STOCHASTIC FUNCTIONAL PARTIAL DIFFERENTIAL EQUATIONS:
EXISTENCE, UNIQUENESS AND ASYMPTOTIC STABILITY
TOM´
AS CARABALLO
Dp o. Ecuaciones Di e enciales y An´alisis Num´e ico.
Uni e sidad de Se illa. Apa ado de Co eos 1160.
41080–SEVILLA, SPAIN
Email: ca aball@nume .us.es
KAI LIU* AUBREY TRUMAN
Depa men o Ma hema ics
Uni e si y o Wales Swansea
Single on Pa k, Swansea
SA2 8PP, UK
Email: [email protected]
ABSTRACT
Exis ence and uniqueness o s ong solu ions o a class o s ochas ic
unc ional di e en ial equa ions in Hilbe spaces a e es ablished. Su -
icien condi ions which gua an ee he ans e ence o mean squa e and
pa hwise exponen ial s abili y om s ochas ic pa ial di e en ial equa-
ions o s ochas ic unc ional pa ial di e en ial equa ions a e s udied.
The s abili y esul s de i ed a e also applied o s ochas ic o dina y di -
e en ial equa ions wi h he edi a y cha ac e is ics. In pa icula , as a di-
ec consequence ou main esul s imp o e some o hose om Mao and
Shah [13] in which i is p o ed ha unde ce ain condi ions pa hwise ex-
ponen ial s abili y is ans e ed om nondelay equa ions o delay ones i
he cons an ime lag appea ing in he p oblem is su icien ly small, while
in ou ea men he ans e ence ac ually holds o a bi a y bounded
delay a iables no only in ini e bu in in ini e dimensions.
Keywo ds: S ochas ic pa ial di e en ial equa ion; S ochas ic unc ional pa ial di e en-
ial equa ion; Mean squa e and pa hwise exponen ial s abili y.
AMS 1991 Classi ica ions: 60H15, 34K40.
* Au ho o Co espondence.
1
1. INTRODUCTION
The s udy o s ochas ic unc ional di e en ial equa ions is mo i a ed by he ac ha when
one wan s o model some e olu ion phenomena a ising in Physics, Biology and Enginee ing
e c., some he edi a y cha ac e is ics such as a e e ec , ime lag and ime delay can appea
in he a iables (see, o example, Kolmano skii and Myshkis [8], Mohammed [15]). On
he o he hand, one o he mos impo an and in e es ing p oblems in he analysis o
s ochas ic unc ional di e en ial equa ions is hei s abili y, he heo y o which (mainly
o ini e dimensional sys ems) has been g ea ly de eloped o e he las se e al yea s.
As is well known, in he case wi hou any he edi a y ea u es, Lyapuno ’s echnique
is a ailable o ob ain su icien condi ions o he s abili y o solu ions o s ochas ic (pa -
ial) di e en ial equa ions. Howe e , in he case o s ochas ic di e en ial equa ions wi h
he edi a y p ope ies, o ins ance, e en wi h cons an ime delays, Lyapuno ’s me hod
becomes di icul o apply e ec i ely as K aso skii [10] poin ed ou o he s udy o s a-
bili y o o dina y di e en ial equa ions, and as Kushne [11] and El’sgol’ s and No kin [5]
(among o he s) did o s ochas ic di e en ial equa ions. The main eason is ha i is much
mo e di icul (o e en impossible in some cases) o cons uc p ope Lyapuno unc ions
(o unc ionals) o s ochas ic unc ional di e en ial equa ions han o hose wi hou any
he edi a y cha ac e is ics. As a consequence, a compa ison echnique has been de eloped
by a ious au ho s such as K aso skii [10] and Mao and Shah [13] (among o he s). Le us
illus a e his poin o ou mo i a ion in mo e de ail.
Conside he ollowing s ochas ic unc ional di e en ial equa ion
x( ) = Z
0
(s, x(s), x(s−h1)) ds +Z
0Z0
−h2
g(s, x(s+ ))h( )d dw(s), > 0,(1)
whe e h1>0, h2>0, o equi alen ly,
x( ) = Z
0
(s, x(s), x(s)) ds +Z
0
g(s, x(s)) dw(s)
+Z
0h (s, x(s), x(s−h1)) − (s, x(s), x(s))ids
+Z
0hZ0
−h2
g(s, x(s+ ))h( )d −g(s, x(s))idw(s).
(2)
We can ega d (1) as he pe u bed sys em o he co esponding s ochas ic di e en ial
equa ion wi hou he edi a y cha ac e is ics
x( ) = Z
0
(s, x(s), x(s)) ds +Z
0
g(s, x(s)) dw(s), > 0.(3)
Clea ly, i he ime lag scales h1>0, h2>0 a e su icien ly small, he pe u ba ion e m
Z
0h (s, x(s), x(s−h1)) − (s, x(s), x(s))ids
2
+Z
0hZ0
−h2
g(s, x(s+ ))h( )d −g(s, x(s))idw(s),
could be expec ed o be so small ha he pe u bed equa ion (1) would beha e asymp-
o ically as Equa ion (3) does. Fo ins ance, we could expec ha i Equa ion (3) is
exponen ially s able and he ime lags h1>0, h2>0 a e small enough, hen Equa ion (1)
will emain exponen ially s able. So, in o de o ind ou whe he he unc ional equa ion
(1) is exponen ially s able, one can check he exponen ial s abili y o he equa ion (3) and
hen compu e whe he he ime lags h1>0, h2>0 a e su icien ly small. In o he wo ds,
he di icul p oblem o s abili y o unc ional equa ions would ha e been ans e ed o
an easie one ( he s abili y o equa ions wi hou he edi a y cha ac e is ics).
Mo i a ed by he in ui i e ideas desc ibed abo e, Mao and Shah [13] ob ained some
su icien condi ions o he p- h momen exponen ial s abili y (and also pa hwise s abili y)
o s ochas ic o dina y di e en ial delay equa ions. Fo example, conside he ollowing one-
dimensional s ochas ic delay di e en ial equa ion
dx( ) = ( , x( ), x( −h)) d +g( , x( )) dw( ), > 0,(4)
whe e h > 0, o equi alen ly,
dx( ) = ( , x( ), x( )) d +g( , x( )) dw( )
+ [ ( , x( ), x( −h)) − ( , x( ), x( ))] d . (5)
I was p o ed in [13] ha unde some ci cums ances pa hwise exponen ial s abili y is
ans e ed om he nondelay equa ion (i.e., h= 0 in (5)) o he delay one (4) i he
cons an ime lag h > 0 appea ing in he p oblem is su icien ly small.
Ne e heless, i is wo h poin ing ou ha he esul s de i ed in [13] a e somewha
es ic i e o many p ac ical applica ions. In ac , he si ua ion u ns ou o be a he
complica ed when one conside s he gene al unc ional di e en ial equa ions, e en he usual
s ochas ic delay di e en ial sys ems. Fo ins ance, he e exis a wide a ie y o in e es ing
p oblems in which i is possible o ensu e ha i nondelay equa ions a e exponen ially
s able, hen delay ones emain exponen ially s able wha e e he delay in e al could be,
wha is mo e e en i , he delays a e no cons an s. In his wo k, by a comple ely di e en
app oach om ha in [13] we shall ca y ou a much mo e delica e in es iga ion. Fo
ins ance, by applying some gene al esul s o be de i ed in Sec ion 4 o he equa ion
(1), we can p o e ha unde some ci cums ances mean squa e and pa hwise exponen ial
s abili y o (1) a e ans e able om he equa ion (3) o a bi a y delay cons an h1>0,
bu o he ime lag h2>0 which mus be su icien ly small.
One o he main aims o his pape is o gi e su icien condi ions which con ain as a
special case he co esponding esul s in ini e dimension ( ha is, o s ochas ic o dina y
di e en ial equa ions) o ans e he exponen ial s abili y o s ochas ic pa ial di e en ial
equa ions o s ochas ic unc ional pa ial di e en ial equa ions. The p oblem we a e e e -
ing o is de o ed o he conside a ion o an in ini e dimensional e sion o (1) in which
has he ollowing o m:
( , x, y) = A( , x) + 1( , y),
3
wi h he amily o (non-linea ) ope a o s A( , ·) sa is ying some kinds o coe ci i y condi-
ions (see Sec ion 2) as well as 1sa is ying Lipschi z con inuous ones. We would also like
o men ion ha , in some sense, a sui able coe ci i y condi ion implies he (exponen ial)
s abili y o solu ions in mean squa e (and also pa hwise exponen ial s abili y) in nondelay
cases (see Ca aballo and Liu [3] and Chow [4]). In addi ion o his, we will be able o
assu e exponen ial s abili y in mean squa e (and, as a consequence, pa hwise exponen ial
s abili y) o a g ea numbe o ini e dimensional s ochas ic unc ional di e en ial equa-
ions while he esul s o Mao and Shah [13] only gi e his kind o s abili y o ce ain
delay sys ems in which he delay mus be cons an and su icien ly small.
He e, we shall analyze only he second momen o solu ions. Al hough we should
emphasize ha his s udy can be ex ended o he p- h momen (p≥2), which is impo an
i i pe mi s us o ob ain some in o ma ion abou he s abili y o sample pa hs. We
also ema k ha , as is well known, mean squa e exponen ial s abili y, ene gy equali y
and Bo el–Can elli’s lemma could imply pa hwise exponen ial s abili y (see, o ins ance,
Ca aballo and Liu [3], Mao [12]).
In Sec ion 2 we begin wi h some p elimina y esul s. We ha e no seen a gene al
ea men on exis ence and uniqueness o s ong solu ions o s ochas ic unc ional di e en-
ial equa ions in in ini e dimensions in he li e a u e. In Sec ion 3 we shall i s es ablish
a esul , which is easy o e i y in many si ua ions, o exis ence and uniqueness o s ong
solu ions o a class o s ochas ic pa ial unc ional di e en ial equa ions. The esul s o
exponen ial s abili y a e s udied in Sec ion 5. Finally, wo examples a e gi en in Sec ion
6 o illus a e he heo y de i ed in he p eceding sec ions.
2. PRELIMINARIES
Fi s o all, we in oduce he amewo k in which ou analysis is going o be ca ied ou .
Le V, H, K be eal, sepa able Hilbe spaces such ha
V ,→H≡H0,→V0,
whe e V0is he dual o Vand he injec ions a e con inuous and dense. In pa icula , we
also assume bo h Vand V0a e uni o mly con ex. We deno e by k · k ,| · | and k · k∗ he
no ms in V,Hand V0 espec i ely; by h·,·i he duali y p oduc be ween V0, V , and
by (·,·) he scala p oduc in H.
Le w( ) be a Wiene p ocess de ined on a ce ain comple e p obabili y space (Ω,F, P)
and ake alues in he sepa able Hilbe space K, wi h inc emen al co a iance ope a o W.
Le (F ) ≥0be he σ-algeb as gene a ed by {w(s),0≤s≤ }, hen w( ) is a ma ingale
ela i e o (F ) ≥0and we ha e he ollowing ep esen a ion o w( ) :
w( ) =
∞
X
i=1
βi( )ei,
whe e {ei}i≥1is an o hono mal se o eigen ec o s o W,βi( ) a e mu ually indepen-
den eal Wiene p ocesses wi h inc emen al co a iance λi>0, Wei=λieiand W=
P∞
i=1 λi<∞( deno es he ace o an ope a o , see Pa doux [16]).
4
Fo an ope a o G∈ L(K, H), he space o all bounded linea ope a o s om Kin o
H, we deno e by kGk2i s Hilbe -Schmid no m, i.e.
kGk2
2= (GWG∗).
Gi en h≥0, p≥2 and T > 0, we deno e by Ip(−h, T;V) he space o all V– alued
p ocesses (x( )) ∈[−h,T ](we will w i e x( ) o sho ) measu able ( om [−h, T]×Ω in o
V), and sa is ying:
(1). x( ) is F -measu able almos su ely in (in he sequel, we will w i e a.e. .), whe e
we se F =F0 o ≤0;
(2). ERT
−hkx( )kpd < +∞.
I is no di icul o check ha he space Ip(−h, T;V) is a closed subspace o Lp(Ω ×
[−h, T],F ⊗ B([−h, T]), dP ⊗d ;V),whe e B([−h, T]) deno es he Bo el σ–algeb a on
[−h, T]. We also w i e L2(Ω; C(−h, T ;H)) ins ead o L2(Ω,F, dP;C(−h, T;H)), whe e
C(−h, T;H) deno es he space o all con inuous unc ions om [−h, T] in o H.
Le C=C([−h, 0], H) be he space o all con inuous unc ions om [−h, 0] in o
Hwi h sup-no m kψkC= sup−h≤s≤0|ψ(s)|,ψ∈C,Lp
V=Lp([−h, 0]; V) and Lp
H=
Lp([−h, 0]; H).
Gi en a s ochas ic p ocess x( )∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)), we associa e
wi h an Lp
V∩C- alued s ochas ic p ocess x : Ω →Lp
V∩C, ≥0, by se ing x (s)(ω) =
x( +s)(ω), s∈[−h, 0].
The i s pu pose o his pape is o es ablish an exis ence and uniqueness esul o
a class o nonlinea s ochas ic pa ial unc ional di e en ial equa ions o he o m
(dx( ) = (A( , x( )) + ( , x ))d +g( , x )dw( ), ∈[0, T]
x( ) = ψ( ), ∈[−h, 0],(6)
whe e, in gene al, he ope a o s a e assumed o be nonlinea . In ac , we a e in e es ed in
he case in which A( , ·) : V→V0is a amily o nonlinea mono one and coe ci e ope a o s,
( , ·) : C→Hand g( , ·) : C→ L(K, H) a e Lipschi z con inuous. I is wo h poin ing
ou ha , in many applica ions, Ausually deno es a pa ial di e en ial ope a o (linea o
nonlinea ), while and ga e i s o de pa ial di e en ial ones (c . [3][16][17]). We will
i s es ablish he desi ed esul s by a a ia ional ype o a gumen , which is simila o
ha one ca ied ou by Pa doux’s [16] o a case wi hou delays, bu subjec o necessa y
changes o make ou scheme go h ough when ( , ·) : C→Hand g( , ·) : C→ L(K, H).
Then we will ea he mo e gene al case wi h ( , ·) : L2
V→Hand g( , ·) : L2
V→ L(K, H)
by using a Gale kin app oxima ion echnique.
5

3. EXISTENCE AND UNIQUENESS OF SOLUTIONS
Le A( , ·) : V→V0be a amily o (nonlinea ) ope a o s de ined a.e. . and p≥2. Assume
he ollowing hypo heses:
(a.1) Coe ci i y: ∃α > 0, λ, ν ∈R1such ha :
−2hA( , x), xi+λ|x|2+ν≥αkxkp,∀x∈V , a.e. .;
(a.2) Mono onici y:
−2hA( , x)−A( , y), x −yi+λ|x−y|2≥0,∀x, y ∈V , a.e. .;
(a.3) Boundedness: ∃γ > 0 :
kA( , x)k∗≤γkxkp−1,∀x∈V , a.e. .;
(a.4) Hemicon inui y:
θ∈R1→ hA( , x +θy), zi ∈ R1is con inuous ∀x, y, z ∈V , a.e. .;
(a.5) Measu abili y:
∈(0, T)→A( , x)∈V0is Lebesgue −measu able ∀x∈V , a.e. ..
Le ( , ·) : L2
H→Hbe a amily o nonlinea ope a o s de ined a.e. ., and sa is y:
( .1) ( , 0) = 0 ;
( .2) Lipschi z condi ion: ∃k1=k1(h)>0 such ha
| ( , η)− ( , ξ)| ≤ k1kη−ξkC,∀η, ξ ∈C , a.e. .;
( .3) Measu abili y: ∈(0, T)→ ( , η)∈His Lebesgue–measu able, ∀η∈L2
H.
And le g( , ·) : L2
H→ L(K, H) be ano he nonlinea ope a o amily de ined a.e. . and
sa is y:
(g.1) g( , 0) = 0 ;
(g.2) Lipschi z condi ion: ∃k2=k2(h)>0 such ha
kg( , η)−g( , ξ)k2≤k2kη−ξkC,∀η, ξ ∈C , a.e. .;
(g.3) Measu abili y: ∈(0, T)→g( , η)∈ L(K, H) is Lebesgue–measu able ∀η∈
L2
H.
Gi en an ini ial alue
ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) ,
6
he objec i e in his sec ion is ha unde he condi ions desc ibed abo e, we hope ully
ind a unique p ocess x( )∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) such ha













x( ) = ψ(0) + Z
0
[A(s, x(s)) + (s, xs)] ds
+Z
0
g(s, xs)dw(s), P −a.s., ∀ ∈[0, T],
x( ) = ψ( ), P −a.s., ∀ ∈[−h, 0].
(∗)
Rema k. (1) Fi s , i is wo h men ioning ha al hough he esul s can be p o ed o
p > 1, he in e es ing si ua ions in he applica ions appea when p≥2. Because o his,
we con en ou sel es wi h he analysis o he case p≥2.
(2). We obse e ha i x∈L2(0, T;C), hen in iew o ( .1)–( .3), (x)∈L2(0, T;H)
whe e (x)( ) = ( , x ).Mo eo e , he mapping x∈L2(0, T;C)7→ (x)∈L2(0, T;H)
is con inuous and so measu able. Since η∈C7→ ( , η)∈His con inuous a.e. ., i
ollows ha i x( ), ∈[−h, T] is an H– alued and F –adap ed s ochas ic p ocess, so is
( , x ), ≥0. In addi ion, i x∈L2(Ω ×(0, T); C), hen (x)∈L2(Ω ×(0, T); H) .
Finally, i xnis a bounded sequence in L2(Ω×(0, T); C), (xn) is bounded in L2(Ω×
(0, T); H) once again.
Simila esul s a e deduced om (g.1)–(g.3) o g:L2(0, T;C)→L2(0, T;L(K, H))
de ined by g(x)( ) = g( , x ).These ema ks imply ha he in eg als appea ing in (∗) a e
well de ined.
(3). In o de o a oid unnecessa y echnicali ies in he ollowing s abili y analysis, we
con en ou sel es wi h he conside a ion o Equa ion (∗) ins ead o a mo e gene al one.
Howe e , i is wo h poin ing ou ha unde some simila condi ions, i is possible o
ex end he esul s de i ed he e o mo e gene al s ochas ic sys ems in ol ing coe icien s
such as ( , x( ), x ) and g( , x( ), x ) as well as o emo e Condi ions ( .1), (g.1).
3.1. Uniqueness o solu ions
Now we shall p o e ha he e exis s a mos one solu ion o (∗). This esul will be deduced
mainly om (a.2) and I ˆo’s o mula.
Theo em 1. Assume he p eceding hypo heses hold. Then, he e exis s a mos one
solu ion o (∗)in Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) .
P oo . Suppose ha x, y ∈Ip(−h, T ;V)∩L2(Ω; C(−h, T;H)) a e wo solu ions o (∗).
Then, applying I ˆo’s o mula o (∗) and aking in o accoun (a.2), we ob ain
|x( )−y( )|2= 2 Z
0
hA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids
+ 2 Z
0
( (s, xs)− (s, ys), x(s)−y(s)) ds
7
+ 2 Z
0
(x(s)−y(s),(g(s, xs)−g(s, ys)) dw(s))
+Z
0
kg(s, xs)−g(s, ys)k2
2ds.
≤λZ
0
|x(s)−y(s)|2ds
+ 2 Z
0
|x(s)−y(s)|| (s, xs)− (s, ys)|ds
+ 2 Z
0
(x(s)−y(s),(g(s, xs)−g(s, ys)) dw(s))
+Z
0
kg(s, xs)−g(s, ys)k2
2ds.
Now, i ollows om ( .2) and (g.2) ha o any ∈[0, T ]
Esup
0≤s≤
|x(s)−y(s)|2≤(|λ|+ 1) Z
0
E|x(s)−y(s)|2ds + (k2
1+k2
2)Z
0
Ekxs−ysk2
Cds
+ 2Esup
0≤s≤ Zs
0
(x( )−y( ),(g( , x )−g( , y ))dw( )).
(7)
Howe e , by Bu kholde -Da is-Gundy’s inequali y, we ha e
Esup
0≤s≤ Zs
0
(x( )−y( ),(g( , x )−g( , y ))dw( ))
≤3Ensup
0≤s≤
|x(s)−y(s)|hZ
0
kg(s, xs)−g(s, ys)k2
2dsi1/2o
≤1
4Esup
0≤s≤
|x(s)−y(s)|2+KZ
0
Ekg(s, xs)−g(s, ys)k2
2ds
≤1
4Esup
0≤s≤
|x(s)−y(s)|2+K·k2
2Z
0
Ekxs−ysk2
Cds
(8)
o some posi i e cons an K > 0. On he o he hand, since x(s) = y(s) o s≤0, we
easily ge
Z
0
Ekxs−ysk2
Cds =Z
0
Esup
−h≤ ≤0
|xs( )−ys( )|2ds
=Z
0
Esup
−h≤ ≤0
|x(s+ )−y(s+ )|2ds
≤Z
0
Esup
0≤ ≤s
|x( )−y( )|2ds.
(9)
8
Thus, i ollows om (7)–(9)
Esup
0≤s≤
|x(s)−y(s)|2≤2h|λ|+1+k2
1+k2
2+2k2
2KiZ
0
Esup
0≤ ≤s
|x( )−y( )|2ds, ∀ ∈[0, T].
Now, G onwall’s lemma ob iously implies uniqueness.
Rema k. (1) Obse e ha i we assume he ollowing mono onici y hypo hesis
(a.2)’ Fo all ξ, η ∈Lp(−h, T;V) wi h ξ0=η0such ha
−2hA( , ξ( ))+ ( , ξ )−A( , η( )) − ( , η ), ξ( )−η( )i+λ|ξ( )−η( )|2
≥ kg( , ξ )−g( , η )k2
2a.e. ∈[0, T],
ins ead o (a.2), uniqueness would ha e been easily deduced. Indeed, no ice ha in his
case, I ˆo’s o mula and (a.2)’ imply
E|x( )−y( )|2≤λZ
0
E|x(s)−y(s)|2ds ∀ ∈[0, T],
o a bi a y wo solu ions x, y o he p oblem. Mo eo e , i is su icien o assume an
in eg al e sion o (a.2)’, namely,
(a.2)” Fo all ξ, η ∈Lp(−h, T;V) wi h ξ0=η0such ha
−2Z
0
hA(s, ξ(s))+ (s, ξs)−A(s, η(s)) − (s, ηs), ξ(s)−η(s)ids
+λZ
0
|ξ(s)−η(s)|2ds ≥Z
0
kg(s, ξs)−g(s, ηs)k2
2ds a.e. ∈[0, T].
(2) Con e sely, i is no di icul o p o e by ca ying ou simila compu a ions o he
ones in (8) ha (a.2), ( .2) and (g.2) imply (a.2)” (o cou se, wi h di e en pa ame e λ
om ha one in (a.2)).
3.2. Exis ence o s ong solu ions
Fi s o all, we s a e a heo em on exis ence and uniqueness o solu ions o s ochas ic
e olu ion equa ions. Nex , by means o his esul we will p o e he desi ed exis ence o
solu ion o (∗).
Theo em 2. Assume (a.1)–(a.5) hold wi h λ= 0 . Then, he e exis s a unique p ocess
x∈Ip(0, T;V)∩L2(Ω; C(0, T;H)) such ha
x( ) = ψ(0) + Z
0
[A(s, x(s)) + 1(s)] ds +M( ), P −a.s. , ∀ ∈[0, T],
9
By i ue o (a.2), we ge
−2EZT
0
hA(xn)−A(z), xn−zids +λE ZT
0
|xn−z|2ds ≥0 (36)
o all z∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H) . Ne e heless, (27), (30) and (31) allow
us o ake limi s in (36) and, i ollows
−2EZT
0
h −A(z), x −zids +λE ZT
0
|x−z|2ds ≥0.(37)
Now, i we se z=x−θz2( o θ > 0, z2∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H) ), we ge
−2EZT
0
h −A(x−θz2), θz2ids +λθ2EZT
0
|z2|2ds ≥0.(38)
In (38), we di ide by θ, ake limi as θ→0 and hen use he hemicon inui y (a.4) o
ob ain:
−EZT
0
h −A(x), z2ids ≥0,∀z2∈Lp(Ω ×(0, T); V)∩L2(Ω ×(0, T); H),(39)
and he e o e =A(x). Since (32) is ue wi h =A(x), he p oo o Theo em 3 is now
comple e.
4. EXISTENCE AND UNIQUENESS BY A GALERKIN APPROXIMATION
Fi s o all, we would like o poin ou ha in many si ua ions, i is con enien o conside
ano he no m k·kL2
Hins ead o k·kC o ini ial da um spaces in (∗). The a gumen s in
he las sec ion s ill ca y h ough when he no m k · kCis eplaced by k · kL2
Hin ( .2) and
(g.2). In his sec ion we shall in es iga e an exis ence and uniqueness esul o (∗) in a
mo e gene al si ua ion. P ecisely, le us assume hypo heses (a.1)–(a.5) o he amily o
ope a o s A( , ·). Suppose ( , ·) : L2
H→His a amily o nonlinea ope a o s de ined a.e. .
and sa is ying:
(F.1) ( , 0) = 0 ;
(F.2) Lipschi z condi ion: ∃k1=k1(h)>0 such ha
| ( , η)− ( , ξ)| ≤ k1kη−ξkL2
V,∀η, ξ ∈L2
V,a.e. .;
(F.3) Measu abili y: ∈(0, T)→ ( , η)∈His Lebesgue–measu able, ∀η∈L2
V.
And le g( , ·) : L2
H→ L(K, H) be ano he nonlinea ope a o amily de ined a.e. . and
sa is ying:
16

(G.1) g( , 0) = 0 ;
(G.2) Lipschi z condi ion: ∃k2=k2(h)>0 such ha
kg( , η)−g( , ξ)k2≤k2kη−ξkL2
V,∀η, ξ ∈L2
V,a.e. .;
(G.3) Measu abili y: ∈(0, T )→g( , η)∈ L(K, H) is Lebesgue–measu able ∀η∈
L2
V.
Theo em 4. In addi ion o (a.1)–(a.5), (F.1)–(F.3) and (G.1)–(G.3), suppose he wo
ollowing hypo heses hold:
(C) The e exis α > 0, λ, ν, τ ∈R1such ha o all ξ∈Lp(−h, T;V)
−2hA( , ξ( ))+ ( , ξ ), ξ( )i+λ|ξ( )|2+τkξ0k2
Lp
V+ν
≥αkξ( )kp+kg( , ξ )k2
2a.e. ∈[0, T];
(M) Fo all ξ, η ∈Lp(−h, T;V)wi h ξ0=η0,
−2hA( , ξ( )) + ( , ξ )−A( , η( )) − ( , η ), ξ( )−η( )i+λ|ξ( )−η( )|2
≥ kg( , ξ )−g( , η )k2
2a.e. ∈[0, T].
Then, o each ψ∈Ip(−h, 0; V)∩L2(Ω; C(−h, 0; H)) he e exis s a unique solu ion o
he p oblem (∗)in Ip(−h, T;V)∩L2(Ω; C(−h, T ;H)).
P oo . Uniqueness ollows immedia ely om I ˆo’s o mula, Assump ion (M) and G onwall’s
lemma. Indeed, le x, y ∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)) be wo solu ions o (∗).
Then, i is easy o ob ain
E|x( )−y( )|2= 2 Z
0
EhA(s, x(s)) −A(s, y(s)), x(s)−y(s)ids
+ 2 Z
0
E( (s, xs)− (s, ys), x(s)−y(s)) ds
+Z
0
Ekg(s, xs)−g(s, ys)k2
2ds.
≤λZ
0
E|x(s)−y(s)|2ds,
om which uniqueness ollows by means o G onwall’s lemma.
As o he exis ence, we shall spli he p oo in o he ollowing ou s eps.
STEP 1. Fini e-dimensional app oxima ion
Le { 1, 2, ..., n, ...}be an o hono mal basis o Hwhe e i∈V o all i≥1. Le
Vn=Hn=V0
ndeno e he ec o space gene a ed by { 1, ... n}. Le Pn∈ L(H, Hn) be
17
he o hogonal p ojec ion om Hon o Hn. Then, Pncan be ex ended o an ope a o ˜
Pn
om V0on o V0
nin he ollowing way
˜
Pnu=
n
X
i=1
hu, ii i, u ∈V0.
Le {l1, l2, ..., ln, ...}deno e an o hono mal basis in K, and le πn∈ L(K, Kn) be he
p ojec ion om Kon o Kn= span{l1, ..., ln}.
Now we conside he p oblem
(∗1) 




d(xn( ), i) = hA( , xn( )) + ( , xn
), iid
+ ( i, g( , xn
)d(πnw( ))),1≤i≤n,
xn( ) = Pnψ( ), ∈[−h, 0].
This equa ion can be ew i en in an equi alen way as ollows. Le An( , ·) deno e he
amily o ope a o s om Vnin o V0
nde ined as An( , x) = ˜
PnA( , x), x ∈Vn. Assume
n( , ·) : L2
Hn→Hngi en by n( , ξ) = Pn ( , ξ) o ξ∈L2
Hn,gn( , ·) : L2
Hn→ L(Kn, Hn)
de ined by gn( , ξ) = Png( , ξ) o ξ∈L2
Hn, and, inally le Wn( ) deno e he Kn– alued
Wiene p ocess de ined by Wn( ) = πnw( ). Then, Eq. (∗1) can be ew i en as
(∗2) (dxn( ) = (An( , xn( )) + n( , xn
)d +gn( , xn
)dWn( )
xn( ) = ψn( ) = Pnψ( ), ∈[−h, 0].
Al hough Eq. (∗2) can be conside ed as an I ˆo s ochas ic di e en ial equa ion in Rn, we
can no apply he classic esul s on exis ence and uniqueness o solu ions since Andoes no
sa is y a Lipschi z ype o condi ion. Howe e , we can apply o his si ua ion he esul s
p o ed in he p eceding sec ion, i.e., Theo em 3 wi h k · kC eplaced by k · kL2
Hin ( .2) and
(g.2). Indeed, i is easy o check ha An, n, gn, Wnand ψnsa is y he assump ions in
Theo em 3 by eplacing V,H,V0by Vn,Hn,V0
n. The e o e, o each na u al numbe n≥1,
he e exis s a unique xn∈Ip(−h, T;Vn)∩L2(Ω; C(−h, T;Hn)) which is he solu ion o
(∗2). Owing o he na u al injec ions, we ha e ha , in ac ,
xn∈Ip(−h, T;V)∩L2(Ω; C(−h, T;H)).
STEP 2. A p io i compu a ions
As in he p oo o Theo em 3, we se xn:= xn(s), An(xn) := An(s, xn(s)), n(xn) :=
n(s, xn
s) and gn(xn) := gn(s, xn
s).
The ene gy equali y implies
|xn( )|2=|ψn(0)|2+ 2 Z
0
hAn(xn) + n(xn), xnids
+ 2 Z
0
(xn, gn(xn)dWn(s)) + hhZ.
0
gn(xn)dWn(s)ii ,
(40)
18
and consequen ly,
|xn( )|2=|ψn(0)|2+ 2 Z
0
hA(xn) + (xn), xnids
+ 2 Z
0
(xn, g(xn)dWn(s)) + hhZ.
0
g(xn)dWn(s)ii .
(41)
On he one hand, since
hhZ.
0
g(xn)dWn(s)ii =Z
0
kg(xn)πnk2
2ds ≤Z
0
kg(xn)k2
2ds,
we immedia ely ge om Condi ion (C) ha
|xn( )|2≤ |ψn(0)|2+ 2 Z
0
hA(xn) + (xn), xnids
+ 2 Z
0
(xn, g(xn)dWn(s)) + Z
0
kg(xn)k2
2ds
≤ |ψn(0)|2+ν + τkψnk2
L2
Hn
+λZ
0
|xn|2ds
−αZ
0
kxnkpds + 2 Z
0
(xn, g(xn)dWn(s)),
(42)
which, a e aking expec a ions, yields ha
E|xn( )|2+αZ
0
Ekxnkpds ≤E|ψ(0)|2+ν + τEkψk2
L2
H+λZ
0
E|xn|2ds. (43)
Consequen ly, he e exis posi i e cons an s c1, c2such ha
sup
−h≤ ≤T
E|xn( )|2≤c1(44)
EZT
0
kxn( )kpd ≤c2,(45)
and, as p≥2, he e exis s c3>0 such ha
EZT
0
kxn( )k2d ≤c3.(46)
On he o he hand, (42) immedia ely yields ha
Esup
0≤ ≤T
|xn( )|2≤E|ψ(0)|2+νT +TτEkψk2
L2
H+|λ|ZT
0
E|xn( )|2d
+ 2Esup
0≤ ≤TZ
0
(xn, g(xn)dWn(s)).
(47)
19
E alua ing he las e m in (47) by applying Bu kholde -Da is-Gundy’s inequali y (c . see
[12]), (G.2) and aking in o accoun (46), we ha e
2Esup
0≤ ≤TZ
0
(xn, g(xn)πndw(s))≤6E
 ZT
0
|xn|2kg(xn)k2
2ds!1/2

≤1
3Esup
0≤ ≤T
|xn( )|2+c4ZT
0
Ekg(xn)k2
2ds
≤1
3Esup
0≤ ≤T
|xn( )|2+c4k2
2ZT
0
Ekxn
sk2
L2
Vds
≤1
3Esup
0≤ ≤T
|xn( )|2+c5+c6Ekψk2
L2
V.
(48)
Hence, he e exis s a posi i e cons an c7such ha
Esup
0≤ ≤T
|xn( )|2≤c7.(49)
So, we ha e inally p o ed ha
{xn}n≥1is bounded in Ip(−h, T;V)∩L2(Ω; C(−h, T;H)),
{A(xn)}n≥1is bounded in Lp0(Ω ×(0, T); V0),
{ (xn)}n≥1is bounded in L2(Ω ×(0, T); H),
{g(xn)}n≥1is bounded in L2(Ω ×(0, T); L(K, H)),
whe e A(xn), (xn), g(xn) a e de ined in he ob ious way and p0deno es he conjuga e o
p.
STEP 3. Taking weak limi s
Owing o he las asse ions in S ep 2, we can ensu e ha he e exis s a subsequence
{xnk}o {xn}such ha
xnk* x in Ip(−h, T;V) and weakly s a in L2(Ω; L∞(−h, T;H)),
xnk(T)* ξ in L2(Ω; H),
A(xnk)* χ in Lp0(Ω ×(0, T); V0),
(xnk)* σ in L2(Ω ×(0, T); H),
g(xnk)* ζ in L2(Ω ×(0, T); L(K, H)).
Le θ:R1→R1be de ined as θ( ) = 0 i < 0
1 i ≥0. I ϕis a unc ion om [0, T] in o R1,
we can de ine ano he unc ion ϕ: (−ρ, T +ρ)→R1(whe e ρis a posi i e ixed numbe )
in he ollowing way:
ϕ( ) = nϕ( ) i ∈[0, T]
0 o he wise.
20
This pe mi s us o ew i e Eq. (∗1) (wi h n=nk) as ollows
(xnk( ), i) = (ψ(0), i)θ( )−(xnk(T), i)θ( −T)
+Z
0
hA(xnk) + (xnk), iids
+Z
0
( i, g(xnk)πnkdw(s)),∀ ∈(−ρ, T +ρ), i = 1, ..., nk.
(50)
Obse e ha , as he map φ∈L2(Ω×(0, T); L2(K, H)) 7→ R.
0φ(s)dw(s)∈L2(Ω×(0, T); H)
is linea and con inuous, hen i is weakly con inuous (whe e L2(K, H) deno es he space
o all Hilbe -Schmid ope a o s om Kin o H). Now, we shall p o e ha g(xnk)πnk* ζ,
as k→ ∞, in L2(Ω ×(0, T); L2(K, H)). Indeed, his con e gence is equi alen o
EZT
0
(Q∗g(xnk)πnk)d →EZT
0
(Q∗ζ)d ,
o all Q∈L2(Ω ×(0, T); L2(K, H)), and also o
EZT
0
(g(xnk)πnkQ)d →EZT
0
(ζQ)d .
The e o e, i is su icien o p o e ha Qπnk→Qin L2(Ω ×(0, T); L2(K, H)). Bu his
is an immedia e consequence o Theo em I. 2.3 in Pa doux [16].
Now, we can ake weak limi s in (50) and ob ain:
(x( ), i) = (ψ(0), i)θ( )−(ξ, i)θ( −T) + Z
0
hχ+σ, iids
+Z
0
( i, ζ)dw(s),∀ ∈(−ρ, T +ρ),∀i≥1,
(51)
so i ollows ha
ξ=x(T)
dx( )=(χ( ) + σ( ))d +ζ( )dw( ), ∈[0, T],(52)
x( ) = ψ( ), ∈[−h, 0].(53)
The e o e, i emains o p o e ha χ+σ=A(x) + (x) and ζ=g(x). This will be done
in he nex s ep.
STEP 4. Final s ep: he mono onici y me hod
Conside ∈Lp(Ω ×(−h, T); V)∩L2(Ω ×(−h, T); H) and se
unk=−2EZT
0
e−λ hA(xnk) + (xnk)−A( )− ( ), xnk− id
+λE ZT
0
e−λ |xnk− |2d −EZT
0
e−λ kg(xnk)−g( )k2
2d .
(54)
21

No e ha unk≥0 due o Assump ion (M). On he o he hand, we can ake limi s in he
e ms o (54) excep o he ollowing e m
ynk=−2EZT
0
e−λ hA(xnk) + (xnk), xnkid
+λE ZT
0
e−λ |xnk|2d −EZT
0
e−λ kg(xnk)k2
2d .
(55)
Bu , (41) immedia ely yields ha
E|xnk( )|2=E|Pnkψ(0)|2+ 2EZ
0
hA(xnk) + (xnk), xnkids
+E hhPnkZ.
0
g(xnk)dWnkii .
(56)
In pa icula , (56) p o es ha he unc ion 7→ E|xnk( )|2is absolu ely con inuous and
hence
de−λ E|xnk( )|2+λe−λ E|xnk( )|2=e−λ dE|xnk( )|2.(57)
Now, i can be ob ained ha
e−λT E|xnk(T)|2≤E|Pnkψ(0)|2−λZT
0
e−λ E|xnk( )|2d
+ 2 ZT
0
e−λ EhA(xnk) + (xnk), xnkid
+ZT
0
e−λ Ekg(xnk)k2
2d ,
(58)
and he e o e,
ynk≤E|ψ(0)|2−e−λT E|xnk(T)|2.(59)
As an immedia e consequence, i ollows ha
lim sup
k→∞
ynk≤E|ψ(0)|2−e−λT E|x(T)|2.(60)
Applying I ˆo’s o mula o Eq. (52), we can ge
e−λT E|x(T)|2=E|ψ(0)|2−λZT
0
e−λ E|x|2d
+ 2 ZT
0
e−λ Ehχ+σ, xid +ZT
0
e−λ Ekζk2
2d .
(61)
So
lim sup
k→∞
ynk≤ZT
0
e−λ E2hχ+σ, xi+λ|x|2− kζk2
2d , (62)
22
and inally
0≤lim sup
k→∞
unk≤ − 2EZT
0
e−λ hχ+σ−A( )− ( ), x − id
+λE ZT
0
e−λ |x− |2d −EZT
0
e−λ kζ−g( )k2
2d .
(63)
I we ake =xin (63), i ollows ha ζ=g(x) and, also
−2EZT
0
e−λ hχ+σ−A( )− ( ), x − id +λE ZT
0
e−λ |x− |2d ≥0.(64)
In o de o inish he p oo , we only need o use hemicon inui y (a.4). Indeed, we no ice
ha he unc ion also sa is ies a simila p ope y and i is easy o deduce om (F.2)
ha he map θ∈R17→ ( ( , η +θξ), x)∈R1is con inuous o all η, ξ ∈L2
V, x ∈
Hand a.e. ∈[0, T]. Now, in (64) se ing =x−θu o θ > 0 and u∈Lp(Ω ×
(−h, T); V)∩L2(Ω ×(−h, T) : H), di iding by θand le ing θ end o 0, we hen ge
∀u∈Lp(Ω ×(−h, T); V)∩L2(Ω ×(−h, T); H)
−2EZT
0
e−λ hχ+σ−A(x)− (x), uid ≥0.(65)
Consequen ly, χ+σ=A(x) + (x) and he p oo o he heo em is comple e.
5. STABILITY OF STRONG SOLUTIONS
In his sec ion we shall show ha unde sui able condi ions exponen ial s abili y can be
ans e ed om equa ions wi hou ime lags o hose wi h ime lag ones. Since we a e
mainly in e es ed in exponen ial s abili y p oblems o he second momen o solu ions, we
will assume he e exis s a p ocess
x∈I2(−h, T;V)∩L2(Ω; C(−h, T;H)) ,∀T > 0,
which is he s ong solu ion o he ollowing p oblem:
(dx( ) = [A( , x( )) + ( , x )] d +g( , x )dw( ), ≥0,
x( ) = ψ( ), ∈[−h, 0].(66)
In o he wo ds, x( ) sa is ies he ollowing in eg al equa ion (in V0):
x( ) = ψ(0) + Z
0
[A(s, x(s)) + (s, xs)]ds
+Z
0
g(s, xs)dw(s), P −a.s., ≥0,
(67)
23
and x( ) = ψ( ), ∈[−h, 0]. In pa icula , in his sec ion we suppose all he condi ions
in Sec ion 3 hold so ha he e exis s a unique s ong solu ion o he s ochas ic unc ional
di e en ial equa ion (66). Fo simplici y, we also suppose in he sec ion ha he coe icien s
A, and ga e con inuous wi h espec o ime .
Fi s o all, we in es iga e he case wi hou he edi a y cha ac e is ics. In o he wo ds,
conside Eq. (67) wi h h= 0 and hus k1(h) = k1>0, k2(h) = k2>0 in ( .2), (g.2), hen
he equa ion (66) educes o
(dx( ) = [A( , x( )) + ( , x( ))] d +g( , x( )) dw( ), ≥0,
x(0) = x0.(68)
I i is possible o know he exis ence o some Lyapuno unc ion, we could ob ain
mean squa e s abili y o solu ions. Indeed, assume he e exis ∈C2(H;R+) and posi i e
cons an s ci,1≤i≤4 , such ha 0(x)∈V o all x∈Vand
c1|x|2≤ (x)≤c2|x|2,L (x)≤ −c3 (x),| 0(x)| ≤ c4|x|,
o all x∈V, whe e Lis he associa ed di usion ope a o de ined as
L (x) = hA( , x) + ( , x), 0(x)i+1
2 [ 00(x)g( , x)Wg∗( , x)],∀x∈V,
we can ge (applying I ˆo’s o mula o unc ion ec3 (x), x∈Hand Equa ion (68))
ec3 (x( )) = (x(0)) + c3Z
0
ec3s (x(s)) ds
+Z
0
ec3shA(s, x(s)) + (s, x(s)), 0(x(s))ids
+Z
0
ec3s( 0(x(s)), g(s, x(s)) dw(s))
1
2Z
0
ec3s [ 00(x(s))g(s, x(s))Wg∗(s, x(s))] ds.
Taking expec a ions and obse ing ha L (x)≤ −c3 (x) , we ha e
ec3 E (x( )) ≤E (x(0)) + c3Z
0
ec3sE (x(s)) ds +Z
0
ec3sEL (x(s)) ds
≤E (x(0)),
and consequen ly
E (x( )) ≤e−c3 E (x(0)) ,∀ ≥0.
¿F om he assump ions on , we easily deduce ha
E|x( )|2≤c2
c1
e−c3 E|x(0)|2,∀ ≥0
24
which means mean squa e exponen ial s abili y o he i ial solu ion o (68).
Al hough, as we ha e men ioned be o e, he cons uc ion o Lyapuno unc ions is
no , in gene al, a i ial p oblem, he e exis s a condi ion ha makes (x) = |x|2become
a na u al Lyapuno unc ion. This is he ollowing hypo hesis:
(H): he e exis s a posi i e cons an γ > 0 such ha
2hA( , x) + ( , x), xi+kg( , x)k2
2≤ −γ|x|2,∀x∈V.
Indeed, on his occasion
L (x) = 2hA( , x) + ( , x), xi+kg( , x)k2
2
≤ − γ|x|2,
he e o e, se ing c3=γ, we ob ain exponen ial s abili y in mean squa e sense.
Rema k. Obse e ha in a a ie y o p ac ical si ua ions, he ollowing assump ion (H)0
(which seems easie o check) implies (H):
(H)0: he e exis s a posi i e cons an α > 0 such ha
−2hA( , x), xi ≥ αkxk2,∀x∈Vand −α+ 2k1β2+k2
2β2<0,
whe e k1,k2bo h a e nonnega i e cons an s in ( .2), (g.2) and β > 0 deno es
he cons an sa is ying
|x| ≤ βkxk,∀x∈V .
Indeed, no e ha
2hA( , x) + ( , x), xi+kg( , x)k2
2
≤ − αkxk2+ 2( ( , x), x) + [g( , x)Wg∗( , x)]
≤ − αkxk2+ 2| ( , x)||x|+ [g( , x)Wg∗( , x)]
≤ − αkxk2+ 2k1β2kxk2+k2
2β2kxk2
≤[−α+ 2k1β2+k2
2β2]β−2|x|2,
and deno e γ= [α−2k1β2−k2
2β2]β−2, he assump ion (H) holds.
In wha ollows, we shall show ha he same hypo heses as abo e (mainly ( .2),(g.2)
and (H)0) imply mean squa e exponen ial s abili y o he i ial solu ion o he s ochas ic
unc ional di e en ial equa ion (66). Howe e , i is pa icula ly wo h poin ing ou ha on
his occasion he cons an s k1,k2a e gene ally dependen on he ime lag cons an h > 0.
This ac simply means ha in o de o ob ain exponen ial s abili y, he ime lag mus be
su icien ly small. Howe e , as will be shown by Examples 1, 2 below, on some occasions
such as he ime delay case, he cons an k1o k2could be independen on h > 0 so ha
he s abili y is ue o any h > 0, a esul which imp o es ha o Mao and Shah [13] in
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