2024年1月10日发(作者：)

Acta

Mathematica

Scientia,

2019,

39B(3):

874-914

/10.1007/sl0473-019-0315-2

©Wuhan

Institute

Physics

and

Mathematics,

Chinese

Academy

of

Sciences,

2019数学物理学报http:

act

〃SOME

RECENT

PROGRESS

ON

STOCHASTIC

HEAT

*EQUATIONSYaozhong

HU

(胡耀忠)Department

of

Mathematical

and Statistical

Sciences,

University

of

Alberta,

Edmonton,

T6G

2G1,

Canada

E-mail:

yaozhong@ualberta.

caAbstract

This

article

attempts

to

give

a

short

survey

of

recent

progress

on

a

class

of

elementary

stochastic

partial

differential

equations

(for

example,

stochastic

heat

equations)

driven

by

Gaussian

noise

of

various

covariance

structures.

The

focus

is

on

the

existence

and

uniqueness

of

the

classical

(square

integrable)

solution

(mild

solution,

weak

solution).

It

is

also

concerned

with

the

Feynman-Kac

formula

for

the

solution;

Feynman-Kac

formula

for

the

moments

of the

solution;

and

their

applications

to

the

asymptotic

moment

bounds

of

the

solution.

It

also

briefly

touches

the

exact

asymptotics

of

the

moments of

the

words

Gaussian

random

field;

Gaussian

noise;

stochastic

partial

differential

equation

(stochastic

heat

equation);

Feynman-Kac

formula

for

the

solution;

Feynman-

Kac

formula

for

the

moments

of

the

solution;

chaos

expansion;

hypercontrac­tivity;

moment

bounds;

Holder

continuity;

joint

Holder

continuity;

asymptotic

behaviour;

Trotter-Lie

formula;

Skorohod

integral2010

MR

Subject

Classification

60G15;

60G22;

60H05;

60H07;

60H10;

60H15;

28C20;35K15;

35R60Contents1

Introduction

8752

Heat

Equation

and

Brownian

Motion

8762.1

Semigroup

and

Duhammel

8762.2

Lie-Trotter

8772.3

Heat

kernel

and

Brownian

motion

................................................................................8802.4

Feynman-Kac

8812.5

Girsanov

8813

Stochastic

Integral:

L2

theory

8843.1

Gaussian

noise

and

.8843.2

Deterministic

8853.3

General

integrand

............................................................................................................888*

Received

October

14,

201&

revised

December

19,

2018.

Y.

Hu

is

supported

by

an

NSERC

grant

and

a

startup

fund

of

University

of

Alberta.g

Springer

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION8758914

Stochastic

Heat

Equation

with

Additive

Noise

5

Stochastic

Heat

Equation

with

Multiplicative

Noise

8925.1

Some

.8935.2

Covariance

8945.3

Existence

and

uniqueness

of

the

solution

....................................................................8975.4

Feynman-Kac

formulas for

the

moments

of

the

solution

...........................................8985.5

Feynman-Kac

formulas

for

the

solution

.......................................................................8995.6

Moment

bounds

...............................................................................................................9025.7

Discussion

of

the

proof

of

the

moment

9045.8

Joint

Holder

9096

Asymptotics

9111

IntroductionThere

are

large

amount

of

research

papers

on

stochastic

partial

differential

equations.

This

paper

attempts

to

give

a

short

survey

on

some

special

problems

on

a

class

of

simplest

stochastic

partial

differential

equations:

stochastic

heat

equations

driven

by

a

class

of

Gaussian

noises

(see

(5.1)

below

for

more

specifics):

dfU^t,

x)

=

x)

+

uW,

where

dt

denotes

the

partialderivative

with

respect

to

t,

△

denotes

the

Laplacian,

and

W

=况篇：為^

"

is

the

Gaussian

noise.

This

equation

is

one

of

the

simplest

stochastic

partial

differential

equations,

for

which

one

may

obtain

some

more

precise

properties

of

the

solution.

On

the

other

hand,

this

equation

has

its

own

importance

because

it

is

relevant

to

the

parabolic

Anderson

localization.

It

is

also

related

to

the

KPZ

(Mehran

Kardar,

Giorgio

Parisi,

and

Yi-Cheng

Zhang)

equation,

which

is

the

field

theory

of

many

surface

growth

models,

such

as

the

Eden

model,

ballistic

deposition,

and

the

SOS

model.

We

shall

concentrate

on

the

existence

and

uniqueness

of

the

solution

for

various

noise

covariance

structures;

Feynman-Kac

formula

for

the

moments

and

for

the

solution

itself;

asymptotics

of

the

solution

when

time

t

is large

and

so

forth.

There

are

certainly

many

other

exciting

areas

which

are

not

covered

in

this

survey.

Let us

mention,

for

example,

one

result

in

[2]

concerning

with

the

above

stochastic

heat

equation

when

the

spatial

dimension

d

=

1,

the

noise

W

is

space

time

white,

and

the

initial

condition

is u(0,

x)

=

6(z),

the

Dirac

delta

function.

To

state

the

result

in

that

mentioned

work,

let

us

introduce10glRMDenotewhere

p(t,

x)

=

—==e~^

.Ai(£)=丄

/

cos

(£t3

+

祝)dt;k（7（h、y）=o-t.m

=

a

_

:二环，

a

=

a(s)

=

s

-

log

V2nf;

Zoou(t)

Ai(x

+

t)

Ai(y

+

t)dt;”ooKt

=

2-”3戸/3,

C

=

{eie}^警

U{rr+

±i}x>0

.

g

Springer

876ACTA

MATHEMATICA

SCIENTIAVol.39

,

the

probability

distribution

of

u(T,

x)

can

be

represented

([2])

byFt(s)

=

P(F(D

+

鲁

W

s)

=

/

罟e-"

tlet

(/

—心“)厶2宙才g，

where

det

denotes

the

determinant

of

the

operator

on

the

Hilbert

space

oo).

Thereare

also

many

other

important

work

which

is

omitted

in

this

article

because of

the

page

limi-

t

of

the

results

in

this

article

are

known.

However,

we

present

them

in

a

style

different

than

in

the

literature.

For

example,

we

establish

the

Girsanov

formula

through

the Trot

ter­Lie

formula

which

gives

a

unified

proof for

both

Feynman-Kac

formula

(Formula

(2.23))

and

Girsanov

formula

(Theorem

2.7).For

the

upper

moment

bounds

of

the

solution,

we

present

three

different

approaches:

Chaos

expansion

with

hypercontractivity;

Burkholder-Gundy-Davis

inequality;

and

the

Feynman-Kac

formula

for

the

moments.

The

first

approach

(Chaos

expansion

with

hypercontractivity)

is

valid

only

for

multiplicative

noise

(as

it

is

the

main

concern

of

this

work),

but

the

noise

does

not

need

to

be

white

in

time.

This

approach

is

also

easy

to

carry

out.

The

second

approach

(Burkholder-Gundy-Davis

inequality)

works

for

more

general

nonlinear

stochastic

partial

differ­ential

equations

x)

=

|

x)--(y{u)W,

but

the

noise

W

needs

to

be

white

in

time.

Bothapproaches

cannot

yet

give

the

lower

bounds

for

all

moments.

The

t

hird

approach

(Feynman-

Kac

formula

for

the

moments)

can

be

used

to

give

both

the

upper

and

lower

bounds.

It

is

also

used

to

obtain

the

exact

asymptotics.

But

to

obtain

the

upper

bound

more

effort

is

needed

than

the

first

two

approaches

(see

Section

5.7,

Method

3).Because

it

is

a

survey

article,

we

emphasize

more

connections

between

various

concepts

and

so

on

at

the

expense

of

strict

mathematical

rigour

in

the

proofs.2

Heat

Equation

and

Brownian

Motion2.1

Semigroup

and

Duhammel

principleLet

4

be

a

linear

operator

from

a

separable

topological

linear space

H

(Hilbert

space,

Banach

space)

to

itself

[In

the

following,

when

we

say

an

opera

tor,

we

always

mean

a

linear

operator

from

a

Banach

space

to

itself].

Consider=加(◎

,

0

S

t

S

T,

"o

is

given,

(2.1)where

u

=

u(t)

:

[0,

T]

—

H

is

H-valued

function.

The

solution

can

be

formally

written

as

u(t)

=

etAu().

If

4

is

a

bounded

operator

from

a

separable

Banach space

to itself,

then

etA

is

given

asIf

A

is

an

unbounded

operator

on

some

Banach

space,

then

one

can

use

the

Hille-Yosida

theory

(through

the

resolvant)

to

construct

the

semigroup

Tt =

etA

associated

with

(generated

by)

the

operator

A

(see

[34,

69]).If

we

want

to

find

the

solution

to

the following

(nonhomogeneous)

equationdu—=Au

Gt,

uq

is

given

,dtg

Springer

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION877we

can

use

the

Duhammel

principle:u =

u(t)

=

TtUQ

十where

Tt

is

the

semigroup

generated

by

A.

This

equation

can

be

applied

to

the

case

when

Gt

=

G(t,

u).

This

means

that

we

want

to

solve

the following

equation:-77

=加

+

G(t,u),uq

is

given.(2.2)The

solution

is

given

byu(t)

=

Ttuo

+I

Ti_sG(s,u(s))ds,0(2.3)which

is

another

equation,

called

the

mild

form

of

equation

(2.2).

In

probability

theory,

this

is

the

motivation

to

introduce

the

concept

of

mild

solution.

It

is

interesting

to

emphasize

the

case

when

G(t,

u)

=

Vtu

is

linear

in

u.

This

means

we

would

like

to

solve

the

following

equation:—=Au

+

Vtu

,

uq

is

given.

di

(2.4)Equation

(2.3)

becomesu(t)Ttu（）+Ti_s%u(s)ds

.(2.5)Iterating

this

identity,

we

have

Dyson-Phillips

series

expansion

of

the

solution

([34,

55,

64]):u(t)

=

TtUQ

+

/

Tt-sVsu(s)ds

JoTtUo

+I

Tt-sVsTsuQds0Tt-sVsTs-.rVrU^drds,Tt-SnVsnTsn-Sn

一

1 •…几2

-

Si

(2.6)whereIt

=

{0

<

Si

<

•

•

•

<

sn

<

^}

,

ds

=

dsi

•…dsn

.This

motivates

the

chaos

expansion

method for

the

stochastic

heat

equation.2.2

Lie-Trotter

formulaIf

we

consider

the

evolution

equation(2.7)d?/,石=(4

+

B)u

(2.8)for

two

linear

operators,

then

the

solution

can

be

written

as

u(t)

=

TtUQ,

where

Tt

=When

A

and

B

are

unbounded

operators,

there

are

difficulties

to

define

eA+B

because

A

and

B

may

even

be

defined

on

different

domains

(see

for

example

[27]).

Even

when

A

and

B

are

bounded,

because

they

may

not

commute,

we

usually

haveeA+B

丰

eAeBThe

Lie

product

formula,

named

for

Sophus

Lie

(1875),

statesg

Springer

878ACTA

MATHEMATICA

SCIENTIAVol.39

em

2.1

Define

||4||

=

sup

d

x

d

matrices

(or

bounded

operators),

thenZe—>ooas

the

operator

norm

of

A.

If

A

and

B

are

twolkll

operator

norm

.ProofLetCk

=

e"+B)“

=

?

+

(A

+

B}/k

+

V

S

kn-⑺畫andooDk=eA/keB/k

=711=0kniUil712=0OO

1Ani十

Bn2AniBn21

+

(j4

+

jB)/fc

+

£-

£n=2

7ij

+ri2=nNotice

that for

the

operator

norm,

we

have

||AB||

<

||X||||B||

and

||A

+

B||

<

||t4||

+

||B||.

Thus,I©

-

Dk

<_n=2

(MH

+

||B||)"

n!

n=2

ni+n2=M厶;

—

y-

||4門|B|严]nn^.<1~

k2

—

n!_n=0十(同+冋)"

=Aell^ll

+

I|B||On

the

other

hand,

by

the

definition

of

Ck

and

Dg

we

haveIlCfcll

<

1

+

(Pll

+

B)/k

+

£

n=2

(皿豐『)"=e(A

+

B}/k*and

Pfcll

+

M

+

||B||)/fc

+<

eAkeBk

§

e(||A||

+

||B||)/fcMIQWI

严nn2Thus,k

||,+BeA/keB/k]

||

=

||C£

-

D钏

S

工

C^Ck

-

Dt)D汁，||J=ok-1J=o<2_pMII+I|B|Ik

—]k2J=o＜暂阿+阿）,which

converges

to

0

as /c

—>

00.□We

also

need

similar

formulas

for

the

time

dependent

matTices.

型

Springer

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION879Proposition

2・2

Let

4(t),

B{t)

: [0,

T]

—>

be

two

d

x

d

matrices

which

are

continuous

in

t

C

[0,

T].

Consider

the

equation=

(A(r)

+

s

<

r

<

T,

u(s)

=

v

.

drThen,

for

any

0

<

s

<

i

<

T,

we

have

u(t)

=

U(t,

s)v,

where(2.9)U(t,s)=

lim

TT

exp

仇too

丄丄

n

n

J

exp

n

n

J,

(2.10)and

where

the

convergence

is

in

operator

norm.

Here,

we

use

the

convention

thatn[]4，—

Am

•

=m(pay

attention

to

the

ordering

of

the

product

(since

the

operators

may

not

commute).Proof

We

give

a

sketch

of

the

proof.

Denote

力=s

+

打s).

We

consider

the

approxi­mation

of

the

equation

(2.9)

on

[s,

t] byf

如¢)

=

(4(切

+

£(切)un(r),

arIt

is

easy

to

see

that

un(t)

=

s)v

and

(2.11)o

_

t

$

'U(t,

s)

=

lim

Un(t、s)

=

lim

TT

exp

------(4

(切

+

£

(切)

n—»oo

n—>oo

.丄丄

ni=n—l

(2.12)LIf

Cn

is

an

d

x

d

matrix

so

that

Cn

=

0(右),then

using

(2.4)

and

(2.5)

(or

(2.6)),

we

havet

—

sU(t,

s)

=

lim

Un(t,

s)

=

lim

TT

exp------(4

(切

+

3

(切)+

G

•

n—+oo

n—*oo

丄丄n(2.13)

Recall

the

definition

of

adjoint

actionadA(B)

:=

AB

—

BA

.We

can

define

ad宾

recursively

and

then

we

can

also

define

/(adA)>From

[36,

p.205],

we

havelog(eAeB)

=

4

+

g(adA)(E)

+

O(B2)

=

A

+

B

+

O(A2)

+

O(B2),where

g(x)

=

This

identity

can

also

be

written

aseAeB

=

exp

{A

+

B

+

O(A2)

+

O(B2)}.We

apply

the

above

identity

to

4

=令4(切

and

B

=

^^£(切.Then,

we

obtaint

—

s

t

—

s

t

—

sexp

------A

(ti)

exp

------3

(切

=exp

------(A

(ti)

+

B

(^))

+

Cn

,n

」

L

71

J

[

nwhere

Cn

=

O(A2)

+

O(B2)

=

0(占).Inserting

this

into

(2.13),

we

havet

一

sU(t,

s)

=

lim

Un(t,

s)

=

lim

TT

exp

—(切

exp—~—E

(tj

•

n—^oc

n—+oo

丄丄Thus, this

proves

Proposition

2.2.

(2.14)□

880ACTA

MATHEMATICA

SCIENTIAVol.39

k

2.3

(1)

Equation

(2.11)

can

also

be

written

as1n―limn

expi=nnexpn(2.15)(2)

There

are

many

extensions

of

the

above

formula.

See

for

example

[26,

27,

56,

65-67].

We

cite

one

result

from

[66].

Let

B

be

a

complex

Banach

space

with

norm

||

-

||

and

let

L(B)

be

the

set of

all

bounded

linear

operators

from

B

to

itself

with

operator

norm

topology:

||t4||

=

sup ||Arr||.

Let

,

An

be

continuous

functions

from

[0,

T]

to

L(E).

ConsiderxEB=—力绑切⑴，t

S

(s,T],

u(s)

=

v

.

7=1(2.16)Then,

u(i)

=

U也

s)o,

where0

N

「t

_

S

-I51+…+打(肚)=lim

TI

TT

exp-------Aj

fs

+

-(t

-

s))

.

n—^oo

A

x

x

-1-

n

n

/(2.17)7=n—1j=l

1-

」Unbounded

operator

cases

are

also

discussed

in

the

above

mentioned

work.2・3

Heat

kernel

and

Brownian

motionWhen

A

is

the

Laplacian

operator

△=刀 冷(unbounded

operator

from

L2(Rd)

to

itself),

i=l

iwe

have

the

classical

heat

equation3

]—x)

=

-Au(i,

x),

t

> 0,

a:

W IK";9t

2

u(0,

X)=

Uq(x).d(2.18)To

solve

this

equation,

let

us

assume

that

uq

6

L2(Rd)

Pl

厶i(R")

and

let

us

introduce必=

I

u(t,x)ebX^dx

.丿RdThen,

the

equation

becomes舊必=一扌罔2弘

必(0：g)=必o(E), (2.19)whose

solution

is讹,0

=广晔％(£).Inverting

the

Fourier

transform

yieldsu(t,

x)

=

/

p©

—

y)u°(y)(ly、(2.20)where

Ptx)

is

the

inverse

Fourier

transform

of

e

,.

_

IgptPt{x)=We

also

write碑dg

=

(27Tt)r/2g_邸.⑵21)u(t,

x)

=

ptUQ^x)

=

•鱼

Springer

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION881Pt(£、y)

=

pt(x

—

y)

is

also

the

transition

probability

density

function

of

the

standard

Brownian

motion

(Bt,t

>

0)

on

some

probability

space

(Q.

77.

P)

(We

denote

by

B(

the

Brownian

motion

starting

at

0

and

by

the

Brownian

motion

starting

at

x):Ptu()(a?)

—

IE"o(Pf)

=

).2・4

Feynman-Kac

formulasLet

us

now

consider

the

following

equationQDenote

V(f)

the

multiplication

operator

by

V(i,

x):

V(t)f(x)

=

V(Z,

x)f(x)

and

tj

=

jt/n.

Using

formula

(2.15),

we

haveo

u(t)

=

lim

I

I

exp冷

GXP

U°n—*oo

2n7=n

—

1{(2.22)u(0,

x)

=

uo(z)

•丄丄=lim

exp±Aexp

-V

(fn_i)

-

-

-

exp

—A

exp

-V

(Zo)

"on—>oo2nn2nn=

nlimpt/nexp

…p“”exp

-V(io)如.Using

the

Markov

property

of

the

Brownian

motion,

we

haveu(t^

x)

=

lim

Ex

<

expn—»oo

'

y

(^n-l,民

1

)t…expnnexp卩(to,

Btn)tn"0(%)exp土

応

u0(Bt)j=iV(t

-

s,x

-Bs)ds

u0(x

+

Bt) >

.

(2.23)Uo J

JRemark

2・4

The

above

formula

(2.23)

is

called

the

Feynman-Kac

formula

for

the

solution

to

⑵22).2.5

Girsanov

formula=E

<

expFirst,

let

us

consider

the

following

initial

value

problem

for

linear

(time

independent

variable

coefficient)

first

order

hyperbolic

partial

differential

equation

(we

denote

u

=字=〈b(z),兽)，t

€

[s,

oo),

u(s,x)

=

h(x).

(2.24)Here,是

denotes

the

gradient.

It

is

easy

to

verify

that

the

solution

to

the

above

equation

is

given

byProposition

2・5

Fix

s

>

0.

Assume

that

6

:

solution

to

(2.24)

is

given

byu仏

z)

=

is

continuously

differentiable.

Thez)),

(2.25)where

7(t,

x)

solves

the

following

ordinary

differential

equationx)

=

b(t,*

rr)),

?(s,

x)

=x.

(2.26)g

Springer

882ACTA

MATHEMATICA

SCIENTIAVol.39

Differentiating

(2.26)

with

respect

to

x

and

denoting

Z(t,

x)

=

dxi是丁

(t，©

=(2.27)

/

l

obtaind

=

where0Z(s,©=/,and

I

is

the

d

x

d

identity

matrix.

This

means盒b)

(丁(t,z)))歼(Snthat1

(

dxjis

the

fundamental

solution

to

(2.27)

as

an

equation

for

the

other

hand,

differentiating

(2.26)

with

respect

to

t,

we

obtain~Y(t,x)=诰b(丁(£,z))Y(t,z),

y(s,x)=冷(t,叽=$

=

b(7(s,”))=

b(x), (2.28)where

Y(t,

x)=第沁、x).

Because

the

solution

to

(2.28)

can

be

represented

by

its

fundamental

solution

(the

solution

to

(2.27)),

we

see

that

Y(t,

x)

=

Z(t,

x)b(x).

Therefore,

we

obtain务(冷(g)迪).Now,

for

u

defined

by (2.25),

we

have(2.29)鲁u(t,z)=〈九(池,工)),冷=〈九(v(t,z)),(乔丁(t,z)3Thus,

u(t,x)defined

by

(2.25)

is

the

solution

to

(2.24)

because

u(s,

x)

=

/1(7(5,

x))

=

h(x).

□Proposition

2・6

Let

x)

be

defined

as

in

previous

proposition

(Proposition

2.5).

As­sume

that

X

is

a

(/-dimensional

standard

Gaussian

random

variable

and

£,6

>

0

are

small

(e

<

6).

Then

for

/

:

>

R

nice

(for

example

continuous

and

bounded),

we

haveE[fh(£,z

+

dX))]

=E

/(2：

+

dX)exp{

|〈b仗+

dX),oX〉(2.30)where

o

denotes

the

Wick product

(see

for

example

[36,

53]).Proof

From

the

definition

of

7,

we

have7(s,x

+

6y)

=

z

+

69

+

eb(x

+

Sy)

+

O(£2)=⑦

+

5u(y),

where

tr

:

—>

]Rd

is

given

byu(y)

=

y+|b仗+

5/)

+

0The

inverse

of

u

(the

solution

to

z

=

u(y))

isy

=

z-V{z),g

Springer

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION883where一拓（7

+

和）+

0

o=-76((3：

+

6z)

—

dV(z))

+

O

0=—+

6z)

+

0=—7&(x

+

dz)

+

00+

6z)卩(z)

+

O(2.31)[?/ is

a

function

of

z].

Thus,=

[一

+

6y)Vzy

+

OOr7zy

=(/

+

£bx

+

Sy))-1

=

I

-

ebx

+

6y)

+

O(€2)and

consequently,V2V(z)

=

—ebx

+

Sy)Vzy

+

O=—ebx

+

6y)

+

e2

{bx

+

Sy))2

+

OThe

Carleman

determinant

of

/

+

W

(see

for

example

[36,

Definition

6.11]

and

references

therein)

can

be

computed

as

follows:det2(/

+

W)

=

exp

Tr

log

{

/

—

sb'{x

+

8y)

+

e2

(bx

+

Sy))2+

e

Trbr{x

+

6y)

—

e2!¥0@

+

切))2)+0(£3)Thus,

by

the

change

of

variable

formula (see

for

example

[36,

Equations

(6.4.16)

and

(6.4.17)]),

we

haveE/(7(s,

x

+

5

X)

=

E/(x

+

5u(X))=E7•仗+

6X)|det2(/

+

V/)|exp<

-〈U(X),X)-a^(x)-||v(x)|2j>j=Ef(x

+

6X)

exp

{|〈b(z

+

6X),

X)

—edib^x

+

dX)

—

|b(x

+

SX)2

+

O=E+

dX)

exp

{I

(b(x

+

<5X),

oX〉

—丽卩(①

+

sx)2 +

og

Springer

884ACTA

MATHEMATICA

SCIENTIAVol.39

the

last

identity follows

from

[36,

Example

6.13].

This

proves

Proposition

2.6.

Now,

we

consider

the

following

partial

differential

equation□云x)

=

x)

+

b(t、x),dt

2

(2.32)

^(0,

x)

=

”o@),where

b

:

x is

assumed

(to

simplify

the

argument)

to

be

smooth

with

boundedderivatives

and

▽“(£

x)=(急"(t,

□?),•••,池u(t、x))T

is

the

gradient.

Denote

the

vector

field

6(t)

=

6(f,

a?)V.

Denote

e

=

t/n^(5

=虫

and

let

X

be

a

standard

d

dimensional

Gaussian

random

variable.

We

also

let

7^(5,

x)

be

the

solution

to£x(s,r)

=

b伙&张(s卫))，

7fc(0,a:)

=

by

Proposition

2.6,

the

Lie-Trotter

formula

(2.15)

yieldsu(t,

x)

=

lim

e盖△吕b(匸眇…e佥“e寻b(Puo(x)71—>OO=lim

e盖訥咛勻…詁紗(警)e島％0(了”一；i(£,a;))n—>oo=lim

€舟△長风也評1)…总金△羔％知IE"o(%_i(£,龙+

6X))=lim

(

n

}

)

...

uq(x

+

8X)71—>OO(a-

f

c-2

4-•

exp

<

-(6(

—,

a;

+

6XoX)

—

a;

+

6X)2

+

OI

0

n

2dz

n=

lim

E

uq{x

+

Bt)

exp工〈b(

k=0+

Bfct

),o(Bfct

一

))Tl

n2n

n

=E

uq(x

+

Bt)

exp[b(t

-

*+

Bs)

—㊁

/

— s, x

4-

|2dsTheorem

2.7

The

solution

to

(2.32)

can

be

represented

as"(£,©)=

IE

uq{x

H-

Bt)

exp[b(t

—

s^x

Bs)dBs

—

—

/

|b(t

—

s,

a;

+

£s)Fds}](2.33)where

B

is

a

¢/-dimensional

standard

Brownian

motion

starting

at

0.3

Stochastic

Integral:

L2

theory3・1

Gaussian

noise

and

CovarianceAssume

that

{W(r,?/),r

>

0

,y

e

}

is

a

centered

Gaussian

field

with

covariance

E

{W(r,

y)W(s,

z))

=

s)q(y,

z),

r,

s

>

0

,

y,zeRd

.

帝爲去W(r,y).

Thus,

we

have(3.1)We

shall

also

use

糸W(r,y),

dyW(r,y}切蔦如叭")，and

W{r,y)

:=

^dyW(r,y)=

E

[W(r,

y)W(s,

z)]

=

g(s,

r)q(y,

z),堑

Springer(3.2)

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION885(3.3)(3.4)E

[dyW(r,

y)dzW(s,

z)

=

g(s,

r)A(y

一

z),

E

[W(r,

9)W(s,

z)

=

?(s,

r)A(g

—

z),

where护

g2d7(S)

r)=航恥,r),

A(y-^)=

dyddzd^

z)

•

(3.5)In

the

above

expression,

we

assume

that

the

noise

in

spatial

variable

is

homogeneous.

We

shall

further

assume

thatA(z)

=

j

£一吹“((1£),

where

t

—

T.

(3.6)丿]Rd3・2

Deterministic

integrandFirst,

we

aim

at

defining

the

stochastic

integral

I(K)

:=

fRd

/(r,

y)W(dr,

dy)

for

a

deter­ministic

function

f.

This

is

a

(stochastic)

double

integral.

We

can

integrate

dy

first

then

dr,

or

dr

first then

dy.

So,

if

we

have

more

regularity

on

g,

then

we

can

impose

less

restriction

on

A

and

vice

versa.

To

find

a

wide

class

of

functions

such

that

the

above

stochastic

integral

/(/)

exists,

we

consider

the

following

approximation

of

the

noise•

1

/S+£

.

1(s,y)

=

-/

W(r,

y)dr

=

-

(%W(s

+

£,y)

-

dyW(s,y)),亿E

J

s

*Let

OSaVbVoobe

two

real

numbers

and

approximate

the

stochastic

integral

1(h)

byI占打：=[J

a[M「y)W£(dr,dy)

丿Rd1eW

y)

[dyW(r

+

£,

9)

—

dyW(r,

y)]

drdy.(3.7)Let

0

another

two

real

numbers.

We

are

going

to

compute

the

covariance

of

le(f')

and

4(/i)

:=

fRd

h(r,t/)]V£(dr,

dy).

We

have-g{r

+

£,

s)

+

p(r,

s)

/(r,

y)h(s、z)A@

一

z)drdsdydz

,where

T

=

[a,

b

x

[c,

d].

Denote/c(r,

s)=I

/(r,

y)h(s,

z)A(9

-

z)dydz.R2d(3.8)Then,已匕（门厶仇）］=g(r

+

£,

s

+

£)—

g(r

s

+

e)

—

g(z*

+

£,

s)

+

s)

fc(r,

s)drds.

(3.9)We

have

the

following

simple

integration

by

parts

type

formulas:I

g(r

+

£,

s

+

e)fc(r,

s)drds

T2/*b+£•d--e

g(u、v)k{u

—

£,

q

—

£)d"du

a+£

Jc+£r

fb

y»d+£I

g(u,u)/c("

—

—

£)dudo

+

I

I

g(^u,

v)k^u

—

£,v

—

e)dudva+£

J

dg

Springer

886ACTA

MATHEMATICA

SCIENTIAVol.39

(u,

v)k(u

—

g(u、v)k(u

—

—

e)d”do

+—

e'jdudv

—g(匕，v)k(u

—

£,

o

—

£)dudQg(u,

v)k(u

—

— e'jdudv(3.10)g(®

v)k(u

—

£,v

—

e)dudv

.In

a

similar

way

we

haveI

g(r

+

€,

s)fc(r,

s)drdsT2v)k(u

—

e,

o)dud°

+dg(u,

v)k^u

—

e,dg(u,

v)k(u

—

s,

v)dudvy

I

g(g

s

+

e)/c(G

s)drds*T2g(u,

v)k(u^

v

—

e)dudv

+g(u,

v)k(u,

v

一

e)dudv

.Combining

the

terms,

we

haveg(®

v)/c(u,

v

—

e)d"dog(r

+

£，s

+

£)—

g(厂,s

+

£)—

g(r

+

£,

s)

+

g(r,

s)

fc(r,

s)drds£

Jt2=

/1+/2+/3

+

/4

+

/5+/6,(3.11)where1g(u,

v)

[fc(u

—

—

s)

—

k(u,

v

—

£)—

k(u

—

£,

u)

+

fc(u,

v)]

dudv,g(u,

q)/c(u,

v

—

£)dudvh&11g(",

v)k(u

—

£,

o

—

£)d"df

—g

仏

v)k(u

—

£, u

—

e)dudo —g(",

v)k(u

—

£,

v)dudvg(u,

v)k(u

—

e,v

—

£)dudQg(u,

v)k(u

—

e,

v)dudv

—h41g(“,

o

—

£)dudo

—

/Ja'rb

a+£g(u,

v)k(u

—

£, q

—

£)dudog(u,

v)k(u

—

£,

q

—

e)dudv

.1g(u,

v)k(u

—

—

e}dudv

—First,

we

assume

that

k

:

T2

~R

is

twice

continuously

differentiable.

It

is

easy

to

see

that

when£

t

0,

we

haveAg

Springerf

d2kI

2

g(",o)^^(SQ)dudu,(3.12)

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION887h

=d+£v)

[k(^u

—

e.v

—

e)

—

k(u,

v

—

£)]

dudvg(u,

v)k{u

—

£, q

—

e)dudvfb

y

g(u,d)页(u,d)du

—g(a,d)E(a,d),泳(3.13)(3.14)hAZdbad

Bkg(b,

W)—(b,

u)du

-

g(b,

c)k(b,

c),g(a,

V)—(a,

u)du

+

g(a,

c)k(a,

c),(3.15)(3.16)(3.17)hg(”，

u,

c)do

+

g(a,

c)上(a,

c),h

-*

g(b,

d)k(b,

d)

-

g(a,

c)k(a,

c).Thus,g(r

+

£,

s

+

e)

—

g(r.

s

+

£)—

g(r

+

£,

s)

—

g(r,

s)

k(r,

s)drdst2

L

Bk仃畑)蘊("gd—

rb

I

g(s

d)苑(a,

d)du

一

g(a,

d)k(a,

d)

严

BkI

9(b,v)

—

(b,v)dv

-

g(b,

c)fc(&,

c)

+f

g(a,o)||(a,Q)do+广

QkI

g仏

c)

—(u,

c)dv

+

g(b,

d)fc(6,

d)

+

g(a,

c)k(a,

c).(3.18)With

some

simple

arrangement, we

can

write

the

above

identity

as

lim

g

g(r

+

£,

s

+

£)—

g(r,

s

+

£)—

g(r

+

e,

s)

—

g(r.

s)

k(r,

s)drds£->0=[g(a,

c)

一

g(b,

c)

一

g(a,

d)

+

g(b、d)]

fc(a,

c)

严

BkI

[g(a,

v)

-

g(a,

d)

-

g(b,

v)

+

g(b,

d)]

—

(a,

u)do

++广

dkI

[g(u,c)

-

g(u,d)

一

g(b,c)

+

g(b,d)]

—

(u,c)duf

d2k+

J

2

b(u，u)

—

9(u,

d)

-

g(b,

v)

+

g(b,

d)]

(tz,

v)

the

above

identity,

we

used

the

facts

that(3.19)rdI

g(a,d)

—

(a,

u)du

=

g(a,

d)

[fc(a,

d)

-

fc(a,

c)]and

some

other

similar

identities.

By

a

limiting

argument,

we

can

have

the

above

identity

for

general

k.

We

summarize

the

above

to

the

following

proposition

(see

[47]

for

a

more

thorough

discussion

in

the

case

when

f

=

h).Proposition

3.1

Define

kf^h

as

the

k

given

by

(3.8).

Ifd2kI

b(",

u)

一

b)

一

g(a,

v)

+

g(b,

6)|

(u,

v)

dudv

<

oodudvT2(3.20)g

Springer

888ACTA

MATHEMATICA

SCIENTIAVol.39

is

a

Cauchy

sequence

(as

s

—>

0)

in

L2.

In

this

case,

we

say

that

the

stochasticintegral

exists

and

we

define

/(/)

=

lim£^o

Moreover,

1(f)

is

a

Gaussian

variable

withmean

0

and

the

covariance

is

given

byE(Z(/)I(/i))

=

[g(a,c)-g(b,c)-g(a,d)+g(b,d)]/i：(a,c)

4-I

[g(a,

V)-

g(a,

d)

一

g(b,

u)

+

g(b,

d)]

—

(a,

v)dv

fd

dk+rb

3kI

[g(u,

c)

-

g(u,

d)

-

g(b,

c)

+

g(b,

d)]

—

(u,

c)du

f

d2k+

j

[g(u,u)-g(u,d)-g(b,u)

+

g(b,d)]^^(u,o)dudu.

(3.21)Remark

3・2

If

g

is

twice

continuously

differentiable,

then

we

can

write

the

corresponding

terms

in

the

above

(3.21)

asd^g(r,

s)drdsfc(a,

c)

+'T2

drds舄

Ss)drds

筈(s)d°+舄Ss)drds筈(以)血+厶Eg)心))=f(S

v)k{u,

v)dudv

.A

simple

application

of

Fubini

theorem

to

integrate

u

and

v

first

yields(3.22)The

above

identity

is

also

easy

to

be

obtained

directly

from

(3.9).

This

argument

can

be

used

backward

to

prove

(3.21)

by

assuming

that

g

is

twice

continuously

differentiable

combined

with

a

limiting

3.3

If

7(5,

r)

=

following

isometry

formula:r)

is

locally

integrable,

then

by

(3.22)

we

have

the/(r,

y)h(s,

2)7(r,

s)A(y

—

z)dydzdrds

.(3.23)[See

for

example

Equation

(2.1)

of

[41].] There

are

various

other

definitions

and

formulas

for

I(h).

See

[14,

42,

48-50]

and

references

3.4

If

c

=

0

and

d

=

t

and

if

g

=

h,

then,E

[1(h)2]

=

[g(0,0)

— g(s,0)

—

g(0,t)

+

g(s,t)]/:(0,0)rl

3k+

I

[g(0,

u)

-

g(0,

t)

-

g(s,

v)

+

g(s,

t)]

—

(0,

u)du++fs

dkI

[p(w,

o)

-

g(u,t)

-

g(s,0)

+g(s,t)]

—

(u,0)du洋k[g(“,o)

—

g(u,t)

一

g(s、v)

+g(s,t)]

-^^(u,v)dudv.

(3.24)If

the

limit

exists

when

t

—>

oo,

then

we

can

define

x]Rd

h(s,

y)W(ds,

dy).3.3

General

integrandAfter

we

define

the

stochastic

integral

for

deterministic

kernel

1(h)=人十

xRd

h(s,

9)

W(ds,

dy),

we

can

introduce

a

Hilbert

space

HI

with

scalar

product

as(g,恤=

E[Z(g)W)]-型

Springer(3.25)

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION889Another

way

to

introduce

this

Hilbert

space

IK

is

as

follows.

Denote

by

£

the

vector

space

of

all

step

functions

on

[0,

T]

x

On

this

vector

space

£,

we

introduce

the

following

(reproducing

kernel

Hilbert)

scalar

product

for

indicate

functions:〈l[o,t]x[o@],

1[o,s]x[o，p]ihi

=

g(t,s)q(=,y).〉In

the

above

formula,

if

<

0

we

assume

by

convention

that

l[o,叭]=—1

[一叭,()]•

This

scalar

product

can

be

extended

to

all

elements

in

£

by

(bi-)linearity.

]HI

is

the

completion

of

S

with

respect

to

the

above

scalar

recall

some

basics

on

Malliavin

calculus

(see

[36,

62]

for

more

det

ails).

The

set

of

the

smooth

and

cylindrical

random

variables

F

are

of

the

formwith

G

/

6

C^°(Rn)

(namely

f

and

all

its

partial

derivatives

have

polynomial

growth).

For

this

kind

of

random

variable,

the

derivative

operator

D

in

the

sense

of

Malliavin

calculus

is

the

H-valued

random

variable

defined

byDF

=

W箸(I如…,/(071

))0J

•The

operator

D

is

closable

from

L2(Q)

into

L2(Q;

1H)

and

then

it

can

be

extended

to

more

general

functionals.

We

define

the

Sobolev

space

D)1,2

as

the

closure

of

the

space

of

smooth

and

cylindrical

random

variables

under

the

norm||"||i,2

=

yiE[F2]+E[||DF||i].An

element

u

E

L2(Q,HI)

is called

Skorohod

integrable

(or

in

the

domain

of

the

divergence

operator

6)

if

there

is

a

random

variable

in

L2(Q),

denoted

by

J(u),

such

thatE

=

E

[{DF,讷诃，VFe

D1'2

.(3.26)6

is

t

hen

the

adjoint

of

the

derivative

operator

D.

It

is

called

the

Skorohod

integral of

u

and

we

also

denote

6(")=

人十

fRd

u(s,

y)W(ds,

dy).There

are

also

other

definitions

of

Skorohod

integral.

Now,

we

give

another

definition,

which

is

to

use

Wick

product

([30, 36,

53]).

If

u

=

Gh,

where

G

is

an

element

D1,2

and

h

is

(deterministic)

smooth

function

on

x

IR”

with compact

support.

Then,

we

defineGh(s,

0

W(ds,

dy)=Fo

1(h)

=

GT■仇)-(DG,

h).(3.27)Then,

we

approximate

a

general

element

u

e

L2(Q,

IHI)

by

linear

combination

of

elements

of

the

nform

Gh、namely,

u

—

lim

刀

Gkhg

and

then

define

the

stochastic

integral

人十

fRd

u(s,

“)

W(ds,dy)

by

a

limiting

argument.

This

definition

of

stochastic

integral

is

the

same

as

the

one

definedby

using

divergence

operator

5.

Using

the

identity

E(F/(/i))

=

E

[{DF,

/z)],

it

is

easy

to

see

that

for

any

F

€

D1,2,EGh(s,9)W(ds,dg)=

E[F

{G附)一

(DG,

h}}]

=

E

[FGI(h)

一

F〈DG,

h)]=E

[〈D(FG),

h) 一

F〈DG,

h)]

=

E

[{DF,

Gh}].g

Springer

890ACTA

MATHEMATICA

SCIENTIAVol.39

means

that

Gh

is

in

the

domain

of

8

and

6(Gh)=人十

J^d

Gh(s,

y)W{ds,

dy).

This

explains

that

the

stochastic

integral

defined

by

using

Wick

product

is

the

same

as

the

Skorohod

integral

defined

by

the

divergence

would

like

also

to

introduce

another

way

to

define

stochastic

integral.

This

is

through

the

chaos

any

integer

n

>

0,

we

denote

by

Hn

the

n-th

Wiener

chaos

of

W.

H。is

simply

IR

and

for

n

>

1,

Hn

is

the

closed

linear

subspace

of

L2(Q)

generated

by

the

random

variables

{Hn(!()),

0

E

H,

II^IIh

=

1},

where

Hn(x)

=

(—1)%号是鼻一号

is

the

n-th

Hermite

polyno­mial.

For

any

n

>

1,

we

denote

by

IHI®71

(resp.

IHIOn)

the

n-th

tensor

product

(resp.

the

n-th

symmetrie

tensor

product)

of

HI.

Then,

the

mapping

Zn(0

=

can

be

extended

toa

linear

isometry

between

JHIOn

(equipped

with

the

modified

norm

•

||凹0厲)and

Wiener-Ito

chaos

expansion

theorem

states

that

any

square

integrable

random

variable

F

has

the

following

representation:ooF

=

E[F]

+

^In(/n),

n=l(3.28)where

the series

converges

in

L2(Q),

and

the

elements

fn

€

IHI0n,

n

>

1,

are

determined

by

F.

If

u

e

L2(Q,

IHI),

then

we

can

also

writeoo"=IE

同

+

刀

厶i(如)，

n=l(3.29)where

E(u)

G

HI

and

un

:

]HOn

—>

HI.

We

can

identity

u

as

an

element

in

ft

e

HfEG+i).

Then,

we

define

the

Skorohod

integral

of

u asooU

=

I(E

[u])

+

工

Ai+i(Sym(亦))，

n=l(3.30)where

Sym(五)G

written

as

it

=

is

the

symmetrization

of

u

(see

[36]).

An

element

u

in IHIOn

can

be••-

,tn,x仇).Thus,

an

element

from

L2(Q,

IHI)

can

be

written

asoox)

=

UQ(t,x)

+

n=l…，绘，術)),

(3.31)where

(上1,龙1)厂・・,(七71,力71)：is

the

multiple

Wiener-Ito

integral

with

respect

toun(f,

rr;

Zi,,

••-

,tn,Xn)W(dti,

dxr)

•

•

•

W(dtn,£n).Definition

3.5

(Stochastic

integral)

If

u

is

given

by

(3.31),

then

we

can

definep

oo/

u(t,£)W(dt,dz)

=

f("o(t,e))

+

工人+i(Sym

仏)(如巧，•••，圮+1,龙卄1)),

丿

IR+xM

(3.32)n=lif

each

term

of

the

above

series

exists

and

the

series

is

convergent,

whereSym(iZTi)

(t],

=

,

绘+i，忑仇+])^n)-72十丄.,

2=11

仇+i|

-I〉1

"仇(垸,%,

t],

37]，•…,^n+l,

^n+1

)'

'

'

,

(3.33)g

Springer

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION8914

Stochastic

Heat

Equation

with

Additive

NoiseLet

W

be

the

Gaussian

process

introduced

at

the

beginning of

Section

3.

In

this

section,

we

consider

the

following

stochastic

heat

equation:d

1

A

T-<

dlU=2^U

+

W(4.1)"(0,

X)=

Uq{x).The

Duhamel

principle

says

that

if

the

above

equation

has

a

solution,

thenu(t,

x)=I

Pt(x

-

y)u0(y)dy+Pt—s(£

-

9)W(s,9)dsd"I

Pt(z

—

“)uo(")dg

+Pt_s(£

-

y)W(ds,dy),(4.2):p

If

the

right

hand

side

of

(4.2)

exists

in

L2,

then

we

call

itwhere

pt

(x)

=

(2Tvt)~d^2

ex,the

mild

solution

to

(4.1).

Because

the

first

term

in

(4.2)

can

be

handled

by

classical

analysis,

we

concentrate

on

the

second

term.

Thus,

we

may

assume

uq

=

0.

Applying Proposition

3.1

with

Zi(s,y)

=

pt-s{x

—

y)

(for

fixed

t

and

龙)，we

haveE(u(t,

x)2)

=

[g(0,0)

-

g(t,

0)

-

g(0,

t)

+

g(t,

t)]/

b(o,

y)h(o,

z)A(“

-

z)dydzR2d+[h(0,

y)讐(v,

z)

A(y

-

z)dydzdv

I

[g(0,

v)

一

g(0,

t)

一

g(t,

v)

+

g(t,

t)]

]R2d

也o[

^h(u,

y)h(0,

-

z)dydzdu+

/

[g(u,

0)

-

g(u,

t)

-

g(t,

0)

+

g(t,

t)]

OUJo

.+

/

[g(“,

u)

—

g(%

t)

-

g(t,

u)

+

g(t,

t)]Jt2-[霎h(u,y)A(y

—z)dydzdudu.(4.3)丿展2d

OU

OVThe

most

singular

term

in

the

above

expression

is

the

above

last

term

and

we

use

Fourier

trans-

formation

to

compute

it.

The

Fourier

transform

of

/i(r,

y)

= pt_r(y)

=

(2开(七—r))~d^2e~

2(«-^

isW，g)

=

e-T-Thus,

the

last

term

in

(4.3)

is[

譽h(u、y)讐h(u,

y)人(y

-

z^dydzdudv

I

[g(s

Q)—

g(®

t)

—

g(t,

v)

+

g也

t)]

爬2d

OU

OVT2(4.4)=/

[g(u,

v)

—

g(u,

t)

一

g也

v)

+

g(t,

t)][e-g严

罔纭(dE)2If

we

introduce/

f

(2t-u-u)|g||

2p(C

=

/

e

1

“(d§),2丿Rdthen

(4.4)

is/

[g(u,

v)

—

g(u,

t)

—

g(t,

v)

+

g也

t)]

p(2t

—

u

—

v)dudv

<

oo

.Jt2Assume

that

the

covariance

function

g

satisfies(4.5)|g(",o)

一

g(u,t)

—

g(t,u)

+g(t,t)|

<

Ct-uNv^(4.6)g

Springer

892ACTA

MATHEMATICA

SCIENTIAVol.39

some

/3

>

0,

then|g(u,

v)

一

g(u,

t)

一

g(t,

v)

+

g(t,

t)|

[

Jo

JRde(2t

u—v)|g|2|E|4“(dE)d"d°t-u0

[

e-g起罔4“(d°dudu叨□=C

[

t

—

u^+1

f

JOJRd=C

[

[

|t_iz|0+i€_d

叨日

|gf“(dg)du

丿Rd

JOt|g|2

<

C

/

/

匕一丫來严妙瓜迫血.丿]Rd

JoThis

can

be

used

to

prove

easily

the

following

ition

4.1

If

g

satisfies

(4.6)

and

if

fRd

1+^2/3

/1(¾)

<

oo, then

the

mild

solution

to

(4.1)

exist

in

refer

to

[47]

for

more

e

4.2

If

曉(仏厂)|

<

Cu

—

r|_Q,

then—

g(®t)

—

g(t,°)

+g(t,t)|

-L<

C薯(")〃

+I

r

—

v~adr

4-[t

-

r~adruu

.辻

u

<

vI

r

-

u~adr

+

(i

-

u)^q+1

<

(i

-

u)~a+1u[|r_u|Ydr

+

(t

—iz)-a+i

S

V

+】—

if

v

<

means

thatiffollowing

statement:

If

|

辭吋)|“u

—

r|_Q,

then

(3 =

1

~

a.

Our

proposition

implies

then

ther)|

<

Cu~r~a,

and

if

fRd“(d£)

<

oo,

then

the

mildsolution

to

(4.1)

exist

in

g

is

the

covariance

function

of

the

fractional

Brownian

motion

of

Hurst

parameter

H、

then

0

=

2H.

Thus,

the

condition

becomes厂萌訥d£)<

is

for

any

Hurst

parameter

H

G

(0,1)

regardless

>

1/2

or

<

1/

4.3

The

condition

in

the

proposition

is

also

necessary

(see

for

example

[47]).

It

seems

that

the

first

efforts

to

give

an

optimal

condition

for

the

linear

equation

(4.1)

driven

by

a

Brownian

motion

W

in

time

is

in

the

articles

[29,

63].

We

refer

to

[47]

for

more

discussion.5

Stochastic

Heat

Equation

with

Multiplicative

NoiseIn

this

section,

we

are

going

to

consider

the

stochastic

heat

equation

with

multiplicativenoise:罟=^Au

+

uW

,t

>0,

2：

e

Rd；(5.1)w(o,x)

=

U0(x),

g

Springer

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION893where

VT

is

a

Gaussian

random

field

introduced

in

(3.1).

This

equation is

a

continuous

version

of

the

parabolic

Anderson

model

(see

[10,

31,

32,

59]

and

references

therein

for

the

discrete

version

of

Anderson

model).

It

is

also

related

to

other

systems

in

random

environment

such

as

the

KPZ

equation

[6,

33]

or

polymers

[1,

7].5・1

Some

generalityThere

are

only

two

independent

ingredients

in

equation

(5.1):

the

initial

condition

and

the

noise.

In

this

survey,

we

assume

that

the

initial

condition

uq

is

as

regular

as

one

wishes

to

assume

and

we

focus

on

different

assumptions

of

the

structure

of

the

noise.

However,

we

would

like

also

to

point

out

that

recently,

there

are

studies

to

allow

uq

to

be

more

general,

in

particular,

one

can

allow

the

initial

condition

to

be

a

general

measure

including

the

Dirac

delta

function

(see

for

example

[3,

11.

12,

16-19,

28]

and

references

therein).Definition

5.1

(mild

solution)

Let

u

=

{u(t,x),

0

<

t

<

T,x

E

Rd}

be

a

real-valued

predictable stochastic

process,

such

that

for

all

t

G

[0,

T]

and

x

G

展〃，the

process

{pt_s(x

—

9)”(s,

0

<

s

<

t,y

e

is

Skorohod

integrable,

where

pt^x)

is

the

heat

kernel

onthe

real

line

related

to

|A.

We

say

that

u

is

a

mild

solution

of

(5.1)

if

for

all

t

G

[0,

T]

and

x

e帆

we

haveu(t,

x)=Pt

*

uo(z)

+Pt-s(H

一

y)u(s,

y)W(ds,

dy)a.s.,(5.2)where

the

stochastic

integral

is

understood

in

the

sense

of

Skorohod

or

It6

and

pt

*

“o(z)=

J^d

pt(x

—

9)uo(9)d"

denotes

the

convolution

in

spatial

tion

5・2

(weak

solution)

We

say

that

u

is

a

weak

solution

to

(5.1)

if

for

all

t

E

[0,

T]

and

any

0

€

D(IRd,

R)

(the

set

of

all

smooth

functions

from

to

R

with

compact

support),

we

havef

u(t,x)(/)(x)dx

=

I UQ(x)(f){x)dx

-|--Rd

2u(s,

a;)A0(x)dxds+

/

/

u(s,

z)0(z)W(ds,

dz)

a.s.,Joo

丿Rdwhere

the

stochastic

integral

is

understood

in

the

sense

of

Skorohod

or

Ito.(5.3)Remark

5.3

In

this

survey,

we

always

use

Skorohod

integral.

Equation

(5.1)

is

also

called

the

Skorohod

type

equation.

Let

us

also

mention

that

in

equations

(5.2)

or

(5.3)

if

we

replace

the

stochastic

integral

by

Stratonovich

one.

then

we

say

we

solves

the

Stratonovich

type

equation

(5.1).In

equation

(5.2),

we

can

replace

t

by

s,

z

by

9

and

obtain

an

expression

for

u(s,®).

Substituting

this

expression

into

(5.2)

yieldsu(t.x)

=

pt

*

"o(7)+Pt-s(H

-

y)Ps

*

uo(y)W(ds,

dy)+

///

pt-s(x

-

y)ps_ru(r,

z)W{dr,

dz)iy(ds,dy).

JO

JoContinuing

this

way,

we

have

the

following

solution

candidate

for

the

mild

solution

to

(5.1):00u(t,龙)=工如农

x),

where

un(t,

x)

=

n=0©； •))，(5.4)g

Springer

894ACTA

MATHEMATICA

SCIENTIAVol.39

for

each

(t,

x),

•)

is

a

symmetric

element

in

([0,

t

x

whose

precise

form

is=亦Pt-Sb(n)(%

—

%b(n))…"sgi-So⑴(Zcr(2)

—

xa(l')

)Ps。⑴"0(%

(氏5)where

cr

denotes

the

permutation

of

{1.2,

•…,n}

such

that

0

<

5^(!)<

••-

<

s(T(n)<

t

(see

for

example

[38,

Formula

(2.3)],

[41,

Formula

(3.3)],

and

[49,

Formula

(4.4)]).

As

in

Section

3.3,

we

introduce

the

Hilbert

space

1¾

as

the

completion

of

smooth

functions

from [0,

t]

x

to

R

with

conipact

support

with

respect

to

the following

scalar

product:,

(5.6)where

1(h)

=

f

fRd

h(s,

y)W(ds,

dy).

fn(t,

x-

•)

can

be

considered

as

an

element

in

IK?n.

To

show

the

existence

and

uniqueness

of

the

solution,

it

suffices

to

prove

that

for

all

(i,

x),

we

haveoo刀

nfn{t,x-

・)||詁”

<

oo

-

n=0(5.7)5.2

Covariance

st

ruetureFrom

Proposition

4.1,

we

see

that

if

|g(u,

v)

—

g(u,

t)

一

g(t,

u)

+

g(t,

t)|

<

C|i

-

u

A

and1+；2討（此）＜

*

-(5.8)then

the

mild

solution

to

(4.1)

exist

in

L2.

In

fact,

it

is

argued

in

[47]

that

the

above

condition

is

also

necessary for

the

existence

and

uniequess

of

a

solution

to

(4.1).As

we

shall

see

from

the

chaos

expansion,

the

solution

to

(4.1)

is

the

first

(non

constant)

chaos

term

of

the

solution

to

(5.1)

(when

the

noise

and

initial

conditions

are

the

same).

Because

all

the

terms

in

the

chaos

expansion

are

orthogonal,

we

see

that

the

L2

norm

of

the

solution

to(4.1)

is

dominated

by

the

L2

norm

of

solution

to

(5.1).

Hence,

condition

(5.8)

is

also

necessary

for

the

solution

of

(5.1)

to

exist

in

r,

this

condition is

not

sufficient.

One

example is

when

the

noise

is

white

in

space,

that

is

Hi

=

••-

=

Hd

=

1/2.

In

this

case,

“(d£)=第加

dg

and

/3

=

2Hq.

Condition

(5.8)

becomesL

可沿

(5-9)which

means

>

d/4.

We

can

compare

this

result

to

two

situations

studied

in

[49].

When

d

=

1,

condition

(5.9)

reads

as

Hq

>

1/4,

while

in

[49]

it

was

assuming

Hq

>

1/2

to

find

an

L2

solution

of

(5.1).To

fuTtheT

compare

the

two

equations

with

additive

and

multiplicative

noises,

let

us

denote

the

solution

to

(4.1)

by

ua(i,x)

and

the

solution

to

(5.1)

by

um(^,x).

Assume

that

the

noises

and

the

initial

conditions

are

exactly

the

same.

Then,

um(f,

x)

=

ua(f,

x)

+

u(t,

x)

for

some

random

field

u(t^

x)

which

are

orthogonal

to

ua(t^

E

[ua(t,

x)u(t^

a;)]

=

0.

Hence,E

=

E

[ua(t,x)2]

+

E

[u(t,x)2].This

implies

E

x)2]

>

E

[ua(t,

a;)2].

Thus,

when

the

initial

condition

and

the

noise

arethe

same,

that

the

solution

to

(5.1)

is

in

L2

implies

that

the

solution

to

(4.1)

is

also

in

L2.

The

reverse

is

not

always

true.

Here

is

one

example.0

Springer

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION895Example

5・4

Let

the

noise

W

be

independent

of

time

and

be

white

in

space,

that

is,

)*g(s,r

=

sr

and

A(q

—

z)

=

S(y

— z)

or

“(d§)=(療d

•

In

this

case,

the

following

statements

are

proved

in

[38]

(Assume

the

initial

condition

u()(x)

=

1):(i)

When

d

=

1,

the

solution

to

(5.1)

exists

in

U

for

all

time

i

>

0.(ii)

When

d

=

2,

the

solution

to

(5.1)

exists

in

L2

only

for

all

time

t

e

[0,

to]

and

when

t

>

to

the

solution

is

not

square

integrable.(iii)

When

d

=

3,

each

chaos

un(t^

x)

in

the

formal

chaos

expansion of

the

solution

exists

but

the

chaos

expansion

is

not

convergent

in

L2

(namely,

(5.7)

diverges).(iv)

When

d

=

4,

all

un

(n

>

1)

is in

the

other

hand,

it

is

obvious

that

for

(4.1),

the

solution

exists

in

L1

for

all

i

>

0

when

d

=

1,2,on

(5.1)

is

well

studied

in

the

literature

under

some

conditions

which

we

list

hout

this

article,

we

use

some

codes

to

represent

various

specific

form

of

the

covari­ance

structures.(W-W)

The

Gaussian

noise

is

white

both

in

time

and

space:g(s,厂)=s

A

r

(the

minimum

of

s

and

r)

and(5.10)A®

-

z)

=

6(y

一

2)•

(5.11)(W-F)

The

Gaussian

noise

is

white

in

time

and

is

fractional

in

space.

Namely,

g

is

given

by(5.10)

anda(“

-

z)

=

H

[乩(2皿一1)尬一剧2屁-2]

.

i=ld(5.12)(F-W)

The

Gaussian

noise

is

fractional

in

time

and

white

in

space9(s,r)

=

|(s2H

+

r2H

-

|s

-

r|2H)

and

A(“

一

z)

is

given

by

(5.11).(5.13)(F-F)

The

Gaussian

noise

is

fractional

in

both

time

and

space.

Namely,

g

is

as

defined

by

(5.13)

and

A(y

—

z)

is

given

by

(5.12).(W-G)

The

Gaussian

noise is

white

in

time

and

general

in

space.

This

means

that

g

is

given

by

(5.10)

andA(y

—z)=/

ei(yr)知(此)

(5.14)is

the

covariance

of

a

general

mean

zero

Gaussian

process.(F-G)

g(s,r)

is

given

by

(5.13)

and

A

is

given

by

(5.14).g

Springer

896ACTA

MATHEMATICA

SCIENTIAVol.39

Ser.B(G-G)

g(s,r)

is

a

general

covariance

function

and

A

is

given

by (5.14).

In

this

case,

we

usually

assume

that

noise

in

time is

also

homogeneous:7(s

一『)=•

(5.15)In

the

above

cases,

when

the

noise

is

fractional

in

time,

we

usually

assume

that

the

Hurst

parameter

Hq

is

greater

than

or

equal

to

1/2.

In

some

papers,

we

can

allow

the

noise

to

be

fractional

in

time

with

Hurst

parameter

H

<

1/2.

When

we

do

so,

we

need

to

assume

higher

regularity

in

space

variable

on

the

covariance

function

and

it

is

more

convenient

to

replace

W

in

equation

(5.1)

by

W

(no

spatial

derivative

of

the

noise).

This

was

what

has

been

done

in

the

literature.

However,

when

we

consider

the

moments

of

the

solution,

we

will

use

W

in

Equation

(5.1)

instead

of

磊

W

to

avoid

confusion.

When

we

need

to

represent

the

solution,

we

use

x)=茁性^

"(匸比)，We

shall

rewrite

the

conditions

in

[14,

48]

in

accordance

to

our

not

at

ion.(F-Qi)

The

noise

is

fractional

in

time, given

by

(5.10),

and

the

covariance

strueture

for

the

spatial

variable

is

given

by

q(y、z)

which

is

not

homogeneous:E(W(s,

y)W(r,

z))

=

g(s,

z), (5.16)whereQ©可=讥Q每Z):=切…鳥；「任/血刃

(5-17)satisfies

the

following

properties

for

some

M

<

2

and

7

G

(0,1]:(QI)

Q

is

locally

bounded:

there

exists

a

constaut

Co

>

0

such

that

for

any

JC

>

0,Q(x,y)

+

K)Mfor

any

x^y

such

that

x,

y

<

K.(Q2)

Q

is

locally

7-Holder

continuous:

there

exists

a

constant

Ci

>

0

such

that

for

any

K>0,|Q(z,

y)

—

Q(u,

v)|

<

Ci

(1

+

K)M

(丘—叩

+

协一叩),for

any

x,

y,u,v

G

such

that

x,

y,

|u|,

v

<

K.(F-Q2)

The

noise

is

fractional

in

time,

given

by

(5.10),

and

the

covariance

structure

Q

satisfies

a

different

set

of

conditions.

Namely,

E(VT(s,?/)iy(r,

2))

=

g(s、r)q(y、z),

where

=dxdzq{y,

z)

satisfies

the

following

properties

for

some

M

<

2

and

7

e

(0,1]:(Q3)

For

some

constant

Co

>

0

and

some

a

€

(0,1],Q(x,

x) +

Q(y,

y)

-

2Q(x,

y)

<

C0x

-

y2a,

V

x,

ye

Rd.

(5.18)(Q4)

For

some

(3

6

[0,1),

there

exists

a

constant

Q

>

0

such

that

for

all

Af

>

0,Q(x,

y)

>

CqM?®、V

x,

?/

€

with

min

,d(|xi|

A

|s|)

>

M.

(5.19)

(Q5)

There

exists

a

constant

Ci

>

0

such

that

for

all

Af

>

0,9)|

S

Ci(l

+

M)",

for

all

x,y

with

|3?|,

y

<

M. (5.20)

No.3

YZ

Hu:

STOCHASTIC

HEAT

EQUATION

897Remark

5.5

(1)

The

sentence

that

noise

is

white

is

equivalent

to

say

that

noise

isfractional

with

Hurst

parameter

H

=

1/2.

For example,

when

we

say

that

the

noise

is

white

in

time,

it

means

that

in

(5.13),

H

=

1/2.(2)

In

the

literature,

we

sometime

assume

that

the

noise

W

is

independent

of time.

This

is

equivalent

to

say

that

W

is

fractional

with

Hurst

parameter

H

=

1:

namely,

in

(5.13),

the

function

g

is

given

by

g(s,

r)

=

sr.5・3

Existence

and

uniqueness

of

the

solutionThere have

been

a

number

of

papers

concerning

the

existence

and

uniqueness

of

the

solution

to

the

stochastic

heat

equation

(5.1).

Here,

we

list

some

sufficient

condition

obtained

in

the

literat

m

5・6

The

mild

solution

to

the

stochastic

heat

equation

with

multiplicative

noise(5.1)

exists

uniquely

for

all

time

f

>

0

if

one

of

the

following

conditions

are

satisfied.(1)

([38,

Theorem

2.1])

d

=

1.

The

noise

is

time

independent

in

time

and

white

in

space.(2)

([49,

Proposition

4.3])

d

=

1.

The

noise

is

white

in

space

and

fractional

in

time

with

Hurst

parameter

H

>

1/2.(3)

([41,

Theorem

3.2])

The

noise

is

both

general

in

time

and

in

space.

7

is

locally

integrable

and

a

satisfies[<

©

(5.21)JRrf

1

+

KI(4)

([42,

Theorems

4.3

and

4.5])

The

noise

is

white

in

time

and

fractional

in

space

with

Hurst

parameter

H

>

1/4.(5)

([48,

Theorems

6.2])

The

noise

is

fractional

in

time

with

Hurst

parameter

H

>

1/4

and

the

covariance

in

space

is

given

by

Q.

Namely,

the

covariance

is

given

by

(5.16)),

where

g

is

given

by

(5.13)

with

H

>

1/4

and

Q

satisfies

(QI)

and

(Q2)

with

7

>

2

—

m

5・7

The

mild

solution

to

the

stochastic

heat

equation

with

multiplicative

noise(5.1)

exists

uniquely

up

to

some

positive

time

To

if

one

of

the

following

conditions

are

satisfied.(1)

([38,

Theorem

4.1])

d

=

2.

The

noise

is

time

independent

in

time

and

white

in

space.(2)

([49,

Proposition

4.3])

The

noise

is

white

in

space

and

fractional

in

time

with

Hurst

parameter

H

>

(1/2)

V

(d/4),

where

ab

denotes

the

maximum

value

of

the

real

numbers

a

and

m

5・8

([14,

Theorem

3.6])

Let

the

covariance

of

the

noise

be

given

by

(5.16),

where

g

is

given

by

(5.13)

with

general

Hurst

parameter

H

>

0

and

Q

satisfies

(Q3)

with

ct

>

1

—

2H.

Then,

a

weak

solution

to

(5.1)

exists.

898ACTA

MATHEMATICA

SCIENTIAVol.39

Ser.B5.4

Feynman-Kac

formulas

for

the

moments

of

the

solutionIn

this

work,

when

we

talk

about

the

solution,

we

always

mean

a

square

integrable

solution.

When

the

equation

is

linear

with

additive

or

multiplicative

noise,

we

can

always

have

a

formal

chaos

expansion

for

the

solution

candidate.

With

this

rather

explicit

chaos

expansion

form

of

the

solution,

we

can

also

find

the

solution

in

a

distribution

space.

This

idea

was

done

for

stationary

counterpart

of

(5.1)

in

[40].On

the

other

hand,

in

most

cases,

under

our

conditions,

solution

is

not

only

square

in・

tegrable

but

also

in

Lp

for

any

finite

p

and

we

have

a

Feynman-Kac

formula

for

the

positive

integer

in

this

subsection,

we

summarize

some

known

results

on

the

Feynman-Kac

formula

for

the

moments

of

the

m

5・9

(1)

([41,

Theorem

3.6])

Let

the

noise

be

general

both

in

time

and

inspace,

namely,

g(s,

r)

is

a

general

covariance

function

and

A

is

given

by

(5.14)

such

that

7

defined

by

(5.15)

is

locally

integrable

and

“

satisfies

(5.21).

Then,

the

solution

to

(5.1)

has

moments

of

all

orders

and

the

moments

are

given byE

(u(t,

x)p)

=

E

(

j=i+

x)

expELl

=

(B-7,1,

•

•

•

,

nian

motions.j

=

1,

••-

,p

are

independent

d-dimensional

standard

Brow­(2)

([23,

Theorem

2.2

and

equation

(3.2)])

Let

the

noise

be

of

the

type

(F-Q2),

namely,

E(W(s,

y)W(r^

2))

=

g(s,

2),

where

g

is

given

by

(5.13)

with

Hurst

parameter

H〉0

and

=

dydzq(y^

z)

satisfies

(Q3)-(Q5).

Then,

the

solution

x)

to

(5.1)

hasmoments

of

all

orders

andE

[u(t,x)p]

=

E<

uo(B{

+

x)

L=i牙

H(2H

-

1)l

g

/俨z

]q（硏黔）+

q（硝卫爲）]〃”，（5.23）0q)

—

Q(0“,©u)

—

Q(仇,•where

for

two

functions

0

and

0,-

1Q(s

v,

0,0)

—

~

[Q(0u：*0u)

+

(3)

[43,

Theorem

4.2]

Suppose

that

the

noise

is

white

in

time

and

fractional

in

space

with

Hurst

parameter

|

<

H

<

Then,

the

solution

to

(5.1)

has

moments

of

all

orders

and

for

any

positive

integer

p,£)]

=p

j=

/+x)

exp

(ci,h

'(5.24)

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION899where

c、,h

=吉Y(2H

+

1)

sin(7rH)

and1_2H姿（讯-於）疋甘is

the

limit

in

L2(Q),

as

e

—0,

of

V^'^k

=幷人广引刖疋卜-2H0(卑-於)疋酋.Remark

5・10

(1)

When

the

noise

is

fractional

both

in

time

and

in

space

with

H。、d…，Hd

>

1/2.

Then,

“(dg)

=

Ch

TI

卩一'凤d&

for

some

constant

Ch-

Condition

(5.21)

is

equivalent

toi=ld匸比〉d-l.

i=l(5.25)In

this

case,

the

moment

formula

can

be

written

asE

(u(t,

x)p)

=

E

(

“o(尽+

£)j=ix

expi

B)=

(B-7,1,

■

■

•

,

j

=

1,

■

■

■

,p

are

independent

d-dimensional

standard

Brow-

dnian

motions

and

an

=

n

HQH)—

1).i=0(2)

([49,

Theorem

5.3])

Suppose

that

the

noise is

fractional

with

Hurst

parameter

Hq

>

1/2

in

time

and

white

in

space.

When

d

=

1,

condition

(5.25)

is

satisfied

and

then

we

have,

for

any

positive

integer

p,E

(u(t,

x)p)

=

E

(

JJ

uq(B{

+

x)j=ix

expl

一

B)dsdr*(5.27)where

o^h0

=

H°(2Ho

—

1).

When

d

=

2,

condition

(5.25)

does

not

hold

(the

inequality

becomes

an

equality).

In

this

case,

it

is

proved in

[49,

Theorem

5.3]

that

the

above

formula

also

hold

true

but

for

some

t

G

(0,

tp)

for

some

strictly

positive

tp

depending

on

p.5・5

FeynmaKac

formulas

for

the

solutionIf

W

in

(5.1)

is

replaced

by

a

nice

genuine

function,

then

we

have

a

Feynman-Kac

formula

(2.23).

This

formula is

still

true

for

the

solution of

(5.1)

if

the

covariance

of

W

satisfies

some

additional

properties

more

restrictive

than

the

existence

of

the

solution

or

the

existence

of

the

moments.

For

example,

we

know

that

when

VV

is

a

space

time

white

and

when

d

=

1,

the

solution

to

(5.1)

exists

in

L2

for

all

/

>

0.

But

there

is

no

Feynman-Kac

formula

for

the

solution.

In

this

subsection,

we

give

a

survey

on

the

conditions

such

that

the

Feynman-Kac

formula for

the

solution is

m

5・11

([41,

Theorem

5.3])

Let

the

noise

be

general

both

in

time

and

in

space.

This

means

that

7

is

given

by

(5.15)

and

A

is

given

by

(5.14).

Assume

that

there

are

constant

鱼

Springer

900ACTA

MATHEMATICA

SCIENTIAVol.39

Ser.B/3

G

(0,1)

and

constant

cp、such

that0

<

7(t)

<

厂©

for

all

t

>

0(5.28)(5.29)and“(dg)<

oo

.1

+

罔

2-20Then,u(t,

x)

=

Eb"o(砖)exp—9)W(dg)A(Br

—

Bs)drds(5.30)is

the

unique

mild

solution

to

Equation

(5.1).Now,

assume

that

the

noise

is

both

fractional

in

time

and

in

space

with

Hurst

parameters

(Ho,

…,Hd)

with

all

Hurst

parameters

greater

than

1/2.

This

is

a

particular

case of

the

above

theorem

with

corresponding

parametersd0

=

2

—

,

M(dC)

=

ch

i=

is

easy

to

see

that

condition

(5.30)

becomesd2//°

+〉]

Hi

>

d

+

1.2=1(5.31)Thus,

we

haveCorollary

5・12

([50,

Theorem

7.2])

Suppose

that

the

noise

is

fractional

both

in

time

and

in

space

and

(5.31)

holds.

Suppose

that

uo

is

a

bounded

measurable

function.

Then,

theprocessx)

=

Eb“o(£f)exp—y)W©,d")(5.32)where

an

=

Y[

i=0d—

1),

is

the

unique

mild

solution

to

Equation

(5.1).We

also

have

the

Feynman-Kac

formula

for

the

coefficients

(5.4)

of

the

chaos

expansion

of

the

solution

to

Equation

(5.1).

Namely,

we

have

the

following

ition

5.13

Under

the

condition

of

Theorem

5.11

on

the

covariance

structure

ofthe

Gaussian

noise,

we

prove

that

the

solution

of

(5.1)

is

given

byn=0

where

the

Ito-Wiener

coefficients

hn

can

be

expressed

byz)

=

Eb

[f(Bf

)6(笙—

yi)…—如)]•0

Springer(5.33)

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION901When

the

noise

is

fractional

in

time with

HuTst

parameter

less

than

1/2,

we

need

to

assume

some

more

restrictive

conditions

on

the

space

variable

such

as (F-QJ

and

(F-Q2)-

In

particularly,

we

needs

that

W

is

differentiable

almost

surely

in

the

spatial

variables

,Xd-In

both

cases,

it

is

more

convenient

to

useBy

(5.17),

the

covariance

structure

of

V

isE(V(s,

x)V(t,

y))

=

7(s,

t)Q(x,

y).Equation

(5.1)

can

be

writ

ten

as厉=㊁"祈卩，上二0,

x

G

;(5.34)u(0,x)

=

u0(x),We

use

the

following

approximation

of

the

m

5.14

((F-Qi)

[4&

Theorem

3.4,

Proposition

3.5])

Suppose

that

0

€

Ca

([0,T])

with

7Q

>

1

—

2H

on

[0,

T].

Then,

the

limit

of

幷

(/>s)ds

as

6

—>

0

exists

in

L2

and

is

called

the

nonlinear

stochastic

integral

V(ds,

0S),

where严匕乞)=[Ve(s,

x)ds

,JoMoreover,

we

have1with

V£(s,x)

=

—

(V(s

+

£,

x)

—

V(s

—

e,

x)).(5.35)E+H(2H

—+H(2H

-

1)『v(dzj

沪HTQ(0e,伽)d&(Q(0p,如_”)—

Q(0®,伽))d『d02H-2

(Q(如一r,如—Q(如切))drde.(5.36)Let

7Y

be

the

Hilbert

space

obtained

from

the

completion

of

the

linear

span

of

indicator

functions

l[o,t]x[o,x],

t

W

[0,

T],

x

6

with

respect

to

the

scalar

productl[0,s]x

[0,切〉九=丁(上，S)Q(Z,卩).Theorem

5.15

((F-Qi)

[48,

Theorems

4.1,

6.2])

Let

H

>

*

-

于

and

let

uq

be

bounded.

For

any

t

e

[0,T]

and

x

e

IRd,

the

random

variable

V(ds,

Bf_s)

is

exponentially

integrable

and

the

random field

u

(Z,

x)

given

byu(t,x)

=

EB

Lo(B?)eXp

V(O-|||CIIh

}

©37)is

in

厶p(Q)

for

any

p

>

1,

where

B

=

{Bf

=

H-x,

t

> 0,

x

e

Rd}

is

a

d-dimensional

Brownian

motion

starting

at

a?

e

Rd,

independent

of

W

and

gfx

(r,

z)

:=

l[o,s](r)l[o,B^_r](2)-

It

is

the

unique

mild

solution

to

(5.1).For

the

noise

structure

of

type

(FQ2),

we

have

similar

results

to

Theorems

5.14-5.15.

First

denote

】Q(®

v,

©

0)

=

—

[Q(0u,

*0u)

+

Q(仇,认)-Qgu、^v)

-

Q(机、©u)]

•g

Springer

902ACTA

MATHEMATICA

SCIENTIAVol.39

em

5.16

(F-Q2)

(1)

[15,

Theorem

2.2]

For

all

0

<

i

< T

and

0,

0

W

C^([0,

T])with

a/3

+

H

>

1/2,

the

stochastic

integral

/(0)：=船

V(dsgs)

is

well-defined

in

the

same

way

as

in

Theorem 5.14

andE

[/9)/(0)]

=

H「俨HJ

[Q(切,伽)+

Q@_e,血一

d

deJo—och

[

[

r2H~2Q(3,0

—

ip)

Jo(2)

([15,

Theorem

3.1,

3.6])

Suppose

that

Q

satisfies

(Q3)

with

2H

+

q〉1

and

that

uq

is

bounded.

For

all

t

>

0

and

x

G

lRd,

the

random

variable

J：

V(ds,

B^_s)

is

exponentially

integrable

and

the

random field

u(t,

x)

given

byu(t,

x)

=

EbUo(B^exp

[卩Jo(dsy)

(5.38)is

in

LP(Q)

for

all

p

>

1.

It

is

a

weak

solution

to

equation

(5.1).5.6

Moment

boundsNow,

we

assume

that

the

covariance

of

noise

(we

continue

to

use

the

notation

of

section3.1)

has

some

of

the

following

esis

1

There

exist

positive

constants

co,

Co,

and

0

<

/?

<

1,

such

thatcot~0

<

Co|t|esis

2

There

exist

positive

constants

Ci,

Ci,

and

0

<

77

<

2

A

c/,

such

that<

X{x)

<

C^x^.Hypothesis

3

There

exist

positive

constants

C2,

C2,

and

0

<

^

<

1,

with

工/

<

2,

d

such

thatd

i=l

dS

A(Z)《①口虑厂化i=l

i=lClearly,

Hypothesis

1

and

Hypothesis

2

generalize

the

case

of

Riesz

kernels

and

Hypothesis

3

generalizes

the

case

of

fractional

m

5.17

([41,

Theorem

6.4])

Suppose

that

7 satisfies

Hypothesis

1

and

A

satisfies

Hypothesis

2

or

Hypothesis

3.

Denoteif

Hypothesis

2

holds;a

=<

dif

Hypothesis

3

holds.、i=lLet

u

be

the

solution

to

(5.1).

If

there

exist

two

positive

constants

Lq

and

such

that

0

<

厶0

S

uo(z)

S

厶1

V(X),

thenC

exp(C7

2?a

<

E

[u(t,

x)k^

<

C'

exp

[c't

2~a

)(5.39)for

all

i

>

0,

a;

e

,

and

Zc

>

2,

where

C,

C'

are

constants

independent

of

t

and

k.

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION903Theorem

5・18

([41,

Theorem

6.5])

Suppose

that

the

noise

is

time

independent

and

A

d

satisfies

Hypothesis

2

or

Hypothesis

3.

Set

a

=

77

if

Hypothesis

2

holds,

and

a

=

刀

Q

if

Hypothesis

3

holds.

Suppose

that

there

exist

two

positive

constants

L°

and

Lq

such

that

0

V

厶0 S

^o(^)

S

厶1

<

oo.

Then,

the

solution

u

to

(5.1)

satisfiesCexp^Cf^fc^)

<

E

i=l

exp(c't^k^^

,

V

x

e

fc

>

2

,

(5.40)where

C,

>

0

are

constants

independent

of

t

and

the

spatial

dimension

d

=

1,

we

can

allow

the

space

covariance

to

be

a

Dirac

delta

function,

that

is,

the

noise is

white

in

m

5.19

([41,

Theorem

6.9])

Suppose

that

the

spatial

dimension

is

d

=

1

and

that

the

noise

is

white

in

space,

namely,

A(龙)=d(z).

Then,(1)

If

the

time

covariance

7

satisfies

condition

Hypothesis

1,

thenexp

(<7护-20上3)§

e

x)k]

<

exp

(C^3-2^^3)

(5.41)for

any

a:

e

A;

>

2,

and

Z

>

0,

where

C,

Cz

>

0

are

constants

independent

of

t

and

k.(ii)

If

the

noise

is

time

independent

(formally

it

corresponds

to

the

case

/3

=

0),

thenexp

(Ct3A;3)

<

E

[u(t,

x)k]

<

exp

(CS叹3)

,

(5.42)for

any

x

€

R,

/c

>

2,

and

>

0,

where

C,

Cz

>

0

are

constants

independent

of

t

and

5・20

(1)

If

the

spatial

dimension

¢/

is

1,

the

noise

is

white

both

in

time

andin

space,

then

(5.41)

also

holds

with

/3=1.

Namely,

if

the

spatial

dimension

d is

1,

and

if

the

noise is

white

both

in

time

and

in

space,

then

the

solution

u

to

(5.1)

satisfiesexp

(Ct炉)<

E

[u(t,

x)k]

<

exp

(C7/c3)

(5.43)for

any

a;

G

R,

A:

>

2,

and

Z

>

0,

where

C,

C‘〉0

are

constants

independent

of

t

and

k.(2)

The

statements

of

the

above

three

theorems

(Theorems

5.17-5.19)

hold

true

if

the

product

in

(5.1)

is

replaced

by

the

ordinary

product

(the

equation

is

in

the

Stratonovich

sense).Finally,

when

the

spatial

dimension

d

is

1, we

can

allow

7

to

be

rough.

More

precisely,

we

can

allow

g

to

be

the

covariance

of

fractional

Brownian

motion

with

Hurst

parameter

H

<

1/m

5.21

(1)

[14,

Theorem

3.1]

Suppose

that

the

noise

is

of

the

type

(F-Q2)

andthat

Q

satisfies

(Q3)

with

2H

+

q

>

1

and

suppose

that

uq

is

bounded.

For

alH

>

0

and

x

e

the

random

variable

循

V(ds,

EJ)

is

well-defined

as

the

limit

given

in

Theorem

5.14

and

is

exponentially

integrable.

The

solution

u(t,

x)

given

by

(5.38)

satisfiesIE

[|呃,2；)广]s

Cexp(C7c台

t宰护)for

all

i

>

1

and

x&Rd

(5.44)

for

some

constant

C

=

C(d,

H、at,

||uo||

OO?①)>(2)

[14,

Theorem

4.1]

Suppose

furthermore

that

infx6Rd

uq

>

0,

and

(5.19)

holds

for

some

(3

e

[0,1).

There

exists

some

C

=

C(d,

H,

a,

0,

u0)

>

0

such

that

for

all

x

€

if

either

k

or

t

is

sufficiently

large,

thenE

u^t,x)k]

>

Cexp(CA:耳t攀皆).

(5.45)g

Springer

904ACTA

MATHEMATICA

SCIENTIAVol.39

k

5・22

If

d

=

1

and

Q(x,

y)

is

the

covariance

of

a

fractional

Brownian

motion

{

B®,

a;

6

R}

with

Hurst

parameter

e

(0,1),

that

is,Qd，y)

=

E

=

j

(|评旳

+

|y|2H1

-1^-汕负)，then

it

is

easy

to

see

that

both

conditions

(5.18)

and

(5.19)

are

satisfied

with

a

=

0

=

H

and

the

combination

of

(5.44)

and

(5.45)

becomesCexp

5・7

Discussion

of

the

proof

of

the

moment

bounds.

(5.46)The

proof

of

the

upper

bounds

for

the

moments

of

the

solution

in

Theorems

5.17-5.21

can

be

completed

by

three

different

approaches:

Chaos expansion

combined

with

an

application

of

hypercontractive

inequality;

Burkholder-Davis-Gundy

inequality;

and

Feynman-Kac

formula.

Chaos

expansion

approach

works

for

multiplicative

Gaussian

noise

(one

needs

a

chaosn

expan­sion

of

the

solution);

Burkholder-Davis-Gundy

inequality

approach

works

only

when

the

noise

is

white

in

time

(one

needs

a

martingale)

but

it

works

for

general

nonlinear

"difffusion”

coeffi­cients

(not

necessarily

additive

or

multiplicative

noise).

Feynman-Kac

formula

approach

works

for

multiplicative

general

Gaussian

noise

(one

needs

a

Feynman-Kac

formula

for

the

moments

of

the

solution).

To

explain

how

these

three

different

approaches

work,

we

give

the

above

three

different

types

of

proof

for

one

particular

theorem

(see

below)

which

is

the

special

case

of

Theorem

5.17

when

the

noise

is

white

in

m

5・23

Suppose

that

the

noise

is

white

in

time

and

its

covariance

A

in

space

variables

sat

isfies

the

upper

bound

inequality

in

Hypothesis

2

or

Hypothesis

3.

Namely,

t

here

exist

positive

constants

ci,

Ci

and

0<77<2AcZ,

0<7/i,•■-,耳乙<1,

such

thatdA(£)<

Ci|创一77

or

A(z)

<

C2

JJ

皿厂化

i=l(5.47)Assume

that

there

exists

a

positive

constant

厶

1

with

<

Li

<

00.

Denote77

a

=

{吕if

Hypothesis

2

holds;if

Hypothesis

3

holds.I

2=

1y

rji

Suppose

0

<

a

<

2.

If

u is

the

solution

to

(5.1),

thenE

x)p]

<

C'

exp

((7'切上^)

for

all

>

0, a;

6

,

and

p

>

2,

where

C,

C'

are

constants

independent

of

t

and

p.(5.48)Proof

Without

loss

of

generality,

we

assume

that

Hypothesis

3

is

satisfied

(2

can

be

proved

similarly).

As

we

promised,

we

shall

explain

three

different

appraoches

to

prove

Theorem

1.

Chaos

expansion

with

hypercontractivityFirst

we

use

chaos

expansion.

"

is

a

mild

solution

to

(5.1)

if=

ptuo(x)

+

堑

Springer/

p—sCr

—

")"(s,9)dW(s,9).

JO丿耐(5.49)

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION905Iterating

this

equation,

we

see

that

the

random

variable

u(t,

x)

admits

the

following

Wiener

chaos

expansionoo"(t,%)=刀

£)，

n=0(5.50)whereUn(t,

x)

=

and

for

each

(t,

re),

*n

Sj

,

y①;•))

(5.51)•)

is

given

by,

•

,Sn,27n)=亦'肌-5(7i)(龙—"(n))…Pso(2)-s。⑴(％(2)

-

£(r⑴)几。(1)“0(皎r(l))

Jwhere

a

denotes

the

permutation

of

{1,2,

••-

,

n}

such

that

0

<

Sc(i)<

•

•

•

<

s

t.

Let

us

compute

the

L2

norm

of

un(t^

x)

=

Ai(fn(t,兀；•))•

From

the

Ito

isometry,

we

haveIE(如(t,:r)2)

=

Ti!

/

J[o,t]n丿附通

/

九(t,z;s,y)九(t,a?;s,z)

J~(A(s—

zjdydzds甘S

冗!||呦||烹

/

/

gn{t,

x',

s,

y^g^t,

x;

s,

z)V[

-

z^dydzds,丿[0,t]”丿呼讥

詁where

da;

=

dzi

•…da;n,

the

differentials

dy,

ds,

and

dr

are

defined

similarly

andgn(t卫;s,y)==殖处7心)(0

—

%(n))…Ps°(2)-Sg(i)(%(2)

-

2/(7(1))

-

First,

we

integrate

9(t(i)and

za^.Psg(2)—(5.52)S。⑴(9b(2)

—

%■⑴)Psg⑵一Sg()(Zcr(2)

—

2ct(1))-^(?/ct(1)

—

za(l))d%•⑴dz”⑴i=n

E0爲沙⑴-聲⑺-“⑴+妝-佝-务⑵⑹广=©!,

k=ldwhere

B1

and

B2

are

two

independent

standard

Brownain

motions

and

9^(2)(上)and

Zc(2)(上)

are

the

k-th

coordinates

of

ya(2}

and

2^(2)•

By

using

Lemma

A.l

of

[50],

we

have,

for

a

standard

normal

variable

X,Gi

=

H

fc=idk=d _________________-sct(i))X

+

y(7⑵(町—亏⑵伙)厂""S

JI

以爲⑵<

CI

Sb

⑵-Sm)厂

a/2

•Continuing

this

way

to

integrate

with

respect

to

9^2),

z

•…，and

2/a(n),

2

we

have

(recall

we

use

the

convention

t

=

5CT(n+1)and

sn_|_i

=

t)07i

r

nn<

Cn[I

|sfc+i

-

Sfc|_a/2ds

0

3=1°_________t

(1

-

a

/2)

nr((l

-

a/2)n

+

1)(5.53)

906ACTA

MATHEMATICA

SCIENTIAVol.39

(t,x)2

<__2：__E”

r(i^n+

1)件护心卫From

the

hypercontractivity

inequality,

we

have,

for

any

p

>

2,

un(t,x)p

<

(P

-

l)"/2||u”(te)||2

<

呼”.『(丄晋7l+

1)Now,

from

the

property

of

the

Mittag-Leffler

function

([57,

Equation

1.8.12]),

we

see||u(t,0)||p

pn/2cn=C

exp

{ctp

是}.We

can

also

write

the

above

asx)^

<

Cexp

{(S7pi+岛}

=

Cexp

{(7如緒},which

is

the

conclusion

of

the

theorem for

upper

2・

BuTkholdei*・Gundy-Davis

inequalityFor

the

upper

moment

bound,

this

approach

works for

nonlinear

counterpart

of

(5.1):dtu(t,

x)

=

2

"(0卫)="o(©

,x)

+

x))W(t,

£),

i

>

0

,

x

G(5.54)where

cr

:

R

—>

R

is

a

continuous

measurable

function

with

linear

growth

(|a(u)|

<

ci

|u|

+

C2-

We

assume

the

existence

of

the

mild

solution).

With

a

substitution

u

=>

u

+

1,

we

can

assume

C2

=

0

to

simplify

notation.

The

mild

solution

satisfiesx)

=

z)

+

“2(t,

z)Pt(x

-

9)uo(9)d"

+Pt-s{x

一

9)cr(u(s,9))d"dswhere

we

recall

the

heat

kernel

form

pt(x)

=

(2肮)-"/2

exp

{-胃If

the

initial

condition

satisfies

|“o(z)|

<

then

|ui(t,x)

<

L.

Now,

using

Burkholder-Gundy-Davis

inequality,

we

have,

for

some

constant

C

independent

of p,0Pt-s(x

一

9)<7(M(s,y^pt-s^x

一

z)cr("(s,

z))A(9

-

z)dydzP/2Pt-s(x

-

y)pt-a(x

一

z)|u(s,

y)||u(s,

z)|A(y

-z^dydzFrom

the

Minkowski

inequality,

we

have||"2(t,©)IE

一

y)pt_s(x

-

z)2/PP/2A(y

—

z)dydz

dsPt-s{x

一

y)pt_s(=

一

z)g

Springer

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION907•A(“

—

z)dydz

sup

(E|w(s,

y)p)2^p

ds

yeRdp/2

(

[

(t_s)-a/2

sup

(E|u(s,2/)|p)2^pds

丿o

yeRdP/2Z(s)

=

sup

(E|u(s,^)|p)2/p

.

i/GRdThen,

we

haveZ(t)

+

Cp

[

(t-

s)-"2z(s)ying

[16,

Lemma

A.2

Part

2)]

with

cv

=

2/a,

we

haveThis

means

thatsupE|u(t,x)p

<

CexpXwhich

is

(5.48).Method

3・

Feynman-Kac

formula

for

the

momentsFrom

Theorem 5.9

(when

the

noise

is

white

in

time

the

covariance

function

y(t,s)

in

time

is

replaced

by

the

Dirac

delta

function),

we

seeE

(u(t,

x)p)

=

E

(

uo(Bf

-F

x)

expi

<

(exp

£

f

A(民-於)dswhere

B,

B

j

=

1,

••-

,p

are

independen

t

d-dimensional

standard

Brownian

mot

ions.

By

the

scaling

property

of

Brownian

motion,

we

haveE(u(t,x)p)

<

CpE(

exp

-

V

/"A(B?-Bf)ds2where

tp

=

p^^t.

From

[25,

Theorem

1.1],

it

follows

thatE(u(t,z)P)

<

Cexp

{cptp}

<

Cexp

This

show

the

upper

bound.

|

.口Remark

5.24

(1)

If

A(x)

=

(or

A@)

=

C2

H

也「"')，then

their

Fourieri-lddtransforms

(in

the

sense

of

distribution)

areA(C=C伐卩"

◎

ME)

=

G

i=lH

I护I)

•dHowever,

condition

(5.47)

is

not

equivalent

toia(c)i

<(or

ia(e)isc[]罔"T).

(5.55)幺

Springeri=l

908ACTA

MATHEMATICA

SCIENTIAVol.39

theless,

we

shall

prove at

the

end

of

this

subsection

that

inequality

(5.48)

still

holds

true

when

the

condition

of

the

theorem

is

replaced

by

(5.55),

where

0

<

77

<

2

A

¢/,

0

<

771,

••-

^r]d

<

1,

and

771

+

•

•

•

+

<

2.

In

fact,

when

〃

=

1,

the

two

expressions in(5.55)

are

the

same

and

in

this

case,

we

can

replace

the

condition

77

E

(0.1)

by

77

E

(0,

3/2).

This

is

a

consequence of

a

result

of

[46].

In

fact,

the

symbol

a

in

[46]

is

a

=

d

—

T).

By

[46.

Remark

3.5

(3)],

we

see

that

the

solution

exists

and

has

moment

bound

(5.48)

if

a

>

—1/2

which

is

equivalent

to

77

E

(0,

3/2).(2)

The

equation

can

be

nonlinear:

namely

the

statement

holds

true

for

the

solution

to

警=j

Au

+

cr(u)

IV,

i

> 0,

x

€

;

and

cr

:

R

—>

R

satisfies

0(")

|

<

ci

u

+

c?

for

someconstants

5

and

C2.

However,

if

0(u)|

<

C

|iz|Q

+

c?

for

some

0

<

a：

<

1,

one

may

obtain

a

substantially

different

upper

bounds.(3)

If

“o(z)

>

Z/o

>

0

for

some

positive

constant

L°

andA

(a;)

>

Ci|a;|_T7

or

A(a?)

>

C2

JJ

2=1d(5.56)then

the

lower

moments

bound

holds:E

[u(t,

x)p

>

C'

exp(C‘切)

(5.57)for

all

>

0,

x

e

展",and

p

>

2,

where

C,

Cr

are

constants

independent of

t

and

We

present

the

chaos

expansion

approach

to

prove

(5.48)

under

condition

(5.55).

We

assume

that

the

second

inequality

in

(5.55)

holds

true

(The

same

arguments

works

if

the

nfirst

inequality

in

(5.55)

holds

true).

We

assume

77

e

(0,1).

Set

now

“(d£)三

Yl

“(d&)・

Using

i=

1the

Fourier

transform

and

Cauchy-Schwarz

inequality,

we

obtainE(u„(i,2：)2)

<

n!||iz0||^

[

[

|和”亿

z;

s,

•)(£)$

“(dg)ds

.J[o,t]n

JRndFuTthermore,

it

is

easy

to

see

that

the

Fourier

transform

of

gn

sat

isfieswhere

we

have

set

sCT(n_|_i)=

t.

As

a

consequence,

we

have,

for

a

standard

Gaussian

random

variable

X、/

|Ag”(t,z;s,

•)(£)『

“(dg)<(八

2

11

sup

/

仇!)廿®d丿Rde-(w+»-咲』口

I需(i)G)®"d需⑹J=1d27Td/2

d'

‘

2=1ScrG+1)

—廿©j~T

supEv2(SCT(i_|_i)

—

scr(i))___1-

-----x

-

<帀一1Again

from

Lemma

A.l

of

[50],

we

haveE(un(t,a;)2)

<

―j-

g

Springerf

HG/G+)

—

s”⑴厂@ds

n-丿Er

—

i

No.3Y.Z.

Hu：

STOCHASTIC

HEAT

EQUATION909which

is

the

same

as

(5.53).

Thus,

the

argument

above

proved

(5.48)

under

condition

(5.55).

The

case

d

=

1

and

rj

G

[1,3/2)

is

slightly

more

complex.

But

the

chaos

expansion

approach

still

works.

We

refer

to

[46]

for

,

we

give

a

proof

of

part

(iii)

in

the

above

Remark.

If

uq^x)

>

L°

and

A(£)>

Ci

H

i=ld,

then

from

the

Feynman-Kac

formula,

it

folllows

that

for

any

e

>

0,E

(u(t,

x)p)

=

E

f

"o(耳+

力)expl

(民-於)dssup

sup

Bg

|

O厶expP(P-

I/—七—a2>Lq

exp2Psup

BS

<

J.0

pd)」pd>Lq

exp飞-2

£Psup

I瓦

I

<.0

B

is

a

one

dimensional

Brownian

motion.

By

the

small

ball

estimate

of

the

Brownian

motion

of

the

form

P(

sup

BS

<

£)<

Ce~~^

(see

for

example

p.519,

Lemma

8.1

of

[54]),

we

have

>

Lgexp

认卩-

1)2-陀-〃for

some

universal

constant

g

Maximizing

the

above

right

hand

side

by

taking£

=(2dc2/(2-ia(p_l)))】/d)(which

is

small

when

p

sufficiently

large),

we

haveE(饥(t,e)P)

>

Lgexp

[c3切^]

•This

gives

the

lower

moment

bounds.5.8

Joint

Holder

continuity□The

Holder

continuity

of

the

solution

u(t,

x)

has

been

widely

studied

(see

[5,

46]

and

in

particular

the

references

therein).

This

is

about

the

possibility

if

there

are

a, /3

G

(0,1)

and

(random)

constant

C

such

that

the

following

bounds

hold:忸(t,

x)

一

”(s,

£)|

<

Ct

-

sf

,

|u(t,

x)

一

u(t,

y)

<

Cx

-

y0

,

V

t,

s,

€

[0,

T],

x.y

.The

joint

Holder

continuity

is

a

sharper

inequality

which

is

to

seek

q,0

e

(0,1)

and

(random)

constant

C

such

that|u(t,

y)

-

u(s,

y)

-

u(t,

z)

+

u(s,

z)|

<

Ct

-

s|Q

x

-

y0

,

V

t,

s,

€

[0,

T],

e

Kd

.We

cite

the

following

theorem

for

Holder

and

joint

Holder

continuity of

the

m

5・25

([46])

We

make

the

following

assumptions

on

the

covariance

structure

of

the

noise

W

and

the

initial

condition.

910ACTA

MATHEMATICA

SCIENTIAVol.39

Ser.B(1)

There

are

positive

constants

qq

€

[0,1]

and

C

such

that7o(t)

S

Ct~aQ

for

alH

>

0

.(5.58)We

also

allow

q=

1

and

in

this

case,

we

take

?o(t)

=

6(t)

(the

Dirac

delta

function).。(2)

There

are

constants

m

e

(—1,1)

(i

=

1,

•…，d)

and

C

such

thatd“(£)

S

one

of

these

two

bounds:i=lfor

all

C

e(5.59)If

the

first

bounds

holds,

we

denote

q

=制

+

…+

We

also

assume

thatQ

>

(£

-

2

+

«o)

V

(d

—

2)if

q

<

0;if

q

>

0.(5.60)a

>

d

—

2Here,

we

use

q

>

0

to

denote

Qj

>

0

for

all

z

=

1,

••-

,

c/

and

q

V

0

to

denote

ctj

<

0

for

all

i

=

1,…，d.

We

shall

only

consider

these

two

cases

(although

mixed

cases

may

be

considered

similarly).(3)

The

initial

condition

Uq

satisfiesle|2|u0(C)|dC

<

Cs~0(5.61)for

some

0

V

1

-弩.Let

do

and

a

be

in

[0,1]

such

that_

_

Ot

—

d2^o

+

&

<

2

—

&o---------.[It

is

easy

to

see

that

the

right

hand

side

is

positive

under

our

assumption.]

Then,

there

is

a

positive

constant

C,

such

that

for

every

t

>

r

>0

and

x,

y

eE

u(t,

y)

-

u(r,

y)

-

u(t,

x)

+

u(r,

x}pS

Cexp

(cp2

+

a-dt

2

+

a-d

)仏

_

”皿。他一讷皿，

(5.62)(5.63)E(|u(t,

y)

-

u(r,y)

+

|u(t,

x)

-

u(r,

x)|)p<

Cexp

^cp^+a-dt~2+«-d~)

—

rpa°

+

|z

—

ypa]

.

This

theorem

combined

with

[44,

Theorem

2.3]

givesCorollary

5・26

Let

the

hypothesis

of

the

previous theorem

be

satisfied.

Let

do

and

a

be

in

[0,1]

such

thatcv

—

d/max(2do,

d)

<

2

-

q0

H----q—• (5.64)(i)

For

any

Af

>

0,

there

is

a

random

constant

Cm

such

that

every

i,

r

6

[0,

M]

and

for

every

x,y

eRd

satisfying

x,

y

<

M、|"(t,

y)

一

w(r,

y)

一

x)

+

u(r.

x)

<

Ct

一

r|a°

x

一

y&

.

g

Springer(5.65)

No.3YZ

Hu：

STOCHASTIC

HEAT

EQUATION911(ii)

For

any

M

>

0,

there

is

a

random

constant

Cm

such

that

every

t.r

e

[0,

M]

and

for

every

x,

2/

G

satisfying

|x|,

y

< M,—

u(r,

x)

+

-

u(t,x)

<

C

t

-

ra° +

|龙

一

ya]

.

(5.66)Remark

5.27

The

case

d

=

1

and

W

is

a

space-time

white

noise

corresponds

to

the

case

a。=

1

and

q

=

0.

In

this

case,

the

above

corollary

says

that

if

2a()+

®

V

1/2,

thenu(t,y)

-

u(r,y]

-

u(t,a?)

+

u(r,x)|

<

Ct

-

r^°x

-

y&

(5.67)on

bounded

domain

of

t,

r,

x,

y.

This

coincides

with

the

optimal

Holder

exponent

result

in

[44].

On

the

other

hand,

the

corollary

also

implies

that

in

this

case,

if

ao

<

1/4

and

a

<

1/2, then

on

bounded

domain

of

r,

x,

y,u{t,

x)

—

u(r,

z)|

+

|n(t,

y)

—

u(t,

x)

<

C

—

ra°

+

This

is

the

optimal

Holder

modulus

of

continuity.—

ya]

.

(5.68)6

AsymptoticsThe

moment

formulas

in

previous

section

can

be

used

to

obtain

the

the

exact

moment

asymptotics

and

the

almost

sure

asymptotics

of

the

solution

in

some

still

consider

the

stochastic

partial

differential

equation

(5.1)

with

initial

condition

w(0,

x)

=

1

and

assume

that

the

noise

is

fractional

both

in

time

and

in

space

with

Hurst

parameters

Hq,

H»

…,Hq.

Denoted=

t-s~a°

,

A(z)

=

Y[

J=i(6.1)whereo(.j

=

2

—

2Hj

,

j

=

0,1,

make

the

following

assumptions

on

the

parameters

appearing

0

<

Qo,…，Qd

V

1,

2qo

+

OLi

<2

i=lif

A(a?)

=

口皿厂5,2=10

0

<

q

<

¢/,

2qq

+

Q

<

2d

=

1and

0

<

qq

<

|if

A(x)

=

x~a^迁(6.2)A(z)=九仗)•Let

Denote底〃)be

the

Sobolev

space

of

all

functions g

on

展"such

that

G

L2(Rd).心｛畑)；畑

M聞,側“"V

0

<

s

<

1

and|Vx^(s,

a;)|2dxds

<(6.3)£(qo,

A)=sup〃,7(^,s)A(rc

-

y)g2(s,rr)g

SpringergwAd

912ACTA

MATHEMATICA

SCIENTIAVol.39

Ser.B|Vx^(s,

jr)|2da;dsThe

following

theorem

can

be

found

in

[24].Theorem

6.1

([24,

Theorem

6.1])

Under

assumption

(6.2),

(6.4)A)

is

finite

and(6.5)where

ch

=

H

Hj(2Hj

—

1)

and

we

recall

that

a

=

1

in

the

case

when

丁(z)

=

§o(z)・J=oThere

are

some

other

works

on

the

exact

moment

asymptotics

as

oo

or

as

x

t

oo.

We

refer

to

the

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[20-22]

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