ~(x,t) of two variables.

Exercise 5.1.

Show that if V satisfies

(5.4) and if ~(x,t) is a smooth function such

that for each t, ~(.,t) E ~B' then t t Vt(~(t)) = ~ V (L~(s) + ~ (s))ds + ~ ~ T ~(x,s)S(dxds). o s ~--; OD

(5.5)

Let Gt(x,y) be the Green's function for the homogeneous differential equation.

I If L = ~ 4, D =

R d, then

lyxl 2 Gt(x,y) = (2~t) -d/2 e-

2t

For a general L, Gt(x,y) will still be smooth except at t = 0, x = y, and its smoothness even extends to the boundary:

if t > 0, Gt(x,o) 6 C~(~).

It is positive,

and for • > 0,

lyxl 2 (5.6)

Gt(x,y) ~ Ct- d/2 e-

where C > 0 and 6 > 0. bounded D. in ~(Rd).

6t

(C may depend on ~).

,

x, y 6 D, 0 < t < ~,

This holds both for D =

If D = R d, Gt(x,.) is rapidly decreasing at infinity by (5.6), so it is Moreover,

for fixed y, (x,t) + Gt(x,y)

differential equation plus boundary conditions. Then if # is smooth, Go# = ~.

satisfies the homogeneous Define Gt(~,y) = fDGt(x,y)~(x)dx-

This can be summarized in the integral

equation: t (5.7)

R d and for

Gt-s(#'Y) = ~(Y) + f Gu-s(L~'y)du' s

~ ~ ~B

342

The smoothness

of G then implies that if ~ 6 C (D), then Gt(~,.) 6 LB-

In

case D = R d,, then ~ 6 =S(Rd) implies that Gt(~,,) ~_ =s(Rd)"

THEOREM 5.1. satisfies

There exists a unique process

(5.4).

{Vt, t)0} with values in ~'(R d) which

It is given by t

(5.8)

Vt(~) =

f f 0

T*Gt_s($,Y)M(dyds)R

d

The result for a bounded region is similar except for the uniqueness statement.

THEOREM 5.2 satisfies

There exists a process

(5.5).

this process

{Vt, t ~ 0} with values in ~'(R d) which

V can be extended to a stochastic process {Vt(#), t >__ 0, ~ ~ SB};

is unique.

It is given by t

Vt(~)

(5.9)

f f

=

T*St_s(~,y)M(dy

as), ~ 6 s B.

0 D

PROOF.

Let us first show uniqueness,

which we do by deriving

(5.9).

Choose ~(x,s) = Gt_s(~,x) , and suppose that U is a solution of (5.4). Consider Us(~(s)). so we can apply

Note that U0(~(0)) = 0 and Ut(~(t))

Now Gt_s(~,.) 6 ~B'

(5.5) to see that t

Ut(~) = Ut(~(t))

f

=

t Us(L~(s)

+ ~

(s))ds +

0 But I~ + ~

= Ut(~).

f f

T*~(x,s)M(dx

ds).

0 D

= 0 by (5.7) so this is t =

f fT* 0

~(X,S) M(dx ds)

D t

=

f f

T*Gt_s(~,x)M(dx

as) = vt(¢).

0 D

Existence:

Let ¢ ~ S B and plug (5.9) into the right hand side of (5.4): t

s

t

f [ f f T*Ss_u(L*,y)M(dy au)Jds + f f T**(y)S(dy,du) 0

0

D t

t

0 D

= f f [ ; T'Gs_uCL*,y)ds ÷ T**(~)]M(dy d~). 0 D

u

343

Note that T* Gs_u(L~,y)

and T*~(y) are bounded, so the integrals exist.

By (5.7)

this is t

= f ~ T*Gt_u(~,y)M(dy

du)

0 D

= vt(~)by (5.9). in S B.

This holds for any ~ 6 S B, but (5.9) also makes sense for % which are not

In particular,

it makes sense for ~ £ S(R d) and one can show using Corollary

4.2 that V t has a version which is a random tempered distribution. This proves Theorem 5.2.

The proof of Theorem 5.1 is nearly identical;

just

replace D by R d and S by s(Rd). --B =

Q.E.D.

AN EIGENFUNCTION EXPANSION

We can learn a lot from an examination of the of the case T ~ I. is a bounded domain with a smooth boundary. conditions)

Suppose D

The operator -L (plus boundary

admits a CONS {~j} of smooth eigenfunctions with eigenvalues kj.

satisfy (5.10)

(1+kj) -p < ~

if p > d/2.

J (5.11)

sup "~j"~(1+kj) - p _

< ~

if p > d/2.

J Let us proceed formally for the moment.

We can expand the Green's

function: -k .t

Gt(x,y) = ~ ~j(x)#j(y)e 3

3

If ~ is a test function -k t

Gt(~,y) = [$j~j(y)e

J

3

where ^ ~j= Df~(x)~.(x)dx, 3

so by (5.9) t

vt(,) = f f

-k(t-s)

~ %j,j(y)~

J

MCdyds).

0 D j

Let

t Aj(t) = f0fD~j(y)e

-k,(t-s) 3

M(dyds).

These

344

Then (5.12)

vt($) This will converge

spaces H

n

introduced

eigenfunction

= [ SjAj(t).

for ~ e ~B' but we will show more.

in Ch. 4, Example

3.

H

n

is isomorphic

Let us recall

the

to the set of formal

series

f = j=~laj~j for which

We see from

(5.12)

that V t ~ [, A j ( t ) ~ b j .

3

PROPOSITION process

5.3.

Let V be defined by (5.12).

in H n; it is continuous

with T 5 I.

if t + M t is.

Moreover,

If M is a white noise based on Lebesgue

process

in H

PROOF.

We first bound E{sup A2(t)}. t d, V t is a right continuous

-n

V is the solution

measure

then V is a continuous

for any n > d/2.

t Let Xj(t) = 10fD ~j(x) M(dxds)

3.3,

t -k(t-s) A(t) 3

= f e 0

3

=

where we have integrated

Xt

by parts

dX(s). 3 t -A(t-s) { k e 3 X.(s)ds 3

3

in the stochastic

integral.

t suplAj(t) I < sup t 0,

nVt÷s-VtH2n = ~(Aj(t+s)-Aj(t))2( 1+k j )--n. The summands

are right continuous,

and they are continuous

if M is.

The sum is

dominated by 41 sup A2(t)(1+kJ ) - n 3 j t d/2 - I, not just for n > d/2.

one improves

the estimate

of E( sup t d/2 in

The conditions

For instance,

in (5.13) reduces to f ~2j(x)dxds.

of (5.4), the uniqueness

result

346

Exercise

5.2.

Verify

Exercise

5.3.

Treat

that V

of the s o l u t i o n

THEOREM

5.4.

n

above

involve directly

(5.12))

satisfies

u s i n g the Hermite

are a n a l o g o u s

derivatives.

Here

(5.4).

expansion

to the c l a s s i c a l

of E x a m p l e

Sobolev

is a result w h i c h

2, Ch.4.

spaces,

relates

but they

the r e g u l a r i t y

to d i f f e r e n t i a b i l i t y .

Suppose M is a white

exists a r e a l - v a l u e d exFonent

d

the case D = R

The spaces H don't e x p l i c i t l y

(defined by

process

noise

based on Lebesgue

U = {U(x,t):

I/4 - £ for any E > 0 such that

xED,

t>__0} which

if D d-1

measure. is HDlder

~d-1 ~x2,...,~Xd,

T h e n there continuous

with

then

V t = Dd-Iut .

Note.

This

is of course

w e a k =th d e r i v a t i v e

a derivative

of a f u n c t i o n

implies Note.

One m u s t be careful

but not identical moment.

Theorem

continuous result

spaces

process

Sobolev

as a c o n t i n u o u s

in c o m p a r i n g

H n of Example

5.4 might

Q is the

#,

(-1)l~Iff(x)D~#(x)dx.

the c l a s s i c a l

that V t can be r e g a r d e d

A distribution

f if for each test function

Q(~) = If we let H n denote o

in the weak sense.

space

process

the c l a s s i c a l 3 in C h a p t e r

lead one to guess

of E x a m p l e

this

-d+1 in H °

Sobolev

4.

I Ch.4,

spaces

H~ with related

Call the latter H n3 for the

that V is in H3 d+1,

in H3 n- for any n > d/2 by P r o p o s i t i o n

but in fact,

This

5.3.

is a m u c h

it is a

sharper

if d > 3. This

gives us an idea of the b e h a v i o r ~--v = L v 5t

.Suppose n o w that T is a d i f f e r e n t i a l coefficients,

so TL = LT.

the f o l l o w i n g

exercise

OU ~

a n d suppose

A p p l y T to both sides

= LU + TM.

makes

= LTV

Of course,

it rigorous.

of the e q u a t i o n

+ ~.

operator

~-~ (TV) i.e. U = T V satisfies

of the solution

both T a n d L h a v e c o n s t a n t

of the SPDE:

+ this

argument

is p u r e l y

formal,

but

347

E x e r c i s e 5.4.

Suppose T and L commute.

Let U be the solution of (5.4) for a general

T with b o u n d e d smooth coefficients and let V be the solution for T --- I.

V e r i f y that

if we r e s t r i c t U and V to the space D(D) of C~ functions of compact support in D, that U = TV.

E x e r c i s e 5.5.

Let V solve ~V -5t

I

Describe v(.,t)

REMARKS.

I.

- V + ~~ x W ' -52V 5x 2

=

0 < x < ~, t > 0;

~V 5V ~ x (0,t) = ~ x (~,t) = 0,

t > 0;

V(x,0) = 0 ,

0 < x < ~.

for f i x e d t.

(Hint:

use Exercises 5.4 and 3.5.)

T h e o r e m 5.2 lacks symmetry c o m p a r e d to T h e o r e m 5.1.

p r o c e s s in ~ ( R d) but must be e x t e n d e d slightly to get uniqueness, d o e s n ' t take values in ~'(Rd).

V t exists as a and this extension

It w o u l d be nicer to have a more symmetric statement,

on the o r d e r of "There exists a unique process with values in such and such a space such that

...".

One can get such a statement,

though it requires a litle m o r e

S o b o l e v space theory and a little more analysis to do it. Let ~ n n be the norm of Example ~B in this norm. (H;)' def -n = HB .

I, C h a p t e r 4.

T h e o r e m 5.2 can then be stated in the form:

on D x R + as follows. V(~) =

If ~ = ~(x,t) f 0

Vs($(s))ds

(5.4).

compact support.

(5.5).

E x t e n d V to be a d i s t r i b u t i o n

is in C 0 ( D x (0,~)), let and

TM(~) =

f ~ T* $(x,s) M(dxds). 0D

Then C o r o l l a r y 4.2 implies that for a.e. ~, V and ~ Now consider

there exists a unique

(5.4) for all ~ 6 H Bn-

Suppose that T is the identity and consider

D × (0,~).

Let H ; be the c o m p l e t i o n of

If n is large enough, one can show that V t is an element of

process V w i t h values in H B-n w h i c h satisfies 2.

Here is how.

define distributions on

For large t, the left-hand side vanishes,

The r i g h t - h a n d side then tells us that V(L~ + ~ )

In other words, for a.e. ~, the d i s t r i b u t i o n V(-,~) (non-stochastic) PDE

~-~- ~ 5t

=

for ~ has

+ TM(~) = 0 a.s.

is a d i s t r i b u t i o n s o l u t i o n of the

348

Thus T h e o r e m identity, (5.5)

5.1 follows

the same holds

into a PDE w i l l

from known n o n - s t o c h a s t i c for T h e o r e m

introduce

the t h e o r y of d i s t r i b u t i o n about

SPDE's.

5.2.

boundary

solutions

theorems

In general, terms.

the t r a n s l a t i o n

Still,

of d e t e r m i n i s t i c

on PDE's.

If T is the of

(5.4) or

we s h o u l d keep in m i n d t h a t

PDE's has s o m e t h i n g

to say

C H A P T E R SIX WEAK CONVERGENCE

Suppose E is a m e t r i c space with m e t r i c p. sets on E, and let ( P ) n really m e a n by "P

n

be a sequence of p r o b a b i l i t y m e a s u r e s on E. =

÷ P "? o

it has no unique answer.

approximation,

What do we

This is a n o n - m a t h e m a t i c a l question, of course.

a s k i n g us to make an intuitive idea precise. context,

Let ~ be the class of Borel

It is

Since our intuition will depend on the

Still, we might b e g i n with a reasonable first

see h o w it might be improved, and hope that our intuition agrees w i t h

our m a t h e m a t i c s at the end. Suppose we say: "p This looks promising, don't.

n

+ P

o

if Pn(A) + Po(A),

but it is too strong.

all A e ~."

Some sequences w h i c h should converge,

For instance, consider

P R O B L E M I.

Let P n = 61/n, the unit mass at l/n, and let P o = 6o.

to converge to Po' but it doesn't.

Certainly P n o u g h t

Indeed 0 = lim P {0} # P {0} = I. n o

Similar

things happen with sets like (-~,0] and (0,1).

CURE.

The trouble occurs at the b o u n d a r y of the sets, so let us smooth t h e m out.

Identify a set A with its indicator function IA. Then P(A) = fIAdP.

We "smooth out

the b o u n d a r y of A" by r e p l a c i n g I A by a continuous function f w h i c h a p p r o x i m a t e s it, and ask that ~fdPn+ IfdP.

We may as well require this for all f, not just those

w h i c h a p p r o x i m a t e indicator functions. This leads us to the following.

Let C(E) be the set of b o u n d e d real valued

continuous functions on E.

DEFINITION.

We say P n converges weakl[ to P, and write Pn => P, if, for all

f £ C(E), ffdP n ÷ ffdP.

350

PROBLEM

CURE.

2.

Our n o t i o n of c o n v e r g e n c e

We p r e s c r i b e

def = {P: P is a p r o b a b i l i t y

s y s t e m of n e i g h b o r h o o d s

{P £ ~(E): This be discussing

interesting,

convergence

The first,

THEOREM

is s o m e t i m e s

6. I.

rather

it shortly,

called

The f o l l o w i n g

but

the P o r t m a n t e a u

is itself

extremely some facts.

characterizations

of w e a k

Theorem.

are e q u i v a l e n t

(ii)

f f d P n ÷ fdP,

(iii)

f f d P n ÷ ffdP,

(iv)

lim sup P (F) < P(F), n

all c l o s e d

(v)

lim inf P (G) > P(G), n

all open G;

(vi)

l i m Pn(A)

measure

- but it is

let us first e s t a b l i s h

P

probability

I ..... n.

than of r a n d o m v a r i a b l e s

(i)

n

i =

to fill our needs - for we shall

gives a number of e q u i v a l e n t

=> P ; all b o u n d e d all b o u n d e d

= P(A),

Let E and F be metric

ph-1(A)

may not a p p e a r

fi E C(E),

The reason why it is s u f f i c i e n t

go into

which

on ~}.

is given by sets of the f o r m

of processes,

what we need.

and we shall

convergence,

measure

IIfidP - f f i d P o I < e, i=I ..... n},

notion of c o n v e r g e n c e

in fact exactly

w i t h a topology.

two definitions:

~(E) A fundamental

seems u n c o n n e c t e d

uniformly functions

continous

which are continuous,

and h

P-a.e.;

F;

all A ~ E such that P(~A)

spaces

f;

= 0.

: E + F a measurable

on E, then Ph -I is a p r o b a b i l i t y

measure

map.

If P is a

on F, w h e r e

= p(h-1(A)).

T H E O R E M 6.2

If h

: E + F is c o n t i n u o u s

(or just c o n t i n u o u s

P-a.e.)

and if P

n

=> P o n

E, then P h -I => Ph -1 on F. n Let PI,P2,... Here

is one answer.

in A has a w e a k l y compact,"

be a sequence

in P(E).

Say t h a t a set K C

convergent

~(E)

subsequence.

but we f o l l o w the common

usage.)

When

does such a sequence

is [ e ! a t i v e l y

compact

converge?

if e v e r y s e q u e n c e

(This s h o u l d be " r e l a t i v e l y

sequentially

351

Then

(P) n

converges

(i)

there exists

weakly

if

a relatively

compact

set K C ~(E)

such that Pn e K for

all n. (ii)

the sequence

Since

has at m o s t one limit p o i n t

(i) g u a r a n t e e s

at least one limit point,

in ~(E). (i) and

(ii) t o g e t h e r

imply

convergence. If this c o n d i t i o n criterion

for r e l a t i v e

DEFINITION. KC

is to be useful

compactness.

A set A C ~(E)

This

6.3.

If A is tight,

and complete,

Theorem.

exists a compact

set

> I - E.

it is r e l a t i v e l y

then if A is r e l a t i v e l y

Let us return

is s u p p l i e d by P r o h o r o v ' s

is tight if for each E > 0 there

E such that for each P e A, P{K}

THEOREM

- and it is - we will need an e f f e c t i v e

compact.

compact,

to the q u e s t i o n

Conversely,

if E is s e p a r a b l e

it is tight.

of the s u i t a b i l i t y

of our d e f i n i t i o n

of w e a k

convergence.

PROBLEM

3.

We are i n t e r e s t e d

this all seems

CURE.

a space of, defined

it. say,

k n o w the s o l u t i o n We often

not r a n d o m variables,

so

We just have to stand b a c k far e n o u g h

a process

canonically

right c o n t i n u o u s

functions

on

= (~(t), ~ E Q,

function.

X is then d e t e r m i n e d

this means

that we are r e g a r d i n g simply

~ikes

W i t h this remark, p u t a metric

to this.

define

on Q by Xt(~)

random variable

of processes,

irrelevant.

We a l r e a d y

to r e c o g n i z e

in the b e h a v i o r

on the f u n c t i o n

will t h e n a p p l y to m e a s u r e s The Skorokhod

on a function

[0,~), then a p r o c e s s

for ~, b e i n g an e l e m e n t

of Q,

by its d i s t r i b u t i o n

P, w h i c h

the whole p r o c e s s

as a single

its values the o u t l i n e

space:

if Q is

{Xt: t>0}

is itself

is a m e a s u r e

can be

a

on Q.

But

random variable.

The

in a space of functions. of the theory

becomes

space Q in some c o n v e n i e n t

way.

clear.

We m u s t

The above

first

definition

o n Q.

space D = D([0,1],E)

It is the space of all f u n c t i o n s

is a c o n v e n i e n t

f : [0,1] ÷ E w h i c h

function

space to use.

are r i g h t - c o n t i n u o u s

and h a v e

352

left

limits

is m u c h

at each t £

(0,1].

like a sup-norm,

We will m e t r i z e =D.

but the p r e s e n c e

The metric

is a bit tricky.

of jump d i s c o n t i n u i t i e s

forces

It

a

modification. First, [0,1]

let A be the class

onto itself.

If k e A,

llkll =

(We may have

of s t r i c t l y

then k(0)

sup 08o } and by one

so this is

< e

K-I ~ k=0

(e

< e

K-I ~ k=0

(e

-K60_ 1/K +P{Sk+I-Sk I - e/2 j+1. Choose 6 k ~ 0 such that sup E{W(6k,Xn)}

< e -- k2k+l

n 1 sup P{W(6k,X n) > ~ } ~ £/2 k+1. n

Thus

Let A C ~ be

I A = {~ e ~ : ~(t k) E Kk, w(~,6 k) ! ~ , Thus A has a compact closure

Now lim sup w(6,~) = 0. 6+0 ~EA

k=I,2 .... }.

in ~ by Theorem 6.5.

Moreover

P{X £A} > I - [ P{Xn(t k) £ ~ } n k - ~ P{W(6k,X n) > I/k} k > I - e/2 - c/2 = I - ~, hence

(X) n

is tight.

MITOMA'S T H E O R E M

The subject of SPDE's involves need to know about the weak convergence not metrizable,

the preceeding

However,

distributions of processes

of distribution-valued

simple as that of real-valued processes.

According

to show that a sequence

(X n) of processes

tight,

each ~, the real-valued

processes

are tight.

(xn(~))

Rather than restrict ourselves

E' =

processes

Since ~' is

is almost as

to a theorem of Mitoma,

in order

one merely needs to verify that for

to S', we will use the somewhat more general

,.. ~

H_I~

H0~

HI~

,.. ~

~ n

Hn = E

Hilbert space with norm II Rn, E is dense in each Hn,

m M n ~ n nn+ I and for each n there is a p > n such that , gn topology

We will

Let

i ; Hn~ n

Where Hn is a separable

with values in ~'.

way.

theory does not apply directly.

weak convergence

setting of Chapter Four.

in a fundamental

< " " " HS P

E has the

determined by the norms n ~n" and E' has the strong topology which is

determined

by the semi norms PACf) = sup{ [f(+)l, ~ A }

where A is a bounded set in E. Let ~([0,1],E')

be the space of E'-valued

right continuous

functions

which

359

have left limits in E', and let C([0,1],E') functions.

be the space of continuous E ' - v a l u e d

C([0,1],H n) and ~([0,1],H n) are the c o r r e s p o n d i n g spaces of H n - V a l u e d

functions. If f,g E ~([0,1],E'),

let

dA(f,g) = inf{llkn + sup p A ( f ( t ) - g ( k ( t ) ) , k e A}, t and

d A =( suptf pA(f(t)-g(t)). , g ) Give ~([0,1],E') bounded A

E.

D([0,1],H =

(resp. C([0,1],E'))

the topology d e t e r m i n e d by the d A (resp. d A ) for

They both become complete,

) have already been defined,

separable,

for H

n

c o m p l e t e l y regular spaces.

The

is a metric space. n

We will need two "moduli of continuity". w(6,~;~) = inf

max

sup

i

ti 0 such that sup P{ sup IIX~II > M} < E. n 0 0 such that

11~,m < ~ => sup ~Isuplx?(~)lJ,

(6.5)

i

n

t

P

< e

t

taxi11 = E{IxI~I}

where

TO see this, consider the function

~ E.

F(~) = sup Illsup Xt(#),l , n t Then (i)

F(0) = 0;

(ii)

F(#) ) 0 and F(#) = F(-~);

(iii)

lal < Ibl => F(a~) ~ F(b#);

(iv)

F is l o w e r - s e m i - c o n t i n u o u s on E;

(v)

lim F(~/n) = 0.

Indeed (i)-(iii) are clear. n Ixtc~j~l~, + Ixtc,~l,,1 a.s.

If ~ 3 ÷ ~ in E, xn(#j ) ÷ X~(#)

in probability,

in L °, h e n c e

and lim inf[supIxn(~ )I^I] > s j t n 3 -- t p

Thus

~c~

= sup E{suplxt(,~l^~} < sup lira inf ~{supTx%~l^1} n

t

n

j

t

M} t

(xt(~)) is tight,

so, given ~ and E > 0 there exists

< £/2.

Choose k large enough so that M/k < £/2.

Then

F(~/k) = sup E{sup]X~(~/k)1^1} n

t

< sup [P{~uplx~(*/k)l>M} ÷ ~l -k n

t

< E. Let V = {~: F(#) < £}. absorbing

(by (v)) set.

V is a closed (by (iv)), symmetric

We c l a i m it is a n e i g h b o r h o o d of 0.

(by (ii)),

Indeed, E =

~ nV, so n

by the Baire c a t e g o r y theorem, one, hence all, of the nV must have a n o n - e m p t y interior. of zero. proves

In particular,

I

~ V does.

I

I

Then V C ~ V - ~ V must c o n t a i n a n e i g h b o r h o o d

This in turn m u s t contain an element of the basis,

say {~: n#nm M}

-e ~ 14 0.

I - e, X t lies in B = {x: < n n , hence q HS

P

Choose M and p as in Lemma 6.14.

K C ~([0,1],E')

P n

< HS

-q

Hx~ -p -< M} for all t. U n

.

There exists q > p such

Then A is compact

in H

-p

.

Let

-q

be the set {~ : ~(t) E A, 0 I - e/23 for all n. under the map ~ ÷ {

(xn(e.)) 3

R) such that

Let K~3 be the inverse

: 0~t~I}.

is tight by

image of K.3 in =D([0'I]'E')

By the Arzela-Ascoli

Theorem,

lim sup w(6,~;e ) = 0; 6÷0 ~eK~ 3 3 moreover

Set K' = K n /~ K!. • 3 J

Then P{X n e K'} > I - e - I£/29 = I - 2g.

Nown

lim sup w(6,~,H ) = lim sup ([ 6+0 ~ K ' -q 6*0 ~eK' j

inf max sup 2) I/2 {ti} i tiI Then

Let p < q and

(Xn) converges weakly in D([0,1],H

> M} 0 and define h0(x) = (I + IxlP0) -I, x 6 R d •

If M is a worthy

m a r t i n g a l e m e a s u r e with d o m i n a t i n g m e a s u r e K, define an i n c r e a s i n g process k by (7.1)

k(t) =

f

h0(x) h0(Y) K(dx dy as),

R 2dx

[0,t]

and (7.2)

y(6) = sup (k(t+6) - k(t)) t O.

dr)

dr)

.

We claim the second

term tends to zero. Choose g > 0 and let ~ > 0 be such that if Ix] < D, then second

integral

is bounded

by

If(x) l 0 be such that T

0 and let n > 0 be such that

EI_ 2 and K > 0.

Let g be of Holder

Suppose further that the jumps of M n

Then

{U~, 0 < t < I} has a version which is right continuous

and has left

limits; (ii) there exists a constant E{ sup t I, Corollary

CrK(I + 2r).

1.2 implies that V n has a continuous

there exists a random variable

Z

n

version.

More exactly,

and a constant A', which does not depend on n, such

that for 0 < y < ~ - I/r

It.-v

Vn sup 0 0. inequality,

Dt

LP => W and Z are in L 2p.

(Hint:

By Prop. 8.1 and

Use induction on p = 2 n to

L p for all p, then use (8.8) and Doob's L P inequality as above.)

THEOREM 8.6.

Let (~n' An) be sequence of parameter

values and let (W n, Z n) be the

k corresponding

processes.

initial measure. on =~[0,I],

PROOF.

Let Vn(dx) = k-1/2(n n(dx) - k dx) be the normalized n n

If the sequence

((~n+l)/kn)

is bounded,

then (Vn, W n, Z n) is tight

~'(Rd2+2d)}.

We regard V n as a constant process:

~{[0,I], ~'(Rd2+2d)}.

V n H V n, in order to define it on t

It is enough to prove the three are individually

By Mitoma's theorem it is enough to show that (vn(~)), are each tight for each

which is constant

criterion

(Theorem 6.8b) for the other two.

An(6) ffi (6d/kn)n sup t for all n so that for each t, (W~(~)) respectively.

and (Z~(#)) are tight on

By Theorem 6.8 the processes

(wn(~))

R d and

R

and (zn(~)) are each tight. Q.E.D.

THEOREM 8.7.

If A n + ~, ~nkn ÷ ~, and ~n/kn + 0, then

(V n, Z n, W n) =>

where V 0, Z 0 and W 0 are white noises based on Lebesgue measure on

R d,

Rd x

in

R+ respectively;

V 0 and Z 0 are real-valued

A n + = and ~n/kn + 0, (V n, W n) =>

PROOF.

Suppose k

n

and W 0 has values

Modifications

for non-integral

T o show weak convergence,

we merely need to show convergence

dimensional

and invoke Theorem 6.15.

The initial distribution of k

n

independent

Poisson

motion.)

Rd •

If

is Poisson

(A n

k are trivial.

of the finite-

and can thus be written as a sum

(I) point processes.

~I ^2 Let ~ , D ,... be a sequence of iid copies with k = changed notation:

R dx R+, and

(V 0, W0).

is an integer.

distributions

(V 0' Z 0, W0),

^n these are not the D used in constructing

Then the branching Brownian motion corresponding

I, ~ = ~n"

(We have

the branching Brownian to An, ~n has the same

k distribution

as ~I + ~2 + ... + ^~ n

the obvious way.

Define ~I

,

~2,

.

.. , WAI , ~2 W .... and ~I, ~2 ,... in

Then

Vn = ~I

k + ... + ~ n,

Wn

k ^1 ~ n W +...+W , Zn

x,,7-n We have written

everything as sums of independent

proof, we will call on the classical Let ~I' ~ " convergence

~

n

random variables.

To finish the

Lindeberg theorem.

i ~I. ..... ~p' ~p" ~pii f =S i (R), t 1 < t 2 !

of the vector

k ZZAI+...+^ n

"" --< % "

We must show weak

397

.. ..... Ztp n (#;)) d__ef(vn(#1)''°" ,vn(~p) • Wtln(#;) ..... W~p(~;), Ztln(~i)

Us

This can be written as a sum of lid vectors, and the mean and covariance of the vectors are independent of n (Prop. 8.1). It is enough to check the Lindeberg condition for each coordinate. distribution of ~

does not depend on ~n' so we leave this to the reader.

k n (~[) = ~-I/2 n Ak Fix i and look at Wt. wt(*~). 1 k=1 1 an

The

^k , Now (Wt(#i)) is

Rd - v a l u e d c o n t i n u o u s m a r t i n g a l e , so by B u r k h o l d e r ' s i n e q u a l i t y ~k

~k

~{lwt(,[)l 4} ~ c 4 E{ll

2

}

l

t ~ td C 4

~

E{~(*~)2}ds

Now t < I so by Proposition 8.5 with k = I, there is a C independent of k and k --

n

such

that this is C(~n+ I). For e > 0,

1

by Schwartz.

1

1

1

Use Chebyshev with the above bound: !

[C(I + ~n)]1/2[C(1 + ~n)/k~E2] 1/2

~ C(I + ~n)/kne. Thus k

n

--I/2Ak . -I/2^k 2 E{Ikn Wt.(~i )12; ~ n Wt.(~i)l > E} k=1 i l ~1 2 ~1 2

= E{Iwt(~i) 1

I : Iwt(~i) 1

1 > kn E

}

~ C3(1 + ~n)/kne ÷ 0. Thus the Lindeberg condition holds for each of the W~(~[).~. .~ for the Zn t.(,i). 1

The same argument holds

In this case, while (Z~(~))~ is not a continuous martingale, its

jumps are uniformly bounded by (kn~n)-I/2, which goes to zero, and we can apply Burkholder's inequality in the form of Theorem 7.11(i).

Thus the finite-dimensional

distributions converge by Lindeberg's theorem, implying weak convergence.

398

The only place we used the hypothesis statement,

that A n ~n ÷ ~ was in this last

so that if we only have A n + ~, ~n/kn + 0, we still have

(V n,

w n) => (v°,w°). Q.E.D.

We have done the hard work and have arrived where we wanted to be, out of the woods

and in the cherry orchard.

We can now reach out and pick our results

from

the nearby boughs. Define,

for n = 0, I, ... t Rt(~) =

~

Ut(~) =

f 0

~Rd Gt_s(V~,y)

• Wn(dy

ds)

t

Recall implies

from Proposition

convergence

fd Gt_s(~,y)zn(dy R

ds).

7.8 that convergence

of the integrals.

of the martingale

It thus follows

immediately

measures

from Theorem

that

COROLLARY

8.8.

Suppose A n ÷ ~ and ~n/kn + 0. (V n, W n, R n) => (V 0, W 0, R0);

(i) (ii)

if, in addition,

An~ n + ~,

(V n, W n, Z n, R n, U n) =>

Rewrite

~t($) - k

In view of Corollary

8.9

(i)

(V 0, W 0, Z 0, R 0, U0).

(8.10) as

(8.13)

THEOREM

Then

V(Gt(~,o))

+ ~/~ Ut(~)

8.6 we can read off all the weak

If k

÷ ~ and ~n + 0, then

~t(~)

~0 = v0

for which ~ ÷ 0. k

converges

of the SPDE ~t = ~ ~

limits

- An n

D{ [0,1], __S'(Rd)} to a solution

+ Rt(~).

+ v-~

in

8.7

399

(ii)

If An+ ~, ~n + ~ and ~ n / k n + 0, then

~(%) - kn

converges in ~{ [0,1], ~'(Rd)}

to a solution of the SPDE

/kn~ n

~

1

~0 = 0 .

(iii) If k

2 n + ~, kn~ n + ~ and ~n + c

~($)

- kn converges in

~ 0, then n

D{ [0,1], S'(Rd)} to a solution of t h e SPDE

St = ~ A ~

+ c~ + V.~

~0 = V0" T h e o r e m 8.9 covers the i n t e r e s t i n g limits in w h i c h k + ~ and ~/k + 0. T h e s e are all Gaussian.

The r e m a i n i n g limits are in general non Gaussian.

those in

w h i c h ~ and k both tend to finite limits are trivial enough to pass over here, w h i c h leaves us two cases (iv)

k + ~

and

~/k + c 2 > 0;

(v)

~ + -

and

~/k + =.

The limits in case

(v) turn out to be zero, as we will show below.

Thus

the only non-trivial, n o n - G a u s s i a n limit is case (iv), w h i c h leads to m e a s u r e - v a l u e d processes.

A MEASURE DIFFUSION

T H E O R E M 8.10

Suppose A n + = and ~ n / k n + c 2 > 0.

D__{[0,1], S'(Rd)}

to a p r o c e s s {~t" t ~

1 n Then ~ D t converges w e a k l y in n

[0,1]} w h i c h is continuous and has

measure-values.

There are a number of proofs of this t h e o r e m in the literature

(see the

Notes), but all those w e k n o w of use s p e c i f i c p r o p e r t i e s of b r a n c h i n g p r o c e s s e s w h i c h we don't want to develop here, so we refer the reader to the references for the proof, and ~ i m i t ourselves to some formal remarks. We can get some idea of the b e h a v i o r of the l i m i t i n g p r o c e s s by r e w r i t i n g

400

(8.13) in the form I

(8.14) If (kn,~n)

I

~ Dt(~) = + CUt(~) + ~/E (V(Gt(~'')) is any sequence

+ ~I Rt(~)) "

I (iv), {(vn,wn,zn,Rn,U n, ~

satisfying

n)}

is tight by

n Theorem 8.6 and Proposition

7.8, hence we may choose a subsequence

converges

(V, W, Z, R, U, D)'

weakly to a limit

(8.15)

From

along which it

(8.14)

~t(~) = + cUt(~) t = + C

f 0

fd Gt-S(~'Y)Z(dy'ds)" R

In SPDE form this is 5t

(8.16)

~o(dX)

= dx

We can see several things from this. so is D.

Consequently,

non-Gaussian

~t' being a positive

- Gaussian processes

In particular,

For one thing, n

distribution,

aren't p o s i t i v e

is positive,

is a measure.

hence

It must be

- so Z itself must be non-Gaussian.

it is not a white noise.

Now ~0 is Lebesgue measure,

but if d > I, Dawson and Hochberg have shown

that ~ t is purely singular with respect to Lebesgue measure Roelly-Coppoletta

has shown that ~t is absolutely

To get some idea of what the orthogonal from Proposition

for t > 0.

If d = I,

continuous. martingale

measure

Z is like, note

8.1 that t

=

f ~I 0

which suggests

D~(A)ds, n

that in the limit t t = ~ ~s (A)ds'

or, in terms of the measure

u of Corollary

(8.17)

v(dx,ds) This indicates

statistics

2.8,

= Ds(dX)ds.

why the SPDE (8.16) is not very useful for studying ~:

the

of Z are simply too closely connected with those of D, for Z vanishes

wherever ~ does, and ~ vanishes

on large sets - in fact on a set of full Lebesgue

measure if d > 2.

In fact, it seems easier to study D, which is a continuous

branching process,

than Z, so (8.16) effectively

expresses

state

D in terms of a process

401

which

is even

less understood.

w e r e w h i t e noises,

processes

Nevertheless, involving

a white

made

rigorous.

there

is a h e u r i s t i c is w o r t h w h i l e

understanding

A n d which,

contrasts

with cases

w h i c h we u n d e r s t a n d

noise w h i c h

give some i n t u i t i v e

This

(i)-(iii),

rather well.

transformation giving.

of

to add, w i l l

(8.16)

into an SPDE

T h i s has b e e n u s e d by D a w s o n

of 11, but w h i c h has never,

we h a s t e n

in w h i c h Z a n d W

certainly

to our knowledge,

to

been

not be made r i g o r o u s

here. Let W be a r e a l - v a l u e d

white

noise

that Z has the same m e a n a n d c o v a r i a n c e

on

as Z'

R

d

x

R+

•

Then

(8.17)

indicates

where

t

z't(~) = f

fd ~ s (y) w(dy ds).

0

(If d = singular

measure,

I,

Ds(dY)

R

= Ds(Y)dy,

so /~s(y)

makes

so it is h a r d to see what /~D_ means,

sense.

If d > I, ~ s is a

but let's

not worry a b o u t

it. ) F---

In d e r i v a t i v e

form,

/

Z' = ~? W

, w h i c h makes

it t e m p t i n g

to r e w r i t e

the SPDE

S

(8.16)

as

(8.18)

~= 5t

+

c~

It is not clear that this e q u a t i o n d = I, it is not clear w h a t

its c o n n e c t i o n

limit of the i n f i n i t e p a r t i c l e

system,

~ has any m e a n i n g

is w i t h

if d ~

the p r o c e s s

so it remains

2, a n d even if

D which

is the w e a k

one of the c u r i o s i t i e s

of the

subject.

THE CASE ~

REMARKS. three

One of the f e a t u r e s

sources

In case

completely,

three

measure,

(i), the b r a n c h i n g

distribution

I term ~d

- initial

of T h e o r e m

the d i f f u s i o n

becomes

~, while the noise comes

effects

contribute

8.9 is that it allows us to see w h i c h of the

diffusion,

is n e g l i g e a b l e

and the diffusion.

In case

÷

or b r a n c h i n g

and the noise

- drives comes

(ii) the initial

deterministic entirely

to the noise term.

the limit process.

f r o m the i n i t i a l

distribution

and only c o n t r i b u t e s

f r o m the branching. In case

washes

out

to the drift

In case

(iii),

(iv), the m e a s u r e - v a l u e d

all

402

diffusion, we see f r o m (8.16) that the initial d i s t r i b u t i o n and d i f f u s i o n both b e c o m e deterministic, while the randomness comes e n t i r e l y from the branching. In case w a s h out.

(v), which we will analyze now, it turns out that all the sources

N o t i c e that T h e o r e m 8.6 doesn't apply w h e n ~/k + ~, and in fact w e can't

a f f i r m that the family is tight. way.

Nevertheless, ~ tends to zero in a rather s t r o n g

In fact the u n n o r m a l i z e d p r o c e s s tends to zero.

T H E O R E M 8.11. (i)

Let k ÷ ~ and ~/k ÷ ~.

P A , M {Dr(K) = 0, all t 6

and, if d = (ii)

Then for any compact set K C R d and ~ > 0 [c,I/~]} ÷ I

I,

Pk,~{~t(K) = 0, all t ~ ~} ~

I.

Before p r o v i n g this we need to look at f i r s t - h i t t i n g times for b r a n c h i n g B r o w n i a n motions.

This d i s c u s s i o n is c o m p l i c a t e d by the p r o f u s i o n of particles: m a n y

of them may hit a given set.

To which belongs the honor of first entry?

The type of first h i t t i n g time we have in mind uses the implicit p a r t i a l o r d e r i n g of the b r a n c h i n g p r o c e s s - its p a t h s form a tree, a f t e r all - and those familiar w i t h two p a r a m e t e r m a r t i n g a l e s might be i n t e r e s t e d to compare these w i t h s t o p p i n g lines.

Suppose that {X ~, ~ £ ~} is the family of processes we c o n s t r u c t e d at the b e g i n n i n g of the chapter, and let A C R d be a Borel set.

For each ~, let

~A = inf{t > 0: X ~t e A}, and define T A ~ by T ~ = { ~A A

if ~

= ~ for all ~ < ~, ~ # ~;

otherwise

The time T~ is our a n a l o g u e of a first h i t t i n g time.

N o t i c e that T A m a y be

finite for m a n y different ~, but if ~ ~ ~, T~ and ~A c a n ' t both be finite. for example, the first entrance T E of the B r i t i s h c i t i z e n e r y to an earldom. individual - call h i m ~ - is created the first Earl of Emsworth,

Consider, If an

some of his

d e s c e n d a n t s may inherit the title, but his e l e v a t i o n is the vital one, so o n l y T ~ is E

finite.

On the other hand, a first cousin - call h i m ~ - m a y be c r e a t e d the f i r s t

Earl of Ickenham;

then T~ will also be finite. E

403

In general~

if ~ ~ ~ and if T~A and T~ are both finite,

of X ~ and of X ~ form disjoint independence

families.

(Why?)

of the different particles,

conditionally

independent

By the strong Markov property

the post-T~ and post-T~ processes

cf the branching

rate ~ which starts with a single particle X I at x. (non-branching)

(symmetry),

PROPOSITION

8.12.

x

Rd •

x then, X tI is a n Under PO"

Brownian motion.

it

is more complicated

For any Borel set A ~

arguments

A is compact and ~ has compact support and w r i t e T ~ and ~

it is true for the

R n x R+

, with ~(x,~) = 0,

Rd =

By standard capacity

While

and rates a detailed proof°

Let ~(x,t) be a bounded Borel function

%{ x PROOF.

are

Brownian motion with branching

The following result is a fancy version of (8.4). same reason

and the

given X ~ (T - A) ~ and X~(T~)

Let pX be the distribution

ordinary

then the descendants

(~A),TA) } •

it is enough to prove this for the case where in

R d × [0,~).

We will drop the subscript

instead of T A and t A.

Define u(x,t) = E0{~(X11,t

+ I)}.

is a martingale,

so that we can conclude

5~-~ U + ~I A U = 0.

Thus by Ito's formula

Note that {u(X I I' t ^ I ) , t A T

t >__ 0}

that u ~ C (2) on the open set A c x

R+ and

Yt g=ef ! u(X ~ t ^ T ~) tAT ~' t

= u(x,0) +

[

Vu(x~,s) • ~

f h'XCs) z { s ~(~)}.

independent

on

(X~, ~ )

vanishes

on {t < ~(~)}, and equals v~ =

also vanishes

Thus in all cases E{ 0.

on the set {s
~(~), T ~ _> C(~)}.

< t ^ T ~}

so

(This even holds if T u = ~, since both sides vanish then.)

Thus u(x,0)

= E~{lim Yt } = E~{ ! *(X~'T T~)}" Q.E.D.

REMARKS.

This implies

that the hitting probabilities

of the branching

Brownian

motion are dominated by those of Brownian motion - just take ~ --- I and note that the left hand side of (8.14) dominates

EX{sup ~(XT~,T~)}

implies

(8.14)

that the left hand side of

= pX{T~< ~, some ~}.

is independent

of ~.

We need several results before we can prove Theorem 8.11. treat the case d = I.

Let D be the unit interval H(x)

PROPOSITION

PROOF.

8.14.

H

(x)

= £~

(x -

I - /~)-2

u" = ~ u u(1)

=

R I and put

if

2

x >

I.

is the unique solution of

on

(I,~)

on

(I,~),

I

0 < u < I

since it is easily verified that the given expression Let T = inf T ~.

Let us first

= px{T~ < ~, some ~}.

This will follow once we show that H

(8.19)

in

If x > 1, Proposition

It also

satisfies

8.12 implies

(8.19).

405

(8.20)

pX{TIX < h} = P0{TX < h} = o(h)

as h ÷ 0.

Let ~ be the first b r a n c h i n g time of the process.

Then

+ pX{~ < h, ~ A h < T < --}. IX -The first p r o b a b i l i t y is o(h) by (8.20). to the latter two.

A p p l y the strong M a r k o v p r o p e r t y at ~ ^ h

If ~ > h, there is still only one particle• X I, alive, so T = T I

and the p r o b a b i l i t y equals E {~ > h, HIX(X~Ah)} + o(h), where the o(h) comes from i g n o r i n g the p o s s i b i l i t y that T < ~ ~ h. i n d e p e n d e n t particles, X 11 and X 12 +

- I

I{T (W 0, ~]).

The idea of the proof is the same:

hence so does the stochastic

integral.

7.6 and 7.8, for the integrand is not in ~s. k Define

•

However,

measure

we can't use Propositions

We will use 7.12 and 7.13 instead.

k

0 (V , W 0) and (~ n, ~ n) canonically

denote their probability

the martingale

It

distributions

on D = D([0,1],

by p0 and P respectively.

S'(R2d)),

and

By (8.23) we can

k define Dt

n

on D simultaneously

the stochastic

integrals

zero for all n > 0.

for n = 0,1,2,...

Thus we can also define

for each s,t and x, independent

W01

k n (¢)VP X(x)-~ n d×[0,t]s t-s xs

Hint.

of n.

Show g(o,.,t0)I{s d/2 this is

2 < C2nT*K(¢," ) [1q

--

T is a differential

operator of order k, hence it is bounded from Hq+ k ÷ Hq,

while K maps Hq+k_ 2 ÷ Hq+ k boundedly.

Thus the above is

< C4W ~" 2+k_2 It follows that U is continuous

in probability

4. I, it is a random linear functional

on H

on Hq+k_ 2 and, by T h e o r e m

for any p > q + k - 2 + d/2.

Fix

P a p > d + k - 2 and let n = p + 2.

Then U 6 H_n.

(It is much easier to see

418

that U ~ ~'(Rd). Corollary

Just note that T'K(#,.)

is bounded if # G ~ ( R d) and apply

4.2).

If ~ e

SO" U(A#) =

=

On the other hand, C 0 C continuous

on H

f D

T*K(A~,y)M(dy)

f T*~(y)M(dy). D

~0' and C O is dense in all the H t.

SO the map ~ ÷ 4# ÷ U ( ~ )

p

w h i l e on the right-hand

of H

n

÷ H

p

+

R

U is

is continuous,

side of (9.5)

E{I fd T* ¢~j2} ~C,T.¢,2®~c,¢,~+q. R

L

which tells us the right-hand hence,

side is continuous

by Theorem 4.1, it is a linear functional

in probability

on Hk+q,

on Hd+ k = H n.

Thus

(9.5)

holds for # ~ H n.

Q.E.D.

LIMITS OF THE BROWNIAN DENSITY PROCESS

The Brownian density process

Dt satisfies

the equation

(9.7)

~ = ! A~ + a V.& + bZ at 2

where W is a d-dimensional

white noise and z is an independent

one-dimensional

white noise,

both on

R d x R+, and the coefficients

are constants.

(They depend on the limiting behavior

of ~ and k.)

Let us ask if the process has a weak limit as t ÷ ~. to see that the process blows up in dimensions The Green's Green's

d =

function G t for the heat equation on

function K for Laplace's

(9.8)

equation by

K(x,y) = - f G t ( x , y ) d t 0

and K itself is given by Cd K(x,y) = iy_x id-2 where C d is a constant.

'

The solution of (9.7) is

a and b

It is not too h a r d

I and 2, so suppose d ~ 3. R d is related to the

419

t

Tit(C) = T)oGt(~)) + a 0

t ~Rd VGt_s(~,y)-W(dy

dGt_s(¢,Y)Z(dyds)

ds) + b 0

d~f

~0Gt(~)

and U t are mean-zero

+ a Rt(~)

Gaussian

R

+ b Ut(~).

processes.

The covariance

of R t is

t E{Rt(~)Rt(~) } =

~ 0

fRd(VxGt_s)(~,y)-(VxOt_s)(~,y)dy

if we then integrate

Now VxG = -V G; Y

ds .

by parts

t = -

~0

= -

~ 0

~d AyGt-s(~'Y)Gt-s(~'y)dy

ds

t Id Gt-s(A#'Y)Gt-s (y'(~)dy ds R

= - f 0

G2t_2s(d~,~)ds 2t

by (5.7).

= - ~

d ~(Y)

= - ~

d ~(Y) [-~(x) + G2t(x,~)]dy

Since d ~ 3, G t ÷ 0

f 0

Gu(d#,y)ds

as t + ~

dy

so

I E{Rt(#)Rt(~) } + ~ .

(9.9) The calculation

for U is easier since we don't need to integrate

by

parts: t E{Ut(~)Ut(~)}

=

=

as t ÷ ~.

PROPOSITION

9.3.

2

ds

2t

!

Suppose d ~ 3.

~

G2t-u(0'~)du

÷

~(¢,~)

- 2""

we see:

AS t ÷ ~, / 2 R t converges

to a random Gaussian

tempered

weakly

to a white

distribution

function

(9.10)

E{U(¢)U(~)}

In particular, convergence

~d Gt-s(#'Y)Gt-s($'y)dy

Taking this and (9.9) into account,

noise and / ~ U t converges covariance

f0

Dt converges

of S'(Rd)-valued

= -K(¢,~).

weakly as t ÷ m. random variables

The convergence in all cases.

is weak

with

420

E x e r c i s e 9.1.

DEFINITION.

Fill in the details of t h e c o n v e r g e n c e argument.

The mean zero Gaussian process {U(#): ~ • ~(Rd)} w i t h c o v a r i a n c e

(9.10) is called the E u c l i d e a n free field.

C O N N E C T I O N W I T H SPDE's

We can get an integral r e p r e s e n t a t i o n of the free field U from P r o p o s i t i o n 9.3, for the w e a k limit of / 2 U t has the same d i s t r i b u t i o n as

fd 0

Gs(*'Y)Z(dY as).

R

This is not enlightening; we w o u l d prefer a r e p r e s e n t a t i o n i n d e p e n d e n t of time. on

This is not h a r d to find.

R d (not on

Rd ×

(9.11)

R+ as before)

U(~)

if ~,

~e

ffi

Let W be a d - d i m e n s i o n a l white noise and, for # £ ~(Rd),

define

fRd?K(#,y).W(dy).

~(Rd),

E{U(¢)U(%)} =

f dV~(~,y).VK(%,y)dy R

=

-

fRd K(~,y)~K(~,y)dy

= - f d K(~,y)~(y)dy R

= - K(~,~).

(This shows a p o s t e r i o r i that U(#) is definedl) G a u s s i a n process,

it is a free field.

P R O P O S I T I O N 9.4.

U satisfies the SPDE

(9.12) PROOF.

~ U = V.W U(~)

=

f

VK(~#,y) oW(dy) R

= f

V*(y).W(dy) R

Thus, as U(#) is a mean zero

421

since for ~ E s(Rd), K(A~,y) = ~(y).

But this is the weak form of (9o12). Q.E.D.

Exercise 9.1.

Convince yourself that for a.e.~,

(9.12) is an equation in

distributions.

SMOOTHNESS

Since we are working on

R d, we can use the Fourier transform.

be the Sobolev space defined in Example

la, Chapter 4.

~t st

If u is any

distribution, we say u ¢ H_loc t if for any ~ e CO, ~u E H t

PROPOSITION 9.5.

Let £ > 0.

loc Then with probability one, W E H_d/2_£ and

loc U & H1_d_2_£,/ where U is the free field.

PROOF. ~W(~) =

The Fourier transform of ~W is a function: ~

e-2Ki~'x~ (x)W(dx)

R

and

E{I¢I * I~12~t/2~¢~I 2} = ¢I * I~12~ t ff ~¢x~Cy~e2~iCYx~'~dy~, SO

_< c I c, + l~12~td~ which is finite if 2t < -d, in which case n~wn t is evidently finite a.s. loc Now VoW £ N_d/2_1_e so, since U satisfies AU = V-W, the elliptic loc regularity theorem of PDE's tells us U g Hl_e_d/2.

Q.E.D.

THE MARKOV PROPERTY OF THE FREE FIELD

We discussed L~vy's Markov and sharp Markov properties in Chapter One, in connection with the Brownian sheet.

They make sense for general

422

distribution-valued G*° =D

This

processes,

involves

extending

but one must

space,

tells us that it h a s a trace on c e r t a i n since we w a n t to talk about

works

the S o b o l e v

lower-dimensional

its values

on

on rather

R d, let us define

if ~ is of the f o r m ~(dx)

is s u f f c i e n t l y nice" m e a n s

nice.

embedding

theorem

manifolds.

But

irregular

sets, we will use

= ~(x)dx,

By the c a l c u l a t i o n

U(~)

by

This c e r t a i n l y

and it w i l l c o n t i n u e following

(9.11),

n~li~= - [ [ ~(dx)K(x,y)~(dy)

Let =E+

(9.11).

to work if

"sufficiently

that

(9.13)

=+

~D a n d

direct method. If ~ is a m e a s u r e

E

the o-fields

the d i s t r i b u t i o n .

S i n c e U takes v a l u e s on a S o b o l e v

a more

first define

be the class

of m e a s u r e s

on

R d which


0, all A C R d - B}.

that there be m e a s u r e s

G = ~{U(~): =B

relative

U(A)

of the r e s t r i c t i o n

9.6.

~ ~A ADB A open

follows

open sets in

easily

set of c a p a c i t y

f r o m the b a l a y a g e

for all y, and K(v,y) zero in ~D.

L~vy's

sharp M a r k o v p r o p e r t y

R d.

set a n d if ~ is s u p p o r t e d by D c, there

-K(~,y) ~ -K(~,y)

R d - B}

"

The free field U satisfies

to b o u n d e d

This

~ E ~, ~(A) = 0, all A C

We call

property

exists

= K(~,y)

of K:

a measure

for all y ~

v the b a l a y a g e

if D C

R d is an

v on 5D s u c h D, and all but a

of ~ on ~D.

423

Suppose ~ E ~ and supp ~ C D c .

If

v is the balayage

of ~ on ~D, we

claim that (9.14)

E{U(~)I~

} = U(v). D

This will do it since,

as U(~)

is ~SD-measurable,

the left-hand

side of

(9.~4) must be E { U ( ~ ) I ~ S D }. Note that v e ~ (for ~ is and -K(v, o) ! -K(~,-))

so if k e ~,

supp(k) C 5, E{(U(~)

- U(v))U(A)}

= f[K(~,y)

- K(v,y)]k(dy)

= 0 since K(~,x) = K(v,x)

for a set of capacity

zero, and

k, being of finite energy,

does not charge sets of capacity zero.

Thus the

integrand vanishes k-a.e.

But we are dealing with Gaussian processes,

this implies

(9.14).

on D, except possibly

so Q.E.D.

NOTES

We omitted most references from the body of the t e x t - a c o n s e q u e n c e of p u t t i n g off the b i b l i o g r a p h y till last - and we will try to remedy that here.

Our

r e f e r e n c e s w i l l be rather sketchy - you may put that down to a lack of s c h o l a r s h i p and we list the sources from w h i c h we p e r s o n a l l y have learned things, w h i c h may not be the sources in w h i c h they o r i g i n a l l y appeared.

We apologize in advance to the

m a n y w h o s e w o r k we have s l i g h t e d in this way.

C H A P T E R ONE

The Brownian sheet was introduced by Kitagawa in

[37], though it is u s u a l l y

c r e d i t e d to others, p e r h a p s because he failed to p r o v e the u n d e r l y i n g m e a s u r e was countably additive.

This o m i s s i o n looks less serious now than it did then.

The G a r s i a - R o d e m i c h - R u m s e y T h e o r e m o n e - p a r a m e t e r processes in article

(Theorem 1.1) was p r o v e d for

[23], and was p r o v e d in general in the brief and e l e g a n t

[22], w h i c h is the source of this proof.

This c o m m o n l y gives the right order

of m a g n i t u d e for the modulus of c o n t i n u i t y of a process,

but doesn't n e c e s s a r i l y give

the best constant, as, for example,

The exact m o d u l u s of

in P r o p o s i t i o n

1.4.

c o n t i n u i t y there, as well as many o t h e r i n t e r e s t i n g s a m p l e - p a t h p r o p e r t i e s of the B r o w n i a n sheet, may be f o u n d in Orey and Pruitt

[49].

K o l m o g o r o v ' s T h e o r e m is u s u a l l y stated more simply than in C o r o l l a r y In p a r t i c u l a r ,

the extra log terms there are a bit of an affectation.

curious to see how far one can go w i t h n o n - G a u s s i a n processes. v a l i d for r e a l - v a l u e d processes, processes.

See for example

We just were

Our v e r s i o n is o n l y

but the t h e o r e m holds for m e t r i c - s p a c e v a l u e d

[44, p.519].

The Markov p r o p e r t y of the Brownian sheet was p r o v e d by L. Pitt s p l i t t i n g f i e l d is i d e n t i f i e d in [59]; the p r o o f there is due to S. Orey communication.)

1.2.

[52]. (private

The

425

The propagation of singularities in the Brownian sheet is studied in detail in [56].

Orey and Taylor showed the existence of singular points of the Brownian

path and determined their Hausdorff dimension in [50].

Proposition 1.7 is due to G.

Zimmerman [63], with a quite different proof. The connection of the vibrating string and the Brownian sheet is due to E. Caba~a [8], who worked it out in the case of a finite string, which is harder than the infinite string we treat.

He also discusses the energy of the string.

CHAPTER TWO

In terms of the mathematical techniques involved, one can split up much of the study of SPDE's into two parts:

that in which the underlying noise has

nuclear covariance, and that in which it is a white noise.

The former leads

naturally to Hilbert space methods; these don't suffice to handle white noise, which leads to some fairly exotic functional analysis.

This chapter is an attempt to

combine the two in a (nearly) real variable setting.

The integral constructed here

may be technically new, but all the important cases can also be handled by previous integrals. (We should explain that we did not have time or space in these notes to cover SPDE's driven by martingale measures with nuclear covariance, so that we never take advantage of the integral's full generality). Integration with respect to orthogonal martingale measures, which include white noise, goes back at least to Gihman and Skorohod [25].

(They assumed as part

of their definition that the measures are worthy, but this assumption is unnecessary; c.f.

Corollary 2.9.) Integrals with respect to martingale measures having nuclear covariance

have been well-studied, though not in those terms. in M~tivier and Pellaumeil [46].

An excellent account can be found

They handle the case of "cylindrical processes",

(which include white noise) separately. The measure ~ of Corollary 2.8 is a Dol~ans measure at heart, although we haven't put it in the usual form.

True Dol~ans measures for such processes have been

426

constructed by Huang,

[31].

Proposition 2.10 is due to J. Watkins

[61].

Bakry's example can be found

in [2].

CHAPTER THREE

The linear wave and cable equations driven by white and colored noise have been treated numerous times.

Dawson

[13] gives an account of these and similar

equations. The existence and uniqueness of the solution of (3.5) were established by Dawson [14].

The LP-boundednes and Holder continuity of the paths are new.

See [57]

for a detailed account of the sample path behavior in the linear case and for more on the barrier problem.

The wave equation has been treated in the literature of two-parameter processes, going back to R. Cairoli's 1972 article

[ 9 ] . The setting there is special

because of the nature of the domain: on these domains, only the initial position need be specified, not the velocity.

As indicated in Exercises 3.4 and 3.5, one can extend Theorem 3.2 and Corollary 3.4, with virtually the same proof, to the equation ~V

~2V

at

5x 2

÷ g(V,t) + f(v,t)~,

where both f and g satisfy Lipschitz conditions. systems in which g is potential term.

Such equations can model physical

Faris and Jona-Lasinio [19] have used similar

equations to model the "tunnelling" of a system from one stable state to another. We chose reflecting boundary conditions in (3.5) and (3.5b) for convenience.

They can be replaced by general linear homogeneous boundary conditions;

the important point is that the Green's funciton satisfies (3.6) and (3.7), which hold in general [27].

427

CHAPTER FOUR

We follow and

some u n p u b l i s h e d

lecture

notes

See also

of the Ito here.

[24]

[34].

CHAPTER FIVE

The t e c h n i q u e s elliptic

operator.

lower order pole,

u s e d to solve

(5.1)

In fact the G r e e n ' s so that the solutions

also w o r k w h e n L is a h i g h e r o r d e r

function

for h i g h e r order o p e r a t o r s

are b e t t e r b e h a v e d

has a

t h a n in the s e c o n d - o r d e r

case. We suspect studies

a special

the solution

that T h e o r e m

case in

[33].

can be found in

these a n d similar

5.1 goes b a c k to the mists of antiquity. Theorem

[58].

5.4 and other

See Da Prato

The basic r e f e r e n c e theorem

treatment

here.

self-contained. distributions,

on the sample paths

for another

of

point of v i e w on

theorems.

CHAPTER

Aldous'

[12]

results

Ito

is in

[I], and Kurtz'

Mitoma's Fouque which

on w e a k

theorem [21] has

includes

SIX

convergence criterion

is p r o v e d

in

generalized

the f a m i l i a r

remains

is in

Billingsley's

[42].

spaces

D(Q).

[5].

We f o l l o w Kurtz'

[47], but the article

this to a larger

book

is not

class of spaces

His proof

of

is close to t h a t

of Mitoma.

C H A P T E R SEVEN

It m a y not be obvious

f r o m the e x p o s i t i o n

- b u t the first p a r t of the c h a p t e r The accounts

for its r e l a t i v e l y

Theorems

general

is d e s i g n e d

elementary

enough

to handle

- in fact we took care to h i d e it

to handle

deterministic

integrands.

character. the r a n d o m i n t e g r a n d s

met

in p r a c t i c e

428

seem to be delicate; we were s u r p r i s e d to find out how little is known, even in the classical case.

Our w o r k in the section "an extension" is just a first attempt in

that direction. P e t e r Kotelenez showed us the p r o o f of P r o p o s i t i o n 7.8. due to K a l l i a n p u r and W o l p e r t [57].

[36]°

An earlier,

T h e o r e m 7.10 is

clumsier v e r s i o n can be found in

The B U r k h o l d e r - D a v i s - G u n d y t h e o r e m is surmnarized in its most h i g h l y d e v e l o p e d

form in

[7].

CHAPTER EIGHT

This chapter completes a cycle of results on weak limits of Poisson systems of b r a n c h i n g B r o w n i a n motion due to a number of authors. strong a word,

"Completes"

is p e r h a p s too

for these point in many directions and we have only f o l l o w e d one: to

find all p o s s i b l e w e a k limits of a certain class of infinite p a r t i c l e systems, and to connect t h e m with SPDE's. T h e s e systems were i n v e s t i g a t e d by M a r t i n - L o f non-branching particles

(~ = 0 in our terminology)

w h o c o n s i d e r e d b r a n c h i n g B r o w n i a n motions in results look s u p e r f i c i a l l y d i f f e r e n t since,

Rd

[45] who c o n s i d e r e d

and by Holley and Stroock

[29],

with p a r a m e t e r s k = ~ = I ; their

instead of letting ~ and k tend to

infinity, they rescale the p r o c e s s in both space and time by r e p l a c i n g x by x/~ a n d t by ~2t. d

B e c a u s e of the B r o w n i a n scaling, this has the same effect as r e p l a c i n g k by

2 and ~ by ~ , and leaving x and t unscaled.

~/k = ~

2-d

The critical p a r a m e t e r is then

, so their results depend on the dimension d of the space.

If d > 3, they

find a G a u s s i a n limit (case (ii) of T h e o r e m 8.9), if d = 2 they have the m e a s u r e - v a l u e d diffusion 8.11). [33],

(case (iv)) and if d = I, the p r o c e s s tends to zero (Theorem

The case ~ = 0, i n v e s t i g a t e d by Martin-Lof and, with some differences, [34], also leads to a G a u s s i a n limit

Gorostitza 8.9(iii)

if ~ > 0).

by Ito

(Theorem 8.9 (i)).

[26] t r e a t e d the case w h e r e ~ is fixed and k ÷ ~

(Theorem

H e also gets a d e c o m p o s i t i o n of the noise into two parts, b u t it

is different from ours; he has p o i n t e d out not in fact independent.

[26, Correction]

that the two parts are

429

T h e n o n - G a u s s i a n case

(case (iv)) is e x t r e m e l y i n t e r e s t i n g a n d has been

i n v e s t i g a t e d b y numerous authors.

S. W a t a n a b e

[60] p r o v e d the c o n v e r g e n c e of the

system to a m e a s u r e - v a l u e d diffusion.

Different proofs have been given by Dawson

[13], Kurtz

[53].

[42], and R o e l l y - C o p o l e t t a

D a w s o n and H o c h b e r g

[15] have looked at

the Hausdorff d i m e n s i o n of the support of the m e a s u r e and s h o w e d it is singular w i t h respect to L e b e s g u e m e a s u r e if d > 2.

(Roelly-Copoletta

[53]).

It is a b s o l u t e l y continuous if d = I

A related equation which can be w r i t t e n suggestively as an = ~ A~ + ~ ( 1 - ~ ) w at 2

has been studied by F l e m i n g a n d Viot

[20].

The case in w h i c h ~/k ÷ ~ comes up in H o l l e y and Stroock's p a p e r if d = I. The results p r e s e n t e d here, which are stronger, are joint w o r k with E. Perkins and J. Watkins,

and appear here w i t h their permission.

The noise W of P r o p o s i t i o n 8.1 is

due to E. Perkins who u s e d it to translate Ito's work into the setting of SPDE's relative to m a r t i n g a l e m e a s u r e s

(private communication.)

A more general and m o r e s o p h i s t i c a t e d c o n s t r u c t i o n of b r a n c h i n g d i f f u s i o n s can be found in Ikeda, Nagasawa, and W a t a n a b e

[32].

H o l l e y and Stroock also give a

construction.

The square process Q is c o n n e c t e d w i t h U statistics.

Dynkin and M a n d e l b a u m

[17] showed that c e r t a i n central limit theorems i n v o l v i n g U statistics lead to m u l t i p l e Wiener integrals, and we wish to thank Dynkin for s u g g e s t i n g that our m e t h o d s m i g h t handle the case when the p a r t i c l e s were d i f f u s i n g in time.

In fact

T h e o r e m 8.18 m i g h t be v i e w e d as a central limit t h e o r e m for c e r t a i n U - s t a t i s t i c s e v o l v i n g in time.

We should say a w o r d about g e n e r a l i z a t i o n s here.

We have t r e a t e d only the

simplest settings for the sake of clarity, but there is s u r p r i s i n g l y little change if we m o v e to m o r e complex systems. b r a n c h i n g diffusions,

We can replace the B r o w n i a n p a r t i c l e s b y

or even b r a n c h i n g Hunt processes,

c h a n g i n g the character of the l i m i t i n g process. Paris,

1984]).

for instance, w i t h o u t

(Roelly-Copoletta

One can treat more general b r a n c h i n g schemes.

[Thesis, U. of

If the family size N

430

has a finite

variance,

Gorostitza

the form at ~D = ~I ~D + ~ new effect

+ ~Z + yVoW,

is to a d d a growth

term,

If E { N 2} = ~, however, in certain

cases w h e n

[26] has shown that one gets where

things

~/k has a finite

"random

field"

has b e e n u s e d to cover a l m o s t one-parameter

processes,

limit.

It seems

rate,

is s o m e t h i n g

this

chapter

of a mystery.

is about

spaces,

pressure

potential

theory

and get n > k + d/2 rather

certain

for e l l i p t i c

for e l l i p t i c

than p a r a b o l i c

a p p l y to p a r a b o l i c

systems,

or h y p e r b o l i c

for that matter).

At any

here.

Frankly,

we w e r e under For s o b o l e v

[67]

of m e a s u r e s

can be found in Doob

in P r o p o s i t i o n

and Hormander

[30].

7.1 can doubtless

the S o b o l e v

embedding

[66]. be

in the p r o o f

introduced

by N e l s o n

in

the q u a n t u m

[48].

He p r o v e d

the sharp

field which d e s c r i b e s

He also s h o w e d that it can be m o d i f i e d

to describe

systems. book

[54] is a g o o d reference

systems

in which,

Markov property

it

- a n d some

see F o l l a n d

one can bypass

for a s t r o n g M a r k o v property,

that L~vy's

than one p a r a m e t e r

than n > k + d.

particles.

Rozanov's

finds

machinery

space

a n d u s e d it to c o n s t r u c t

interacting

Evstigneev

At one time or another,

to be u s e d p a r t i c u l a r l y

and the energy

noise,

The free field was

non-interacting

word.

fields.

n of the S o b o l e v

If M is a w h i t e

M a r k o v property,

study.

NINE

[64]; for the P D E theorems,

The exponent improved.

further

(As is the t e r m itself,

random

~ can t e n d to zero

and d i d n ' t have time to work out an e a s i e r approach.

see A d a m s

The c l a s s i c a l

needs

having more

W e h a v e u s e d some h e a v y t e c h n i c a l deadline

For example,

This

is a p o r t m a n t e a u

t h o u g h w h y it s h o u l d be u s e d more often systems

~ = 0 if E { N - 1} = 0, so that the o n l y

do change.

any p r o c e s s

too.

of

~.

CHAPTER

The t e r m

limiting equations

and K u s u o k a

contrary

holds

for L~vy's M a r k o v property. [43] for results

to the claim in

See

which also

[57], one c o m m o n l y

but the sharp M a r k o v p r o p e r t y

does not.

431

C H A P T E R TEN

There is no Chapter Ten in these notes. s t o p p e d us f r o m h a v i n g notes on C h a p t e r Ten. remarks w h i c h didn't fit in elsewhere.

For some reason that h a s n ' t

W e w i l l use this space to collect some

Since the chapter under d i s c u s s i o n doesn't

exist, no one can accuse us of digressing. We did not have a chance to discuss equations relative to m a r t i n g a l e m e a s u r e s w i t h a nuclear covariance.

These can arise when the u n d e r l y i n g noise is

smoother than a white noise or, as often happens, a p p r o x i m a t e d by a smoothed out version.

it is w h i t e noise w h i c h one has

If one thinks of a white noise, as we did in

the introduction, as c o m i n g from s t o r m - d r i v e n grains of sand b o m b a r d i n g a guitar string, one m i g h t think of nuclear covariance noise as coming from a s t o r m of p i n g - p o n g balls.

The solutions of such systems tend to be b e t t e r - b e h a v e d ,

particular, they often give f u n c t i o n solutions rather than distributions. it p o s s i b l e to t r e a t n o n - l i n e a r equations,

equations are u s u a l l y t r e a t e d in a H i l b e r t - s p a c e setting. [11], Da Prato

[12], and Ichikawa

This m a k e s

s o m e t h i n g rather a w k w a r d to do o t h e r w i s e

(how does one take a n o n - l i n e a r function of a distribution?)

and Falb

and in

Mathematically,

these

See for instance Curtain

[68].

There have been a variety of a p p r o a c h e s d e v i s e d to cope with SPDE's driven b y white noise and related processes.

See Kuo

b a s e d on the theory of a b s t r a c t Wiener spaces.

[41] and Dawson

The latter p a p e r reviews the s u b j e c t

of SPDE's u p to 1975 and has e x t e n s i v e references. and Karandikar measures.

[13] for a treatment

Balakrishnan

[3] and K a l l i a n p u r

[35] have used c y l i n d r i c a l B r o w n i a n motions and finitely additive

See also M 6 t i v i e r and P e l l a u m a i l

[46], w h i c h gives an account of the

i n t e g r a t i o n theory of c y l i n d r i c a l processes. o r t h o g o n a l m a r t i n g a l e measures.

G i h m a n and Skorohod

See also W a t k i n s

[61].

Ustunel

[25] i n t r o d u c e d [55] has s t u d i e d

nuclear space v a l u e d s e m i - m a r t i n g a l e s with applications to SPDE's and s t o c h a s t i c flows.

The m a r t i n g a l e p r o b l e m m e t h o d can be a d a p t e d to SPDE'S as w e l l as o r d i n a r y

SDE's.

It has had succes in h a n d l i n g n o n - l i n e a r equations i n t r a c t a b l e to other

methods.

See Dawson

the linear case.

[65] and F l e m i n g and Viot

[20], and Holley and Stroock

[29] for

432

Another type of equation which has generated considerable research is the SPDE driven by a single one-parameter Brownian motion.

(One could get such an

equation from (5.1) by letting T be an integral operator rather than a differential operator.) theory.

An example of this is the Zakai equation which arises in filtering

See Pardoux

[51] and Krylov and Rosovski

[39].

Let us finish by mentioning a few more subjects which might interest the reader: fluid flow and the stochastic Navier-Stokes equation Temam

(e.g. Bensoussan and

[4] ); measure-valued diffusions and their application to population growth

(Dawson

[65], Fleming and Viot

(Kotelenz

[20]); reaction diffusion equations in chemistry

[38]) and quantum fields (Wolpert

[70] and Dynkin

[16])0

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