Stochastic Functional Partial Differential Equations: Existence, Uniqueness and Asymptotic Decay Property

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 25