SOOCHOW JOURNAL OF MATHEMATICS

Volume No pp Octob er

ANALYSIS OF WHITE NOISE FUNCTIONALS

BY

HUIHSIUNG KUO

Dedicated to the memory of Professor TsingHoua Teng

Intro duction

It has b een almost two decades since Professor T Hida intro duced the the

ory of white noise in This theory has b een develop ed rather extensively

and is now regarded as an innite dimensional calculus The b est source of

information on this topic is the b o ok by T Hida H H Kuo J Pottho and

L Streit White Noise An innite dimensional calculus Kluwer Academic

Publishers

In this set of lecture notes I have tried to give an easy intro duction of the

white noise calculus to graduate students while at the same time provide a

quick understanding of the theory for p eople working on other elds Basic

ideas are explained and some elementary techniques are carried out in detail

However most of the theorems are stated without pro ofs The sources of the

pro ofs can b e easily identied from the content and the list of references

In some sense this set of lecture notes is a continuation of my lecture notes

given at So o chow University in see H H Kuo Lectures on white noise

analysis Soochow J Math I have stated recent results

on HitsudaSkorokho d integral sto chastic dierential equations and Fourier

transform Ihave also discussed p ositive generalized functions and white noise

kernel op erators

Received March

AMS Sub ject Classication F J G

HUIHSIUNG KUO

This set of lecture notes is rather incomplete Many imp ortant applica

tions of white noise calculus eg Feynman integrals Dirichlet forms and

quantum eld theory are not even mentioned They can be found in the

HKPS book Other imp ortant applications are innite dimensional rotation

groups and the variational calculus as Professor Hida describ ed in his b eauti

ful lecture on June at Cheng Kung University We hop e that Professor

Hida will complete the white noise building so on as mentioned in his lecture

It is a great pleasure to give my deep est appreciation to Professor

Y J Lee for arranging my visit to Cheng Kung University I would like to

express my gratitude for the hospitality of the Mathematics Department of

Cheng Kung University Iamvery grateful for the nancial supp ort from the

National Science Council of Taiwan

White noise space

We will construct the white noise space in two dierent ways namely

by using the Minlos theorem for nuclear spaces and by applying the Gross

theorem for abstract Wiener spaces

A Nuclear space

Let S IR be the space consisting of all realvalued rapidly decreasing

functions f on IR ie

n k

lim jx f xj n k

jxj

x x

For instance C functions with compact supp ort e e are all

functions in S IR For any n k dene a norm kk on S IR by

nk

n k

kf k sup jx f xj

nk

xIR

Then S IR fk k n k g is a top ological space In fact it is a

nk

nuclear space Let S IR b e the dual space of S IR

ANALYSIS OF WHITE NOISE FUNCTIONALS

Theorem Minlos Let E be a nuclear space with dual space E

A complexvalued function on E is the characteristic functional of a prob

ability measure on E ie

Z

ihxy i

e d x y E y

E

if and only if it satises the fol lowing conditions



 is positive denite ie for any z y E j n any n

j j

n

X

z z y y

j k j k

jk

 is continuous

Remark The measure is uniquely determined by Observe that

E Thus when condition is not assumed then is a nite

measure

Example Let b e a function on S IR given by

exp j j SIR

where jj is the L IRnorm Then it is easy to check that conditions and

are satised Tocheck condition note that

Z

j xj dx j j

IR

Z Z

j xj dx j xj dx

jxj jxj

Z

sup j xj jx xj dx

x

jxj

jxj

Z

dx k k supjy y j

x

jxj

jy j

k k k k

This shows that is continuous Therefore by the Minlos theorem there

exists a unique probability measure on S IR such that

Z

ihx i

e dxexp j j SIR

S IR

HUIHSIUNG KUO

Denition The measure is called the standard Gaussian measure on

S IR The probability space S IR is called a white noise space

B Abstract Wiener space

k

Let b e the standard Gaussian measure on IR Then

k

Z

k

ihxy i

e d x exp jy j y IR

k

k

IR

An obvious extension of this fact to an innite dimensional real separable

Hilb ert space H is

Z

ihxhi

e dxexp jhj h H

H

However such a measure do es not exist To see this take an orthonormal

basis fv n g for H Then wehave

n

Z



ihxv i

n

e dxe

H

Note that hx v i converges to zero as n for every x H Hence by the

n

Leb esgue dominated convergence theorem we get the ridiculous conclusion



e

Theorem Gross Let H be a real separable Hilbert space with

norm j j Suppose B is the completion of H with respect to a measurable

norm on H Then there exists a unique probability measure on B such that

Z

ihxhi

jhj h H e dxexp

B

Fact Let T b e an injective Hilb ertSchmidt op erator of H Then kxk

jTxj is a measurable norm

Example Consider the Hilb ert space H L IR with norm jj Let H

n

b e the Hermite p olynomial of degree n n dened by

n x n x

H x e D e

n

x

Let e b e the Hermite function of order n n dened by

n



p

x n

e x H xe n

n n

ANALYSIS OF WHITE NOISE FUNCTIONALS

Obviously fe n gSIR Moreover this family is an orthonormal basis

n

for L IR Let A b e the following op erator

d

A x

dx

It is a fact that Ae n e n For any p IR dene

n n

p

jf j jA f j f L IR

p

Note that jf j can b e expressed in another way

p



X

p

n f e jf j

n p

n

where is the inner pro duct of L IR

p

Weshow that the op erator A is a Hilb ertSchmidt op erator of L IR if

and only if p as follows

X

p p

jA e j kA k

n

HS

n

X

p

jn e j

n

n

X

p

n

n

Let S IR b e the completion of L IR with resp ect to the norm jj p

p p

It Then L IR S IR is an abstract Wiener space if and only if p

p

follows from the Gross theorem that the supp ort of the white noise measure

is in S IR if p In fact the converse is also true ie if is supp orted

p

in S IR then p

p

WienerIto decomp osition theorem

For simplicitywe will use L to denote the space L S IR over

with norm kk It follows from the denition of that for each S IR

the random variable hi is dened everywhere on S IR and is normally

HUIHSIUNG KUO

distributed with mean and variance j j Supp ose f is in L IR Take a

sequence f g S IR converging to f in L IR It is easy to see that

n

fh ig is a Cauchy sequence in L Dene

n

hfi limit in L of h i

n

The limit is indep endent of the sequence f g Moreover the random variable

n

hfi is normally distributed with mean and variance jf j

Dene the renormalized Hermite p olynomial of degree n b y

x

p p

h x H

n n

n

Let ff f g b e orthonormal in L IR Let n n n Dene

k k

b n

b n

k 

b b

by f the multiple Wiener integral of f

k

b n

b n

k



b b

h hf i h hf i I f f

n n k n

k 

k

The mapping I can b e extended linearly and continuously to the symmetric

n

n n

c

complex L space L IR onIR It can b e checked easily that

p

n

c

n jf j f L IR kI f k

n

WienerIto decomp osition theorem Any in L can be decom

posed uniquely as fol lows

X

n

c

I f f L IR

n n n

n

Moreover

X

kk njf j

n

n

Example The generating function of the renormalized Hermite p olyno

mials is given by

n

X

t



tx t

e h x

n

n

n

ANALYSIS OF WHITE NOISE FUNCTIONALS

Let g L IR jg j Replace x with hgi in the ab ove equation to get

n

X

t



thg i t

e h hgi

n

n

n

n

X

t

n

I g

n

n

n

Now supp ose z and f L IRf Let g fjf j and t z jf j to get

n

X

z



z jf j n z hf i

I f e

n

n

n

z hf i

Notation Dene the renormalization of e by



z hf i z hf i z jf j

e e

Wehave the following equality

n

X

z

n z hf i

I f e

n

n

n

Furthermore from this equalitywe obtain

n

X

jz j

z hf i n

k e k n jf j

n

n

X

n

jz j jf j

n

n

jz j jf j

e

Hence for any z f L IR wehave



jz j jf j z hf i

k e k e

Gelfand triples

A One dimensional space The Schwartz distribution theory on the space

IR concerns with the following Gelfand triple

S IR L IR S IR

HUIHSIUNG KUO

B Innite dimensional space For an innite dimensional distribution theory

we need to construct a similar Gelfand triple The space IR with the

Leb esgue measure is replaced by the white noise space S IR with the

standard Gaussian measure Thus we have already the middle space

L The question is howtointro duce a small space and a big space

L

In order to construct the Gelfand triple we need to reconstruct the

spaces S IR and S IR in the Gelfand triple by another way The same

pro cedure can then be used to construct the Gelfand triple Let A be

the following op erator

d

A x

dx

For each p IR dene

p

jf j jA f j

p

As wesawinx the norm jj is stronger than the L IRnorm jj if p

p

and is weaker than jj if p Thus for p let

S IR ff L IR jf j g

p p

On the other hand for p let

S IR completion of L IR jj

p p

It is easy to see that the dual space S IR of S IR can be identied as

p

p

S IR when we apply the Riesz representation theorem to L IR Thus we

p

have the following continuous inclusions

S IR L IR S IR p

p

p

Example Consider the delta function in S IR It is not in L IR

t

Its expansion in terms of the Hermite functions is given by

X

e e

t t n n

n

X

e te

n n

n

ANALYSIS OF WHITE NOISE FUNCTIONALS

Therefore for p wehave

p

j j jA j

t t

p

X

p

e tn e

n n

n

X

je tj

n

p

n

n

Fact For the supnorm of the Hermite functions wehave

ke k O n

n

Hence j j if p IR for any p ie p Thus S

t p t

p

We can check the following fact which gives another construction of the

spaces S IR and S IR from the space L IR and the op erator A

Fact

S IR pro jective limit of fS IR p g

p

S IR

p

p

S IR dual space of S IR

IR p g inductivelimitof fS

p

S IR

p

p

Now we will apply the same pro cedure as ab ove to intro duce the spaces

and in We use the Hilb ert space L and the following densely

dened op erator A

X

I f

n n

n

X

n

A I A f

n n

n

For p IR dene

p

kk kA k p

HUIHSIUNG KUO

It is easy to check that the norm kk is stronger than the L norm kk if

p

p and is weaker than kk if p Thus for p let

S f L kk g

p p

On the other hand for p let

S completion of L kk

p p

Note that the dual space S of S can be identied as S when we

p p

p

apply the Riesz representation theorem to L Thus we have the following

continuous inclusions

p S L S

p

p

Finallywe dene the space S oftest functions by

S pro jectivelimitof fS p g

p

S

p

p

The space S of generalized functions or Hida distributions is given by

S dual space of S

inductive limit of f S p g

p

S

p

p

It is a fact that the space S isanuclear space Thus wehave the following

Gelfand triple

S L S

Example Supp ose f L IR From the example in x wehave for any

z



z hf i z hf i z jf j

e e

n

X

z

n

I f

n

n

n

ANALYSIS OF WHITE NOISE FUNCTIONALS

and



z hf i jz j jf j

k e k e

z hf i

Consider now the renormalization e forany f Let p IR Then

n

X

jz j

p n n z hf i

jA f j k e k n

p

n

n

n

X

jz j

p n

jA f j

n

n

n

X

jz j

n

jf j

p

n

n

jz j jf j

p

e

Hence we obtain the following relationship for any p IR

z hf i

e S f S IR

p p

This implies that

z hf i

e S f SIR

z hf i

e S f S IR

z hf i

Observe that the unrenormalized exp onential function e is also in S if

p

p However it has no meaning when p

The S transform

Supp ose L Consider the function dened on L IR by

Z

x x y dy

S IR

The following translation formula for the Gaussian measure is wellknown



hxi jxj

dxd e x L IR

Therefore can b e rewritten as

Z



jxj hyxi

dy x y e

S IR

Z

hyxi

y e dy

S IR

HUIHSIUNG KUO

hxi

Now supp ose we restrict the variable x to S IR then e isinthespace

S of test functions Therefore we can use the natural pairing of S and

S to extend the ab oveintegral to S This is in fact the original idea

of Professor Hida to intro duce the space of generalized functions

Denition The S transform of a Hida distribution in S is dened

to b e the following function on S IR

h i

S hh e ii



j j h i

e hhe ii SIR

Observation The S transform can b e regarded as a function dep ending

on and Wehave the following three cases

Case For L its S transform is a function on L IR



jxj hxi

e x L IR Sxe

where is the inner pro duct of L

Case For S itsS transform is a function on S IR



j j h i

S e hhe ii SIR

hxi

Case Note that for x S IR the renormalization e is in S

Hence for S its S transform extends to b e a function on S IR

hxi

Sxhh e ii x S IR

The S transform is easily shown to b e injective It is a very useful to ol in

white noise analysis For instance we can pro duce a lot of Hida distributions

and test functions according to the following characterization theorems

Theorem PotthoStreit L et S Then its S transform

F S satises the fol lowing conditions

 For any SIR the function F IR extends to an entire

function F z z

ANALYSIS OF WHITE NOISE FUNCTIONALS

 There exist positive constants a K and p such that

ajz j j j

p

z SIR jF zjKe

Conversely if F is a function on S IR satisfying the above conditions then

there exists a unique in S such that S F

Theorem KuoPotthoStreit Let S Then its S transform

F S satises the fol lowing conditions

 For any SIR the function F IR extends to an entire

function F z z

 For any p there exists K such that

jz j j j

p

z SIR jF zjKe

Conversely if F is a function on S IR satisfying the ab ove conditions then

there exists a unique in S such that S F

Examples of generalized functions

First weintro duce the renormalization of the tensor of distributions Let

n

x S IR Dene x inductively by

x

x x

n n n

b b

x x x n x Tr

where Tr S IR is dened by

Z

hTrfi f t t dt f SIR

IR

Fact

n

X

n

k

k nk n

b

k x Tr x

k

k

HUIHSIUNG KUO

The multiple Wiener integral can b e expressed in terms of the renormal

ization of the tensor of distributions

n

I f xh x f i

n n n

Example B th iS In general

t

n n

B t I

n

t

n n

h iS

t

The S transform is given by

n n

S B t t SI R

z hy i

Example Let y S IR and z Then the renormalization e

is in S with S transform given by

z hy i z hyi

S e e SIR

z B t

In particular when y wehave the Hida distribution e and

t

z B t zt

S e e SIR

h Example B t iS In general wehave

t

k

k k

iS B t h

t

Its S transform is given by

k k

SB t t SIR

Example white noise Gaussian functions Let y S IRac c

The renormalization



ahy i jj

c

N e

is a generalized function with S transform given by

ac



jj ahy i

c

exp hy i j j SIR S N e

c c

ANALYSIS OF WHITE NOISE FUNCTIONALS

For the next example we need a basic prop erty of the test functions

e

Fact Every S has a unique continuous version

Example Let denote the delta measure at y S IR It is easy to

y

see that the linear map

Z

e e

x d x y S

y

S IR

is continuous Therefore it induces a Hida distribution also denoted by

y

Its S transform is given by



hyi j j

S e SIR

y

In view of Example wehave

 

hy i jj

c c

N e in S

y

as c

Example Donskers delta function B t a can b e written as

B t a hx ia

t

X

B t a

a

t

p p p

e h h

n n

n

t t

t

n

Its S transform is given by

Z

t

p

SIR s ds a SB t a exp

t

t

Dierential op erators and their adjoints

A function is said to have Gateaux derivative at x in the direction y if

the following limit exists

x y x

D x lim

y

In order to get more information ab out the Gateaux derivative D we should

y

regard D as a bilinear form on y and y

HUIHSIUNG KUO

Lemma Let S be given by

n

xh x fi

Then D exists for any y S IR and

y

n

D xnh x hy f ii

y

where hy f i is the natural pairing of y with one variable of f

Pro of Wehave the following binomial expansion

n

X

n

n nk k k

b

x y x y

k

k

By using this expansion we get immediately that

n

b

D xnh x y f i

y

n

nh x hy f ii

From the ab ove lemma we see that if is a p olynomial in S given by

X

n

h x f i x

n

n

then wehave

X

n

D nh x hy f ii x

y n

n

From this expression we get the following estimate for the norm kD k For

y p

the pro of see HKPS

IR and any p q IR Theorem For any polynomial S y S

we have

p

q p pq pN

kD k jy j k N k

y p q pq

where N denotes the number operator

Remark The numb er op erator is dened as follows

X

n

h x f i

n n

ANALYSIS OF WHITE NOISE FUNCTIONALS

X

n

N nh x f i

n

n

We now proveseveral corollaries ab out the op erator D Note that the

y

set of all p olynomials in S isdenseinS in L and in S with resp ect

to their top ologies

Corollary For any y S IR the mapping D extends to be a contin

y

uous linear operator from S into itself

Pro of Let p b e given Cho ose q p such that y S IR Note

q

that

p p

pq pn q pn

n n

p



n

n

n

n

Hence by the ab ove theorem weget

q p

kD k jy j kk

y p q pq

q p

jy j kk

q q

This implies that D can b e extended to S and that the extension is contin

y

uous from S into itself

Notation In particular when y we use the notation D It

t t

t

is called the white noise dierentiation Hence for each t IR the op erator

is continuousfromS into itself Note that for S represented by

t

X

n

h x f i x

n

n

wehave

X

n

x nh x f t i

t n

n

Corollary Let S Then the function y D is continuous from

y

ong topology into S In particular the function t is S IR with str

t

continuous from IR into S

HUIHSIUNG KUO

p

Pro of First weshowthat k Nk kk

p p

p

X

p n

k Nk nnjA f j

n

p

X

n p n

nnjA A f j

n

X

n p n

nn jA f j

n

X

p n

njA f j

n

kk

p

Now from the ab ove theorem the following holds for any p ositive p and q

p

q p pq pN

k k N k jy j kD

y p pq q

p

q p

k Nk jy j

pq q

q p

kk jy j

pq q

This gives the conclusion immediately

Corollary Let S IR Then the mapping D extends to a strongly

continuous linear operator from S into itself

Pro of We need to show that for any q there exists p suchthat

for all p olynomial in S

kD k C kk

p q

where the constant C dep ends only on p q and From the ab ove theorem

for any p q

p

pq ppq N

kD k j j k N k

p pq

q

Now for any q cho ose p q Then as b efore wehave

p

ppq n

n

Therefore

pq

kD k j j kk

p q q

ANALYSIS OF WHITE NOISE FUNCTIONALS

Finally we discuss the adjoint op erators Let T be a continuous linear

op erator from S into itself Its adjoint operator T is dened by

hhT ii hhTii S

The adjoint op erator T is continuous from S into itself

For y S IR we know that the op erator D is continuous from S

y

into itself Therefore its adjoint D is continuous from S into itself In

y

particular for any t IR the op erator is continuous from S into itself

t

Moreover for any S the function t is continuous from IRinto

t

S with strong top ology

Fact For any S

tS SIR S

t

Pro of Note that wehave

X

h i n n

e h i

n

n

By using we can check easily that

h i h i

e t e

t

Therefore

h i

S hh e ii

t t

h i

hh e ii

t

h i

thh e ii

tS

HUIHSIUNG KUO

HitsudaSkorokho d integral

For convenience we will consider integrals over the unit interval in

R

this section Let B t b e a Brownian motion The Itointegral f t dB t

is dened for anystochastic pro cess f t satisfying the following conditions

f is nonanticipating ie for each t f t is measurable with resp ect to

fB s s tg

R

jf t j dt for almost all

There ha vebeenseveral attempts to extend the Itointegral to anticipating

integrands In particular Hitsuda and Skorokho d used the

WienerIto decomp osition to dene integrals for anticipating integrands It

turns out that their integrals can b e expressed in terms of the adjoint op erator

in the white noise analysis

t

Theorem Kub oTakenaka If f L S IR is nonantic

ipating then

Z Z

f t dB t f t dt

t

where B th i

t

R

Note that the integral f t dt do es not require the nonanticipating

t

condition for f This integral called HitsudaSkorokhod integral can be de

ned for anticipating integrands We will dene this integral b elow and sketch

a pro of for the ab ove theorem

In fact we can give a more general denition for S valued measurable

functions In order to motivate the denition let us consider a measurable

function from into S Supp ose that S L for each

SIR and

Z

S t dt

is the S transform of some unique Hida distribution Then naturally wecan

dene

Z Z

t dt S S t dt

ANALYSIS OF WHITE NOISE FUNCTIONALS

Consequentlyforany SIR

Z Z

h i h i

hh t dt e ii hht e ii dt

From this equation we exp ect that the following holds for all S

Z Z

hh t dt ii hhtii dt

Thus we need to assume that for each S the function hh ii is in

L But this is exactly the Pettis integral of

Theorem Pettis integral HKPS Suppose is a measurable map

from into S such that hhii L for al l S Then

R

there exists a unique element denoted by t dt in S such that

Z Z

hh t dt ii hhtii dt S

Remark Supp ose the Pettis integral of exists Then wehave

Z Z

S t dt S t dt SIR

Theorem Bo chner integral HKPS Let be a measurable map

for t S from into S Suppose there exists p such that

p

R R

almost al l t and ktk dt Then the integral t dt exists and

p

Z Z

ktk dt t dtk k

p p

Remark Obviously the existence of Bo chner integral implies the exis

tence of Pettis integral

The last theorem is not very practical since it is often not easy to compute

the S norm Instead we should use the S transform The next theorem

p

in terms of S transform is actually equivalent to the last b eing expressed

theorem

Theorem Suppose there exist K L a and p such

that

aj j

p

jS t jK te SIR

HUIHSIUNG KUO

R

Then the integral t dt exists and

Z Z

hh t dt ii hhtii dt S

For the examples b elow we need the following facts ab out the

S transform

Fact Supp ose S is represented by

X

n

x h x f i

n

n

Then its S transform is given by

X

n

S hf i

n

n

Fact Dene the Hermite p olynomials with parameter as follows

n

d x x

n n

x e e

n

dx

Then wehave

n n n

h x f i hx f i

jf j

R

Example B t dt B

This follows easily from the fact that S B t tand

Z

t dt SB S h i

R

B t dt B B Example

Simply recall that S B t t

R R

n

B t B t dt dt B Example

n n

Note that

Z Z

S B t B t dt dt

n n

Z Z

t t dt dt

n n

ANALYSIS OF WHITE NOISE FUNCTIONALS

and

n

n

SB S h i

n n

h i

Z Z

t t dt dt

n n

R

Example B tB t dt B

Remark In this example weinterpret B tB tas B t in view of the

t

Kub oTakenaka theorem

R

Let B tB t dt Then its S transform is given by

Z Z

t

s dsdt t S

Z Z

s t dsdt

On the other hand note that

B h i

i h

h i

i h

Therefore we get

S B h i

Z Z

s t dsdt

Example We show that

Z

n n

B t dt B

t

t

n

Here the Hermite p olynomials with parameter are dened as ab ove For

instance wehave

B t B t t B t B t tB t

t t

HUIHSIUNG KUO

n

n n n

Now note that B t hx i h x i Therefore

t t t

t

Z

n

B t dt S

t

t

Z

n

n

th i dt

t

Z Z Z

t t

t u u du du dt

n n

Z Z Z Z

t u u

n

n t u u du du dt

n n

Z Z

v

n v v dv dv

n n

Z Z

n

v v dv dv

n n

n

Z

n

t dt

n

n

n n

On the other hand note that B h x i Hence

n

n n

S B h i

Z

n

t dt

Remark Wenow use Itos formula to showthat

Z

n n

dB t B B t

t

n

In view of Eqs and it is reasonable to exp ect the relationship in

the ab oveKuboTakenaka theorem Let

n

f t xx

t

Wehave the following derivatives

n

f t xn x

t

x

n

f t xnn x

t

x

nn

n

f t x x

t

t

ANALYSIS OF WHITE NOISE FUNCTIONALS

The last equality can b e checked from the generating function

n

X

t



tx t n

e x

n

n

Hence by Itos formula weget

df t B t f t B t dt f t B t dB t f t B t dt

t x x

f t B t dB t

x

Therefore

Z

f B f B f t B t dB t

x

ie

Z

n n

B t dB t B n

t

From the ab ove examples we see that in general the HitsudaSkorokho d

R

integral t dt is a Hida distribution But often weneedtoknow whether

t

it is a random variable ie in the space L For r dene

r

kk k N k

r

r

D completion of p olynomials in S w r t k k

r

Fact For f L IR the op erator D and its adjoint D are continuous

f

f

from D into L

Theorem Suppose L D Then the HitsudaSkorokhodinte

R

gral t dt is in L and

t

Z Z Z Z

ktk dt s t dsdt k t dtk

t s

t

where denotes the inner product of L Moreover

Z Z Z

p

k s t dsdt Ntk dt

t s

HUIHSIUNG KUO

Finallywesketch a pro of for Kub oTakenaka theorem ie if L

S IR is nonanticipating then

Z Z

t dt t dB t

t

It is sucient to prove the equalityfor of the form

t t t t

  

where t t t S is F measurable and is F

t t

 

measurable Being F measurable has the expansion

t



X

n

x h x f i

n

n

n

n

b

This implies that where f S IR has supp ort in t

n

D

t t 

 

Similarlywehave

D

t t 



Therefore from the denition of Itos integral weget

Z

t dB tB t B t B t B t

h i h i

t t t t

  

D D D D

t t  t t 

t t  t t 

  

  

D D

t t  t t 

  

On the other hand for the HitsudaSkorokho d integral wehave

Z Z

t dt t t dt

t t t t

  

t t

Z

t t dt

t t t t

  

t t

D D

 t t  t t

  

ANALYSIS OF WHITE NOISE FUNCTIONALS

This shows that the Itointegral and the HitsudaSkorokho d integral coincide

Sto chastic dierential equations

A natural generalization of sto chastic dierential equations in the white

noise setup is the following equation of the HitsudaSkorokho d typ e

Z Z

t t

f s X ds t g s X ds X x

s s t

s

However since the op erator is continuous from S into itself we need

t

only to consider the equation

Z

t

X x f s X ds t

t s

We need to sp ecify the space where we can get a unique solution and to nd

conditions on f such that this equation has a unique weak solution

Notation Let W denote the set of all functions X from into S

satisfying the conditions

p such that X S for almost all t and kX k L

t p

p

The function t X is weakly continuous ie for any S the

t

ii is continuous function t hhX

t

Theorem Suppose f is a functions from S into S satisfying

the fol lowing conditions

 For any X W the function t f t X is also in W

t

 For any S there exist nonnegative numbers a c p and a nonneg

ative continuous function K t z such that

jSf t zjK t z expajz j j j t z SIR

p

Z

K t z dt c expajz j j j z SIR

p

 There exist nonnegative numbers b q and a nonnegative continuous func

tion Lt z such that for any S

jSf t z Sf t zjLt z jS z S zj

z SIR

HUIHSIUNG KUO

Z

Lt z dt b jz j j j z SIR

q

Then for any S the equation

Z

t

X f s X ds t

t s

has a unique weak solution X in W

The idea of the pro of is very simple just use the metho d of iteration

together with the S transform We give some examples to show how the S

transform can be used to obtain solutions even for sto chastic dierentiation

equations of the Itotyp e

R

t

X dB s Example Solve the SDE X x

s t

By the Kub oTakenaka theorem this equation is equivalentto

Z

t

X x X ds

t s

s

Tosolve this equation let F tSX Then for each SIR

t

Z

t

F tx sF s ds

Therefore F t satises the dierential equation F t tF t with F

x The solution of this rst order dierential equation is given by

R

t

s ds

h i

t



F txe xe

Recall that for f L IR wehave

hf i hfi

S e e

Hence we get the solution

h i

t

X x e

t



h i t

t

xe



B t t

xe

ANALYSIS OF WHITE NOISE FUNCTIONALS

R

t

Example Solve the SDE X x X dB s

t s

Again this sto chastic dierential equation is equivalentto

Z

t

X ds X x

s t

s

Let F tbetheS transform of X Then F t satises the dierential equation

t

F t tF t with F x It is easy to get the solution

Z

R R

t

t t

v dv s ds

s 

ds F txe se

As in the previous example wehave

Z

t

 

B t t B tB s ts

X xe e dB s

t

R

t

Example Solve the equation X x X ds

t s

s

The S transform F tofX satises the equation

t

Z

t

F tx s F s ds

The solution is easily found to b e

R

t

s ds



F txe

Now we use the following

R

t



B s ds

c



Fact The renormalization N e is a Hida distribution with

the S transform

R R

t t

 

s ds B s ds

c

c

 

e S N e

In the ab ove case wehave c Therefore the solution X is given by

t

R

t



B s ds





X N e t

HUIHSIUNG KUO

Positive generalized functions

The p ositive generalized functions have b een studied extensively byKon

dratiev Y J Lee Yokoi among others We will give a brief discussion in this

section

For the nite dimensional distribution theory wehave the following

k

Theorem Every positive generalized function in S IR is induced by

a measure

This theorem can be proved very easily by the Bo chner theorem which

is the nite dimensional version of Minlos theorem in x Actually Minlos

theorem is a generalization of Bo chner theorem to a nuclear space

e

Recall from xthatevery S has a unique continuous version

Denition A generalized function S is called positive if hhii

e

forall

Question Howtocharacterize the p ositive generalized functions

Supp ose is induced by a measure ie

Z

e

x d x S hhii

S IR

Then isobviously p ositive On the other hand supp ose is p ositive Let

us try to see howwe can pro duce such a measure from Minlos theorem

Denition The T transform of S is dened by

ih i

e ii SIR T hh

Remark This T transform was used when Professor Hida intro duced

the theory of white noise in Its relationship with the S transform is

given by



j j

S e T i



j j

S i T e

ANALYSIS OF WHITE NOISE FUNCTIONALS

Lemma If is positive then T is positive denite

Remark The converse is also true

Pro of For any z z and any SIR wehave

n n

X X

ih i

j

k

z z T z z hhe ii

j k j k j k

X

ih i

j

hh j z e j ii

j

Here we have used the fact that S is an algebra ie it is closed under

multiplication or S S

Lemma For any S the function T from S IR into is contin

uous

ih i ih i

Pro of It is sucient to show that e converges to e in S as

converges to in S IR Note that

Z

ith i ih i ih i ih i

e dt e e ih ie

Since the multiplication is continuous on S we need only to show that for

any p there exists q such that

Z

ith i

e dt

p

is bounded for all with b ounded j j But this is obvious from the

q

following equalityinx



z hf i jz j jf j

p

k e k e

p

Now supp ose is p ositive Consider its T transform

ih i

T hhe ii SIR

From the ab ove lemmas we see that T is continuous and p ositive denite

Therefore by Minlos theorem there exists a nite measure suchthat

Z

ihx i

e d x SIR T

S IR

HUIHSIUNG KUO

ie

Z

ih i ihx i

hhe ii e d x SIR

S IR

Question Do wehave the following equality

Z

e

hhii x d x S

S IR

e

where is the continuous version of

Remarks Wemake three remarks

e

The ab ove equality implies that L for all S Actually we

e

will showbelowthat L

We need to take the continuous version for Eq This is b ecause

P

I f is dened only ae Thus for instance the delta function

n n

at y S IR ie hh ii y is not welldened unless wetake the

y y

continuous version

In fact establishing Eq is the only nontrivial part in the character

ization of p ositive generalized functions

ih i

Let A b e the set of all nite linear com binations of the functions e

S IR It is a dense subspace of S By Eq wehave

Z

e

x d x A hhii

S IR

Now supp ose S Cho ose a sequence A such that in S

n n

Since the multiplication is continuous on S weget

j j in S

n

as n This implies that

Z

f g

j j d x lim

n m

nm

S IR

f f

Hence the sequence f g is CauchyinL Let denote the limit of in

n n

f f

L Cho ose a subsequence still denoted by such that converges to

n n

ANALYSIS OF WHITE NOISE FUNCTIONALS

f e

ae On the other hand weknow that converges to p ointwise Hence

n

e

ae and wehave

hhii lim hh ii

n

n

Z

f

lim x d x

n

n

S IR

Z

x d x

S IR

Z

e

x d x

S IR

Thus wehave shown that Eq holds for any in S This completes the

proofofthefollowing

Theorem KondratievYokoi A generalized function is positive if

and only if there exists a nite measure on S IR such that

Z

e

x d x S hhii

S IR

e

where denotes the continuous version of

Example The delta function at y S I R is p ositive Its T transform

y

is given by

ihyi

T e SIR

y

hy i

Example Let y S IR The renormalization e is p ositive

and



ihyi j j

T e SIR

In view of the characteristic functional of the standard Gaussian measure

and the ab ove example we see that is induced by the following measure

y the translation of by y

y

p

t Then is p ositive Example Let denote the measure

t t t

and

t

j j

T e SIR t

HUIHSIUNG KUO



jj

c

On the other hand let g denote the renormalization g N e Then

c c

c

j j

c

SIR T g e

c

It is easy to check that g is p ositive if and only if c or c In that

c

case g is induced by the measure

c cc

Example Donskers delta function B t a is p ositive To see this

note that from xwe get easily

Z

t

a a

ih i

t

t t

p

e exp e s ds T B t a j j

t

t

Obviouslytoprove the p ositivityof B t awe need only to showthat

Z

t

exp j j s ds

t

is induced by a nite measure on S IR Consider the measure given by

T

p

I K with K b eing the integral op erator where T is the transformation T

p

asso ciated with the kernel function It is easy to check directly that

t

t

Z

t

j j s ds T exp

t

Thus the Donsker delta function is p ositive In fact wehave

a

t

a

p

T B t a e

t

t

t

Fourier transform

The nite dimensional Fourier transform is dened by

Z

n

ihxy i

b

p

f y e f x dx

n

IR

There are several diculties in attempting to generalize this Fourier transform to innite dimensional spaces

ANALYSIS OF WHITE NOISE FUNCTIONALS

n

p

asn

The Leb esgue measure dx do es not exist when n

ihxy i

The quantity e may not b e welldened

Trick The three quantities ab ove have no meaning individually when

n But their pro duct is meaningful in white noise calculus In fact

we can rewrite Eq in a dimensionfree manner by using the standard

n

Gaussian measure on IR

n

Z



ihxy i jxj

b

f y e f x d x

n

n

IR

Z

ihxy i

e f x d x

y n

n

IR



jxj ihxy i ihxy i ihxy i

is the renormalization of e in y variable where e e

y

Obviously the white noise analogue of the ab ove Fourier transform for

mulation is

Z

ihxy i

b

y e x dx

y

S IR

However this can b e applied only for a small class of functions We rst use

b

this informal denition to nd for and then give the denition of

white noise Fourier transform

b

Example

Recall that from x wehave

n

X

i

ihxy i n n

e h y x i

y

n

n

But we also have the following identity

n

X

n

k

n nk

b

x k x Tr

k

k

ihxy i

Put this identityinto e and change the order of summation and then

y

take the exp ectation in xvariable to get

k

X

k

ihxy i k

E e h y Tr i

x y

k

k

k

HUIHSIUNG KUO

ie

k

X

k

k

b

y h y Tr i

k

k

k

In order to dene the Fourier transform of S we need to take the

S transform of Eq Note that

ihxy i ihx i

S e e

y

Hence up on taking the S transform of Eq informallyweget

Z

ihx i

b

S e x dx

S IR

ih i

i hhe i

ih i

Observe that e is in S for any S IR Thus we can use this for

mulation to dene the Fourier transform On the other hand recall that the

S transform of is given by



j j h i

S e hhe ii

Hence bytheentire extension wehave



j j ih i

hhe ii S i e

b

Therefore S can b e also expressed as follows



j j

b

S S i e

b

Denition The Fourier transform of S is dened to be the

generalized function with S transform given by

ih i

b

S hhe ii SIR

or equivalently



j j

b

SIR S S i e

ANALYSIS OF WHITE NOISE FUNCTIONALS

b

Example



j j

b

From the ab ove denition wehaveS e On the other hand



j j h i

hh e ii S e



j j

e

b

Hence weget This fact together with Eq gives a representation

of as follows

n

X

n

n

h y Tr i

n

n

n

b

Example

This is obvious b ecause



j j

b

S S i e

 

j j j j

e e

ihy i

Example b e

y

This can b e checked easily by computing the S transforms of b oth sides



jj

c

b

Then g g By convention g Example Let g N e

c c c

and g



j j

c

Recall that Sg e By comparing the S transforms wesee

c

b

easily that g g

c c

Note The Wick product of two generalized functions S is

dened by

S S S

Wehave the following identity

ab

g g g c

a b c

a b

In particular

g

HUIHSIUNG KUO

ie g and are inverse to each other with resp ect to the Wick pro duct

The next two theorems give other representations of the Fourier transform

F

Theorem F T S

Pro of For any SIR



j j

b b

T e S i



j j h i

e hhe ii

S

Hence TF S ie F T S

Theorem F iI

Note Here iI is the second quantization of iI ie

X X

n n n

iI h x f i i h x f i

n n

For the S transform wehave

i SIR S iI S

Pro of This is obvious from the denition of Fourier transform and

Eq

Corollary For any S and any p IR

b b

k g k kk k g k kk

p p p p

b

Remark In particular for p wehave k g k kk This reduces

to the Plancherel theorem in nite dimensional spaces

b

Pro of From the ab ove theorem and Eq weget g iI

b

But obviously kiI k kk for any p IR Hence k g k kk

p p p p

The other equalityfollows easily

ANALYSIS OF WHITE NOISE FUNCTIONALS

The translation of a generalized function by SIR is dened by

S S

Remark It is easy to check that if S then

Theorem The Fourier transform has the fol lowing properties

 For any SIR

b

D b ihi

b

ihi b D

ih i

b

b e

ih i

b

e b

 For any y S IR

b

D b iD

y y

b

In particular b i

t t

Theorem HidaKuoObata The Fourier transform is a continu

ous linear operator from S into itself

actually injective and onto This Remark The Fourier transform F is

prop erty is a trivial consequence of the fact that F I which can b e checked

easily from the denition of the Fourier transform

Theorem HidaKuoObata Suppose A is a continuous linear

operator from S into itself such that AD q A q multiplication by

ihi and Aq D A for al l SIR Then A cF for some constant

c

Finallywe consider the following

Question Is the Fourier transform the adjointofsomecontinuous linear

op erator from S into itself

HUIHSIUNG KUO

In nite dimensional distribution theory the Fourier transform F is rst

dened on S IR and then extended to S IR by the duality ie

hF F f i hF F f i f SIR

In other words the extension of F to S IR is simply the adjoint of itself

Note On the other hand if we identify L IR with its dual space as in the

Gelfand triple then F is the extension of the inverse Fourier transform

We now try to nd a linear op erator G from S into itself such that

F G ie for all S and S

b

hhii hh G ii

Take in particular Then by one of the ab ove examples ie b

x x

ihxi

e weget

ihxi

G xhh e ii

Other expressions for G

ihxi

G xhh e ii S ix

Note The S transform S is dened for S and SIR

hxi

But if S then Sxhh e ii is dened for x S IR We

can use the entire extension to dene Szx z

P P

n n n

G f i h x f i hx i

n n

n n

Note This can b e checked easily by using the following expansion

n

X

i

ihxi n n

e h x i

n

n

Theorem HidaKuoObata The mapping G is a continuous linear

operator from S into itself Moreover G F

Theorem HidaKuoObata The operator G satises the fol lowing

properties

 G q D G for al l SIR

ANALYSIS OF WHITE NOISE FUNCTIONALS

 G D q G for al l SIR

Conversely if A is a continuous linear operator from S into itself satisfying

the above conditions then A cG for some constant c

Laplacian op erators

There are four Laplacian op erators in white noise calculus ie the Gross

Laplacian the number op erator N the Levy Laplacian and the

G L

Volterra Laplacian

V

The Gross Laplacian

Let H B b e an abstract Wiener space Supp ose is a twice H dieren

tiable function on B such that is a trace class op erator of H Then the

Gross Laplacian of is dened by

G

x trace x

H

G

IR is an abstract Wiener space for any Recall that from x L IR S

p

p IR It turns Therefore we can dene for functions dened on S

p G

out that we can express in terms of the white noise dierentiation ie

G

Z

dt

G t

IR

Theorem The Gross Laplacian is a continuous linear operator from

G

S into itself

Corollary The adjoint of the Gross Laplacian is a continuous linear

G

operator from S into itself and

Z

dt

t

G

IR

The number op erator

In an abstract Wiener space the numb er op erator is dened by

xhx xi Nx G

HUIHSIUNG KUO

The numb er op erator can b e expressed in terms of the white noise dier

ential op erator and its adjoint as follows

Z

N dt

t

t

IR

Theorem The number operator N is continuous from S into itself It

is also continuous from S into itself Moreover if

X

n

x h x f i

n

n

then

X

n

N x nh x f i

n

n

The Levy Laplacian

First we mention the original idea of P Levy Supp ose F is a twice

dierentiable function on L The Levy Laplacian F of F is dened

L

by

n

X

F xe e F x lim

k k

L

n

n

k

where fe g is an orthonormal basis for L

n

One waytointerpret the Levy Laplacian is to regard F as the S transform

of a generalized function in S From the nonstandard analysis viewp oint

can b e expressed in terms of the white noise dierential op erator by

L

Z

dt

L t

R

Example Let f sB s ds f L Then

Z

f sB s t ds

t s

f tB t

Since B t weget f t Hence

t

t

dt dt

Z

f t dt

L

ANALYSIS OF WHITE NOISE FUNCTIONALS

Now we use the functional derivative to dene the Levy Laplacian

Denition A function F S IR is called an Lfunctional if its

second derivative F is of the form

Z Z

F F t t t dt g u v u v dudv

L

IR IR

b

where F L IR and g L IR

L lo c lo c

It can be checked easily that the decomp osition in Eq is unique

T

F of F The function F is called the Levy part of F The Levy Laplacian

L

L

on a nite interval T is dened by

Z

T

F F t dt

L

L

jT j

T

Theorem for al l L

L

Theorem KuoObataSaito Let fe g be an equal ly dense and

n

uniformly bounded orthonormal basis for L T Then for any Lfunctional

F we have

n

X

T

F e e F lim

k k

L

n

n

k

Denition A generalized function S is called an Lfunction if

T

F istheS transform of its S transform F S isanLfunctional and

L

T

some unique generalized function The Levy Laplacian of is dened

L

to b e

T T

S F

L L



jj

c

Example Consider the Gaussian white noise function g N e c

c

Its S transform F is given by



j j

c

F e

Wehave the following second functional derivative

Z

F F t t dt

c

IR

Z

F u v u v dudv

c

IR

HUIHSIUNG KUO

Therefore we get

T

F F

L

c

T

g g

c c

L

c

Thus g is an eigenfunction of the Levy Laplacian with eigenvalue

c

c

The Volterra Laplacian

R

The Gross Laplacian dt is not dened for generalized func

G t

IR

tions However it has an extension ie the Volterra Laplacian to certain

V

generalized functions

Denition A function F S IR is called a V functional if its second

derivative F is of the form

Z Z

u v u F f t t t dt F v dudv

V

IR IR

where f S IR and F isthekernel function of a trace class

V

op erator of L IR

It is easy to see that the decomp osition in Eq is unique The

function F is called the Volterra part of F The Volterra Laplacian F of

V V

F is dened by

F trace F

V

V

L IR

Denition A generalized function S is called a V function if its

transform F S isa V functional and trace F istheS transform S

V

L IR

of a unique generalized function The VolterraLaplacian of is dened

V

to b e

S F

V V

Example From the last example for the Gaussian white noise function



jj

c

g N e wehave

c

g g

c c

V

G

c

ANALYSIS OF WHITE NOISE FUNCTIONALS

Theorem The restriction of to S is the Gross Laplacian

V G

Finallywemention white noise kernel op erators asso ciated with general

ized kernel functions

j k

Theorem HidaObataSaito For any S IR the white

noise kernel operator

Z

ds ds dt s s t t dt

t t j jk j k k

 s s

k

j 

j k

IR

is a continuous linear operator from S into S

Remark The integral for isasymb olic expression The rigorous

jk

interpretation is that there exists a unique continuous linear op erator

jk

from S into S such that

hh ii h i S

jk

where is the function

s s t t hh ii

j k t t



k s s

 j

Theorem HidaObataSaito The operator is continuous

jk

j k

from S into itself if and only if SIR S IR

j k

Theorem HidaObataSaito If S IR SIR then

extends to a continuous linear operator from S into itself

jk

Tr is continuous from S into itself Example

G

To see this simply observe that Tr S IR and then apply Theorem

Example N Tr is continuous from S into itself and from S

into itself

Note that Tr SIR S IR Hence by Theorem Tr is continuous

from S into itself On the other hand Tr S IR S IR and so by Theorem

Tr extends to a continuous op erator from S into itself

Remark The Levy Laplacian and the Volterra Laplacian are not

L V

white noise kernel op erators

HUIHSIUNG KUO

References

A For a comprehensive list of references see the following b o ok

T Hida H H Kuo J Pottho and L Streit White Noise An innite dimensional

calculus Kluwer Academic Publishers

B For a brief intro duction to white noise calculus see the following survey pap er

H H Kuo LecturesonwhitenoiseanalysisSoochow J Math

C Other pap ers mentioned in the lectures are

I M Gelfand and N Ya Vilenkin GeneralizedFunctionsVol IV Academic Press New

York and London

L Gross AbstractWienerspaces in Pro c th Berkeley Symp Math Stat and Probab

Univ California Press Berkeley

T Hida Stationary Sto chastic Pro cesses Princeton University Press Princeton

T Hida H H Kuo and N Obata Transformationsforwhitenoisefunctionals J Funct

Anal

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noisecalculusNagoya Math J

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Verlag Berlin Heidelb erg New York

H H Kuo N Obata and K Saito LevyLaplacianofgeneralizedfunctionsonanuclearspace

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Department of Mathematics Louisiana State University Baton Rouge LA USA