Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

EXPONENTIALLY STABLE STATIONARY SOLUTIONS FOR STOCHASTIC EVOLUTION EQUATIONS AND THEIR PERTURBATION TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ¨ ORN SCHMALFUSS Abs ac . We conside he exponen ial s abili y o s ochas ic e olu ion equa- ions wi h Lipschi z con inuous non-linea i ies when ze o is no a solu ion o hese equa ions. We p o e he exis ence o a non- i ial s a iona y solu ion which is exponen ially s able, whe e he s a iona y solu ion is gene a ed by he composi ion o a andom a iable and he Wiene shi . We also cons uc s a iona y solu ions wi h he s onge p ope y o a ac ing bounded se s uni- o mly. The exis ence o hese s a iona y solu ions ollows om he heo y o andom dynamical sys ems and hei a ac o s. In addi ion, we p o e some pe u ba ion esul s and o mula e condi ions o he exis ence o s a iona y solu ions o semi-linea s ochas ic pa ial di e en ial equa ions wi h Lipschi z con inuous non-linea i ies. 1. In oduc ion The exponen ial s abili y o s ochas ic pa ial di e en ial equa ions is an impo an p oblem, and i has ecei ed conside able a en ion du ing he ecen decades as he as li e a u e on his opic shows. Ou aim he e is o s udy he exponen ial s abili y o non- i ial s a iona y solu ions o hese equa ions. The in es iga ion o s abili y o cons an s a iona y solu ions o ini e dimensional s ochas ic di e en ial equa ions goes back o Has ´minski˘ı [14] using Lyapuno unc- ions o he gene a o o he Ma ko semi-g oup. These ideas ha e been ex ended by Mao [20] in he ini e dimensional con ex , while non-cons an s a iona y solu- ions ha e been ea ed in Schmal uß [24]. He e we will gene alize some echniques om hese las wo publica ions. Fo in ini e dimensional (pa abolic) s ochas ic di e en ial equa ions he p oblem o exponen ially s able cons an s a iona y solu ions has been conside ed by Ca aballo and Real [5] (see also [3], [4]), Liu and Mao [19], Chow [7], Haussmann [15] and Ichikawa [16] among o he s. In con as o hese cons an s a iona y solu ions we will in es iga e he asymp- o ic exponen ial s abili y o non- i ial s a iona y solu ions. In his espec , we will conside semilinea s ochas ic e olu ion equa ions wi h Lipschi z con inuous non-linea i ies. Unde sui able assump ions we p o e he exis ence o a unique s a- iona y solu ion by using a ixed poin a gumen based on he pullback echnique. This s a iona y solu ion u ns o be exponen ially s able in mean squa e, and also Da e: June, 2004. Key wo ds and ph ases. andom dynamical sys ems, s a iona y solu ions, exponen ial s abili y, s abiliza ion. 1 2 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS in he almos su e sense. Al hough we p o e almos su e con e gence o he s a iona y solu ion, he excep- ional se s depend on he ini ial condi ion, so i is no possible o conside uni o m con e gence wi h espec o a bounded se o ini ial condi ions. Howe e , his p ob- lem will be o e come by using a echnique om andom dynamical sys ems which allows us o s udy he p oblem o excep ional se s independen o he se o ini ial condi ions. I is unknown in gene al i s ochas ic pa ial di e en ial equa ions wi h gene al di usion coe icien s gene a e andom dynamical sys ems. Howe e , i we suppose some kind o commu a i i y on hese coe icien s, hen we a e able o p o e he exis ence o such a andom dynamical sys em. We p o e he exis ence o a andom ixed poin , which is in ac a andom a iable. This andom a iable gene a es he exponen ially s able s a iona y solu ion o he s ochas ic pa ial di e en ial equa ion. Mo eo e , his s a iona y solu ion a ac s bounded se s o ini ial condi ions. S a iona y solu ions in his in e p e a ion co espond wi h single poin andom a ac o s, an impo an objec in he heo y o andom dynamical sys ems, see C auel, Debussche and Flandoli [8], Flandoli and Schmal uß [12] o [23]. Ano he aim o his pape is o analyze pe u ba ions o s ochas ic pa ial di e - en ial equa ions and he ela ion be ween hei s a iona y solu ions. These pe - u ba ions will be gi en by modi ying he non-linea pa o he equa ion. Unde he assump ion ha he pe u ba ions app oach he non-linea pa o he o iginal equa ion and ha he Lipschi z cons an s o he pe u bed ope a o s a e uni o mly bounded and no oo la ge, we ob ain he exis ence o s a iona y solu ions which con e ge o he s a iona y solu ion o ou o iginal p oblem in he mean squa e sense. Fo omega-wise con e gence we o mula e a heo em on he con inuous dependence o andom ixed poin s on a pa ame e . The pape is o ganized as ollows. In Sec ion 2 we in oduce basic concep s o s ochas ic e olu ion equa ions and andom dynamical sys ems. The hi d sec ion deals wi h he exponen ial s abili y o s a iona y solu ions o s ochas ic e olu ion equa ions in he mean squa e sense and almos su ely. In he nex sec ion, his exponen ial s abili y is analyzed om he poin o iew o andom dynamical sys- ems. Then in Sec ion 5, we conside he pe u ba ion p oblems, and illus a e he esul s wi h an example in he inal sec ion. 2. Random dynamical sys ems and s ochas ic e olu ion equa ions In his sec ion we will desc ibe he concep o exponen ially s able s a iona y so- lu ions o s ochas ic non-linea e olu ion equa ions gene a ed by andom ixed poin s. To do his we s a by desc ibing he noise d i ing he di e en ial equa ion. Le (Ω,F,{F } ∈R,P) be a il e ed p obabili y space such ha Fs⊂ F ⊂ F o s≤ . In wha ollows we will conside a wo-sided Wiene p ocess Wwi h alues in some sepa able Hilbe space Uwhe e he co a iance Qis a symme ic ope a o on Uo ace class. Fo ins ance, o he abo e p obabili y space we will choose o Ω he 3 se o con inuous pa hs C0(R, U) which a e ze o a ze o equipped wi h he compac open opology. Fis supposed o be he associa ed Bo el-σ-algeb a and Pis de ined o be he Wiene measu e wi h espec o he co a iance Q. Fo F we se σ{ω(u)−ω( ) : , u ≤ }. Wha we ha e in oduced is he B ownian mo ion me ic dynamical sys em which is he s anda d noise o andom dynamical sys ems gene a ed by s ochas ic di e - en ial equa ions. No e ha he abo e p obabili y space is no comple ed. The comple ion o his p obabili y space is deno ed by (Ω,¯ F,{¯ F } ∈R,P) whe e{¯ F } ∈Rhas o be a no - mal il a ion, see Da P a o and Zabczyk [10] Page 75. We now in oduce on he abo e non-comple ed p obabili y space a measu able low θ={θ } ∈Ron Ω: (1) θ: (R×Ω,F ⊗ B(R)) →(Ω,F), θ +τ=θ ◦θτ, , τ ∈R, θ0= idΩ. The Wiene shi ope a o s which o m he low θ θ ω(·) = ω(·+ )−ω( ), ∈R, ω ∈Ω lea e he Wiene measu e Pin a ian . Mo e p ecisely, Pis e godic wi h espec o θ. In addi ion, wi h espec o he il a ion we ha e ha (2) θ−1 uF =F +u o any , u ∈R, see A nold [1] Page 72. Since he abo e p obabili y space is canonical we ha e o a Wiene p ocess and i s shi ope a o W( , ω) = ω( ), W ( , θsω) = ω( +s)−ω(s) = W( +s, ω)−W(s, ω). I is impo an o no e ha he measu abili y in (1) is no ue i we eplace F by i s comple ion, see A nold [1] Appendix A3. Howe e , o ixed we ha e he measu abili y o θ : (Ω,¯ F)→(Ω,¯ F). Mo eo e , he mapping R3 →θ ω∈C(R, U) is con inuous o a ixed ω∈Ω. We will s udy he quali a i e beha iou o s ochas ic e olu ion equa ions on some sepa able Hilbe space Hwhich ha e he o m (3) dX =AXd + (X)d +B(X)dW, X(0) = u0, whe e Wis he Wiene p ocess on he p obabili y space (Ω,¯ F,{¯ F } ∈R,P) in o- duced abo e. Assume ha he e exis s a Gel and iple V⊂H⊂V0o sepa able Hilbe spaces, whe e V0deno es he dual o V(see Temam [25] Page 55 o mo e de ails). We deno e by k · k,k · kV he no ms in Hand V espec i ely. The inne p oduc in Hwill be deno ed by (·,·), and he duali y mapping be ween V0and V by h·,·i. The andom a iable u0is supposed o be ( ¯ F0,B(H)) measu able. Le us deno e by a1>0 he cons an o he injec ion V⊂H, i.e. a1kuk2≤ kuk2 V, o ∈V, 4 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS and le −A:V→V0be a posi i e, linea and con inuous ope a o o which he e exis s a a2<0 such ha h−Au, ui ≥ −a2kuk2 V, o all u∈V. Then, i is well known (see, o ins ance, Dau ay and Lions [9]) ha Ais he gene a o o a s ongly con inuous semig oup {S( )} ≥0in Hsa is ying ha (4) kS( )kL(H)≤ea , whe e a=a1a2<0. The ope a o is supposed o be Lipschi z con inuous om H o H: k (u1)− (u2)k ≤ L ku1−u2k, u1, u2∈H, and Bis supposed o be Lipschi z con inuous wi h espec o he Hilbe -Schmid no m LQ 2(U, H) o linea ope a o s om U o H: H((B(u1)−B(u2))Q(B(u1)−B(u2))∗=: kB(u1)−B(u2)k2 LQ 2 ≤LBku1−u2k2 o u1, u2∈H. We now need he spaces L2,s := L2(Ω,¯ Fs,P;H), s ∈R. We ha e he ollowing heo em abou he exis ence, uniqueness and egula i y o (3). Theo em 2.1. Suppose ha u0∈L2,0. Then (3) has a unique (up o equi alence) mild solu ion X(·)on [0,∞)which has a con inuous e sion. In addi ion, EZT 0 kX( )k2 Vd < ∞, and Esup ∈[0,T ] kX( )k2<∞, o any T≥0. Fo he exis ence o a mild solu ion see Da P a o and Zabczyk [10] Theo em 7.4. The egula i y asse ion can be ound in K ylo and Rozo skii [17] Chap e 2. We a e in e es ed in s a iona y solu ions ha a e exponen ially a ac ing in he L2sense o almos su ely. S a iona y solu ion means ha he ini e dimensional dis ibu ions o he solu ion Xa e independen o shi s wi h espec o . These exponen ially s able s a iona y solu ions will be gene a ed by exponen ially a - ac ing ixed poin s gi en by an ( ¯ F0,B(H))-measu able andom a iable X∗wi h alues in Hsuch ha i we choose he ini ial condi ion u0(ω) = X∗(ω) we ha e X( , ω) = X∗(θ ω) almos su ely o all ≥0,whe e he excep ional se may de- pend on . This ixed poin is said o be exponen ially a ac ing i he p ocess ( , ω)→X∗(θ ω) (o a e sion o his p ocess) a ac s he solu ion o (3) o any (app op ia e) ini ial condi ion exponen ially as in he L2-sense o almos su ely. By he θ in a iance o Pwe ha e ha P(X∗(θ 1ω)∈B1,· · · , X∗(θ nω)∈Bn) =P(X∗(θ 1+ ω)∈B1,· · · , X∗(θ n+ ω)∈Bn) o ≥0,0≤ 1< 2<· · · , nand B1,· · · , Bn∈ B(H). 5 Ano he ool ha can be used o desc ibe he s abili y beha iou o a s ochas ic e olu ion equa ion a e andom dynamical sys ems. A comp ehensi e p esen a ion can be ound in A nold [1]. A andom dynamical sys em is gi en by a measu able mapping φ: (R+×Ω×H, B(R+)⊗ F ⊗ B(H)) →(H, B(H)), sa is ying he cocycle p ope y: φ( +τ, ω, x) = φ( , θτω, φ(τ, ω, x)), , τ ∈R+, ω ∈Ω, x ∈H, φ(0, ω, x) = x, (5) whe e θis he low o shi ope a o s (Wiene shi ) in oduced abo e. We emphasize ha (5) has o be sa is ied o any ω∈Ω. Howe e i is su icien o eplace Ω by a {θ } ∈R-in a ian se o ull measu e. Ou side o his in a ian se we can ede ine φby he iden i y mapping on H. La e on we will eplace Ω by a smalle {θ } ∈R-in a ian se Ω0∈ F. The measu abili y o (1) emains ue i we eplace Fby i s ace σ-algeb a. The mapping φis ela ed o he solu ion o a s ochas ic o andom di e en ial equa ion. Fo he ollowing we will always suppose ha he mapping H3x7→ φ( , ω, x)∈H is con inuous o any , ω. Al hough i is known ha ini e dimensional s ochas ic di e en ial equa ions gen- e a e andom dynamical sys ems (see A nold [1] Chap e 1), his is no ue in gene al o in ini e dimensional equa ions. Howe e , o pa icula kinds o noise we can apply he ollowing simple lemma o ob ain a andom dynamical sys em. Lemma 2.2. Le φbe a andom dynamical sys em. Suppose ha he mapping T: Ω ×H→Hhas he ollowing p ope ies: Fo ixed ω∈Ω he mapping T(ω, ·) is a homeomo phism on H. Fo ixed x∈H he mappings T(·, x), T−1(·, x)a e measu able. Then he mapping (6) ( , ω, x)→T−1(θ ω, φ( , ω, T(ω, x))) =: ψ( , ω, x) sa is ies (5). Hence ψis a andom dynamical sys em. The measu abili y o ψ ollows because o he p ope ies o T. La e on we will ans o m a s ochas ic e olu ion equa ion con aining a noise e m in o an e olu ion equa ion wi hou noise bu wi h andom coe icien s. A andom a iable Yon (Ω,F,P) wi h alues in His called empe ed i (7) lim →±∞ log+kY(θ ω)k | |= 0, o equi alen ly i → kY(θ ω)khas a sub-exponen ial g ow h o → ±∞, in o he wo ds, o ε > 0 and ω∈Ω he e exis s a 0(ε, ω)≥0 such ha o | | ≥ 0(ε, ω) i holds (8) kY(θ ω)k ≤ eε| |, which means ha he exponen ial g ow h a e o → kY(θ ω)kis ze o. 6 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS Le ω→G(ω) be a se alued mapping om Ω in o he space o non-emp y closed subse s om H. Such a mapping is called a andom se i o any y∈H he mapping ω→in x∈G(ω)kx−yk is a andom a iable. I Gis a andom se hen he e exis s a andom a iable g wi h g(ω)∈G(ω) (see Cas aing and Valadie [6] Chap e 3). A andom se Gis called empe ed i he andom a iable ω→sup x∈G(ω) kxk is empe ed. I is easily seen ha he se o ω o which (7), (8) a e sa is ied is {θ } ∈T-in a ian . An (F,B(H))-measu able andom a iable X∗is called a andom ixed poin in he sense o andom dynamical sys ems i φ( , ω, X∗(ω)) = X∗(θ ω) o > 0, whe e ωis con ained in a {θ } ∈T-in a ian se o ull measu e. X∗is an exponen ially s able andom ixed poin wi h espec o a closed andom se G con aining X∗i o any andom a iable g∈Gwe ha e lim →∞ kφ( , ω, g(ω)) −X∗(θ ω)k= 0 o all ωin he abo e {θ } ∈T-in a ian se o ull measu e wi h exponen ial speed, such ha he excep ional se is independen o . I φis de ined by he solu ion mapping o a s ochas ic/ andom di e en ial equa ion hen ( , ω)7→ X∗(θ ω) is a s a iona y solu ion o a s ochas ic/ andom di e en ial equa ion. Ou s a egy will be o p o e ha , unde ce ain assump ions, a andom dynamical sys em possesses a andom a ac o which is a single ( andom) poin . The ollowing de ini ion can be ound in Flandoli and Schmal uß [12]. De ini ion 2.3. Le Dbe he se o all closed empe ed andom se s in H. A compac andom se A ∈ D is called a andom a ac o i he in a iance p ope y (9) φ( , ω, A(ω)) = A(θ ω) is sa is ied o ω∈Ω, ≥0and i , in addi ion, he pullback con e gence (10) lim →∞ dis H(φ( , θ− ω, D(θ− ω),A(ω))) = 0 holds o D∈ D and ω∈Ω. We no e ha om his con e gence i ollows (l.i.p.) lim →∞ dis H(φ( , ω, D(ω),A(θ ω)) = 0, whe e by (l.i.p.) we deno e limi in p obabili y. Howe e , in gene al i does no imply ωwise almos su e con e gence. Su icien condi ions o he exis ence o a andom a ac o can be ound in [12]. Theo em 2.4. Suppose ha he mapping x→φ( , ω, x)is con inuous o ≥0, and comple ely con inuous o > 0(which means ha he image o e e y bounded 7 se by he mapping x→φ( , ω, x)is ela i ely compac ). In addi ion, suppose he e is a G∈ D such ha o any D∈ D and ω∈Ω he e exis s T(D, ω)>0such ha (11) φ( , θ− ω, D(θ− ω)) ⊂G(ω), o all ≥T(D, ω). Then he e exis s a unique andom a ac o A(in D). I he andom a ac o A(ω), ω ∈Ω,consis s o a single poin hen Ade ines a andom ixed poin which a ac s empe ed andom se s. 3. Exponen ial s abili y s ochas ic e olu ion equa ions In his sec ion we will p o e he exis ence o exponen ially s able (bo h in he mean squa e sense and almos su ely) non- i ial s a iona y solu ions o ou s ochas ic semi-linea pa ial di e en ial equa ion (3). The exponen ial s abili y o i ial s a iona y solu ions (in pa icula , he null solu ion) o s ochas ic PDEs has been ex ensi ely analyzed (see, o ins ance, [4], [19], [16],... and he li e a u e ci ed he ein). Howe e , when ze o is no a solu ion o he equa ion, i is in e es ing o ind ou whe he o no he e exis o he s a iona y solu ions, which a e gene a ed by andom a iables chosen as ini ial alues in ou p oblem, and o analyze hei s abili y p ope ies. This ac , can also be conside ed as a connec ion be ween he classical me hod o he local analysis o he long- ime beha iou o s ochas ic pa ial di e en ial equa ions and he global one p o ided by he heo y o andom dynamical sys ems. In he sequel we conside he p ocess θsW(·, ω) = W(·, θsω) = W(·+s, ω)−W(s, ω), o s∈R, which is also a Wiene p ocess wi h co a iance Q. Fo ≥0 his p ocess is adap ed o he il a ion {¯ Fs+ } ≥0which ollows om (2). We will deno e by Φ(·, s, u0) he solu ion o (3), co esponding o he ini ial alue u0∈L2,s, which is d i en by θsW, and sa is ying he asse ions o Theo em 2.1. The ollowing equali y holds o u0∈L2,0: (12) Φ(·,0, u0)(θs·) = Φ(·, s, us)(·),almos su ely, whe e us(·) := u0(θs·). Bo h sides o (12) a e d i en by he same Wiene p ocess and he same ini ial condi ion and he ac ha solu ions o (3) a e unique. Indeed, by (2) u0(θs·) is ¯ Fs-measu able. Lemma 3.1. i) Fo s∈R, τ ≥0, u0∈L2,s Φ(·, s +τ, Φ(τ, s, u0)) = Φ(·+τ, s, u0),almos su ely. ii) Se (13) µ:= 2a+ 2L +LB. Then EkΦ( , s, u1 0)−Φ( , s, u2 0)k2≤eµ Eku1 0−u2 0k2 o ≥0, s ∈R, u1 0, u2 0∈L2,s. P oo . i) Conside he p ocess Y( , ω) = ½Φ( , s, u0)(ω) : 0 ≤ ≤τ Φ( −τ, s +τ, Φ(τ, s, u0))(ω) : > τ , 8 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS which is con inuous (almos su ely) by he con inui y o Φ. Fo > τ we ha e Y( ) =S( −τ)Φ(τ, s, u0) +Z −τ 0 S( −τ−q) (Φ(q, s +τ, Φ(τ, s, u0)))dq +Z −τ 0 S( −τ−q)B(Φ(q, s +τ, Φ(τ, s, u0))d(W(q+s+τ)−W(s+τ)) =S( )u0 +Z τ S( −q) (Y(q))dq +S( −τ)Zτ 0 S(τ−q) (Y(q))dq +Z τ S( −q)B(Y(q))d(W(q+s)−W(s)) +S( −τ)Zτ 0 S(τ−q)B(Y(q))d(W(q+s)−W(s)). The i s conclusion ollows i we conca ena e he in eg als and use he ac ha o he inc emen s o he Wiene p ocess we ha e W(q1+s+τ)−W(s+τ)−(W(q2+s+τ)−W(s+τ)) =W(q1+s+τ)−W(s)−(W(q2+s+τ)−W(s)). ii) I is no ha d o see by he p ope ies o he coe icien s ha (3) has he ajec- o ies in L2(0, T;V) (see K ylo and Rozo skii [17], o G ecksch and Tudo [13]) which allows us o apply he I o o mula o he p ocess e−µ kX1( )−X2( )k2whe e we ha e deno ed Xi( ) := Φ( , s, ui 0), i = 1,2.Thus e−µ kX1( )−X2( )k2 =ku1 0−u2 0k2−µZ 0 e−µqkX1(q)−X2(q)k2dq + 2 Z 0 e−µqhA(X1(q)−X2(q), X1(q)−X2(q)idq + 2 Z 0 e−µq( (X1(q)) − (X2(q)), X1(q)−X2(q))dq +Z 0 e−µqkB(X1(q)) −B(X2(q))k2 LQ 2 dq + 2 Z 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)) ≤ku1 0−u2 0k2+Z 0 e−µq(−µ+ 2a+ 2L +LB)kX1(q)−X2(q)k2dq + 2 Z 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)). ≤ku1 0−u2 0k2+ 2 Z 0 e−µq(X1(q)−X2(q),(B(X1(q)) −B(X2(q)))dW (q, θsω)). (14) 9 The esul ollows easily by calcula ing he expec a ion i we eplace by ∧TN, whe e TNis a amily o s opping imes TN(ω) = in { ≥0 : kX1( , ω)k2+kX2( , ω)k2≥N} such ha lim N→∞(TN∧ ) = , almos su ely, since X1, X2ha e con inuous pa hs. ¤ Co olla y 3.2. Φ( , s, ·)maps L2,s in o L2,s+ con inuously. We shall now show he exis ence o an exponen ially s able solu ion o (3). Theo em 3.3. Suppose ha he cons an µappea ing in (13) is nega i e. Then he e exis s an exponen ially a ac ing ixed poin X∗∈L2,0gene a ing an exponen- ially s able s a iona y solu ion o (3). In pa icula , he p ocess ( , ω)→X∗(θ ω) has a con inuous e sion gi en by Φ(·,0, X∗). P oo . We show ha (Φ(k, −k, u0(θ−k·)))k∈Nis a Cauchy sequence in L2,0 o u0∈ L2,0. No ice ha o his u0we ha e ha u0(θ−k·)∈L2,−k. Indeed, EkΦ(k, −k, u0(θ−k·)) −Φ(k−1,1−k, u0(θ1−k·))k2 =EkΦ(k−1,1−k, Φ(1,−k, u0(θ−k·))) −Φ(k−1,1−k, u0(θ1−k·))k2 ≤eµ(k−1)EkΦ(1,−k, u0(θ−k·)) −u0(θ1−k·)k2 =eµ(k−1)EkΦ(1,0, u0(·)) −u0(θ1·)k2 He e we ha e applied Lemma 3.1 i), he {θ } ∈R-in a iance o Pand (12). The Cauchy sequence p ope y ollows since µ < 0. Le he limi o his sequence be deno ed by X∗∈L2,0. X∗as an elemen in L2,0is independen o he choice o u0∈L2,0. Indeed, o 0∈L2,0we ha e ha EkΦ(k, −k, u0(θ−k·)) −Φ(k, −k, 0(θ−k·))k2 =EkΦ(k, 0, u0(·)) −Φ(k, 0, 0(·))k2≤eµkEku0− 0k2, which goes o ze o o k→ ∞ such ha he limi o he abo e Cauchy sequence in L2,0is independen o u0. Since X∗∈L2,0 he p ocess Φ(·,0, X∗) sa is ies all o he conclusions o Theo em 2.1. We now show ha Φ( , 0, X∗)(·) = X∗(θ ·) almos su ely o any ∈R+. By he de ini ion o X∗(ω) he andom a iable X∗(θ ·) is gi en by (L2) lim k→∞ Φ(k, −k, u0(θ−k·))(θ ω) which is equal o (L2) lim k→∞ Φ(k, −k, u0(θ −k·))(ω) 16 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS o a su icien ly small ε > 0. Compa ing he coe icien s o (29) we ha e kψ( )k2≤ R2( ) i kψ(0)k2≤R2(0) whe e R2( ) is he solu ion o dR2 d =2(a+ N X j=1 λjbj|z∗ j(θ ω)|+kT(θ ω)kkT−1(θ ω)kL +ε 2)R2 +1 εkT−1(θ ω)k2k (0)k2. (30) Unde he assump ions o he heo em, his equa ion has he unique exponen ially s able s a iona y solu ion →R2(θ ω) de ined by he andom a iable R2(ω) : = Z0 −∞ 1 εkT−1(θ ω)k2k (0)k2× ×exp  Z0 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε 2)dτ d (31) o ω∈Ω, which in u n ollows easily by he a ia ion o cons an s o mula. This andom a iable is empe ed (and ini e), see Lemma 4.6 below. In pa icula he ball in Hgi en by G(ω) := B(0,2R(ω)) is mapped in o i sel : ψ( , ω, G(ω)) ⊂G(θ ω) o ω∈Ω. Indeed his ball is empe ed. In addi ion, his ball has he p ope y (11). To see his we conside he di e en ial equa ion (30) wi h some andom empe ed ini ial condi ion 2(ω). I we eplace ωby θ− ω o he solu ion o (30) a ime we ha e by he a ia ion o cons an s o mula 2(θ− ω) exp  Z0 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε 2)dτ  +Z0 1 εkT−1(θsω)k2k (0)k2× ×exp  Z0 s 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kL +ε 2)dτ ds. This e m ends o R2(ω) as → −∞ which ollows by Bi kho ’s e godic heo em, (23), (28), (25). In pa icula he i s e m ends o ze o. F om Theo em 2.4 we jus ob ain he exis ence o a andom a ac o A={A(ω)}ω∈Ω⊂G. Se ∆ψ( , ω, x1, x2) = ψ( , ω, x1)−ψ( , ω, x2). Then we ha e dk∆ψ( )k2 d ≤2(a+ N X j=1 λjbj|z∗ j(θ ω)|+kT(θ ω)kkT−1(θ ω)kL )k∆ψ( )k2. 17 We can conclude by he in a iance p ope y (9) sup y1,y2∈A(ω) ky1−y2k2≤sup x1,x2∈A(θ− ω) kx1−x2k2× ×exp  Z0 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kL )dτ  (32) Since A ⊂ G, i is empe ed, and we ob ain om he p ope ies o z∗ jand T ha he igh hand side ends o ze o o ω∈Ω, hence A(ω) is a andom ixed poin deno ed by Y∗. To see ha Y∗is exponen ially a ac ing we no e ha sup x∈D(ω) kψ( , ω, x)−Y∗(θ ω)k2= sup x∈D(ω) kψ( , ω, x)−ψ( , ω, Y ∗(ω))k2 ≤sup x∈D(ω) kx−Y∗(ω)k2× ×exp  Z 0 2(a+ N X j=1 λjbj|z∗ j(θτω)|+kT(θτω)kkT−1(θτω)kL )dτ , and he igh hand side ends o ze o exponen ially as o ω∈Ω. ¤ Since we now know ha A(ω) is a single poin we ha e simila ly o (32) and (10) ha Co olla y 4.5. The andom ixed poin Y∗(hence X∗) a ac s empe ed andom se s in he pullback sense. We now p o e he empe edness o R2. Lemma 4.6. Fo ω∈Ω he mapping →R2(θ ω)has subexponen ial g ow h, hence R2is empe ed. P oo . We abb e ia e α(ω) =a+ N X j=1 λjbj|z∗ j(ω)|+kT(ω)kkT−1(ω)kL +ε 2,Eα=: ¯α < 0, β(ω) =1 εkT−1(ω)k2k (0)k2. By he de ini ion o Ω we ha e Z0 α(θτω)dτ ∼¯α| | o ω∈Ω, → −∞. In addi ion →β(θ ω) has sub-exponen ial g ow h o → ±∞ such ha R2(ω) = Z0 −∞ exp µZ0 α(θτω)¶β(θ ω)d < ∞. 18 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS Fo an a bi a y c > 0, 0 < ε < min(−¯α, c 2)/4 and nega i e s < s0(ω, ε) we hen ha e ecs Z0 −∞ eR0 α(θτ+sω)dτ β(θs+ ω)d ≤ec 2sZ0 −∞ eR0 +s(α(θτω)−¯α)dτ−R0 s(α(θτω)−¯α)dτ−¯α +sc 2+log+β(θ +sω)d ≤ec 2sZ0 −∞ e−3ε( +s)−¯α +c 2sd ≤ec 2sZ0 −∞ eε d . Bu he igh hand side ends o ze o o s→ −∞. Simila ly, we ob ain he con e gence o →+∞, see also A nold [1] P oposi ion 4.1.3. ¤ Co olla y 4.7. Suppose ha and Bjcommu e, i.e. i holds T(ω)−1 (T(ω)x) = (x), o ω∈Ω, x ∈H. Then he conclusion o Theo em 4.4 also holds i ins ead o (28) we assume a+L <0. P oo . Since he expec a ion o z∗ jdepends on λjsuch ha E|z∗ j| ≤ 1 pλj , we can choose λjsu icien ly small so ha PN j=1 bjλjE|z∗ j|is also a bi a ily small. ¤ Rema k 4.8. Usually Bjand commu e i hey a e diagonal in some o hogonal basis o H(see Kwiecinska [18] and ou example in Sec ion 6) . The commu a i i y assump ions o Biensu e ha (20) can be ans o med in o a andom di e en ial equa ion. Howe e , he e exis o he ans o ma ions o e y special classes o s ochas ic e olu ion equa ions o be ans o med in o andom di - e en ial equa ions, see e.g. Flandoli and Lisei [11] and Mohammed e al. [21]. Fo he esul ing andom e olu ion equa ion ou heo y can be applied. 5. Non-linea pe u ba ions We now wan o show ha i we change he nonlinea pa o (3) con inuously, hen he ixed poin s also change con inuously. Fo his we s udy he amily o p oblems indexed by n∈Z+gi en by (33) dX =AXd + n(X)d +B(X)dW, X(0) = u0. We suppose ha he cons an s µngi en by (13) co esponding o he unc ions n, sa is y µn<0 uni o mly o n∈Z+. We will also deno e by Φn(·,0, u0), n = 0,1,2, ..., he solu ion o (33), and by X∗ n hei associa ed andom ixed poin s. Then, ou objec i e is o p o e ha he X∗ na e close o X∗ 0i he nis close o 0 in some sense. 19 Theo em 5.1. Conside he amily o s ochas ic e olu ion equa ions (33). Suppose ha µ:= sup n∈Z+ (2a+ 2L n+LB)<0, and, in addi ion, ha lim n→∞ n(u) = 0(u), o u∈H. Then, o he co esponding ixed poin s we ha e (L2) lim n→∞ X∗ n=X∗ 0. P oo . One can ind he idea o he p oo in Zeidle [26] P oposi ion 1.2 . Deno ing k · k2 L2=Ek·k2and aking in o accoun (12), we ha e kX∗ n−X∗ 0kL2=kΦn(1,−1, X∗ n(θ−1·)) −Φ0(1,−1, X∗ 0(θ−1·))kL2 =kΦn(1,0, X∗ n)−Φ0(1,0, X∗ 0)kL2 ≤kΦn(1,0, X∗ n)−Φn(1,0, X∗ 0)kL2+kΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2 ≤eµkX∗ n−X∗ 0kL2+kΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2, so (34) kX∗ n−X∗ 0kL2≤1 1−eµkΦn(1,0, X∗ 0)−Φ0(1,0, X∗ 0)kL2. Now i is no ha d o p o e ha he igh hand side ends o ze o. Indeed, we se Xn( ) = Φn( , 0, X∗ 0), n ∈Z+, co esponding o he solu ion o (3) wi h = n. Then by he I o o mula we ob ain d d EkXn( )−X0( )k2≤µEkXn( )−X0( )k2+Ek n(X0( )− 0(X0( ))k2 EkXn(0) −X0(0)k2= 0. The inequali y k n(X0( ))k2≤2 sup n∈Z+ k n(0)k2+ 2LkX0( )k2, L := sup n∈Z+ L n<∞ allows us o ind an in eg able majo an o k n(X0( )) − 0(X0( ))k2such ha poin wise con e gence o n(u) o 0(u) and he a ia ion o cons an s o mula yield he con e gence o he igh hand side o (34). ¤ We now conside a amily o e olu ion equa ions (20) wi h = n. To ob ain a amily o equa ions o he o m (26) we can apply he ans o ma ion Twhich is independen o n. Theo em 5.2. Suppose ha he Lipschi z cons an s o na e uni o mly bounded by L, ha a+ N X j=1 bjλjE|z∗ j|+LΠN j=1E(kSBj(−z∗ j)kkSBj(z∗ j)k)<0, and ha lim n→∞ n(x) = 0(x) o x∈H. Le X∗ n, n ∈Z+,be he andom ixed poin s o (20) wi h nins ead o . Then lim n→∞ X∗ n(ω) = X∗ 0(ω), o ω∈Ω. 20 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS P oo . As in he p oo o Theo em 4.4 we in es iga e he andom dynamical sys ems ψngene a ed by (26) wi h = nand ixed poin s Y∗ n(ω) a ac ing empe ed se s, see Co olla y 4.5. We ha e supn∈Nk n(0)k<∞, so, as in he p oo o Theo em 4.4, he e exis s a empe ed se G(ω) con aining all ixed poin s Y∗ n(ω). This se is gi en by a ball B(0,2R(ω)) whe e R2is a s a iona y solu ion o he one-dimensional a ine di e en ial equa ion (30) whe e L has o be eplace by L= supn∈Z+L n and k (0)kby supn∈Z+k n(0)k. By he assump ions R2(ω) exis s. We ha e kY∗ n(ω)−Y∗ 0(ω)k=kψn( , θ− ω, Y ∗ n(θ− ω)) −ψ0( , θ− ω, Y ∗ 0(θ− ω))k ≤ kψn( , θ− ω, Y ∗ n(θ− ω)) −ψn( , θ− ω, Y ∗ 0(θ− ω))k +kψn( , θ− ω, Y ∗ 0(θ− ω)) −ψ0( , θ− ω, Y ∗ 0(θ− ω))k. Since he second ac o on he igh hand side o (32) is independen o ni we eplace L by L, and Y∗ 0, Y ∗ na e con ained in he empe ed ball G, we ha e o any ε > 0, ω ∈Ω and ha he i s e m on he igh hand side is less han ε/2. In addi ion, o his simila o he p oo o Theo em 5.1 he second e m on he igh hand side is also less han ε/2 i nis la ge. We ob ain kY∗ n(ω)−Y∗ 0(ω)k< ε o la ge n. Since he ans o ma ion Tis a homeomo phism we ha e he conclusion. ¤ 6. An example Le us ake H=L2(O) and V=H1 0(O) whe e Ois a bounded domain in Rd wi h a smoo h bounda y. Conside he Laplace ope a o A= ∆ wi h he Di ichle bounda y condi ion. Theo em IX.31 in B ezis [2] ensu es he exis ence o a sequence o eal numbe s {νn}n≥1such ha 0 < ν1< ν2<··· < νn<· · · , and νn→+∞ (namely, he eigen alues o he Laplacian), and a sequence {en}n≥1⊂V∩C∞(O) o associa ed eigen ec o s (i.e. −∆en=νnenon O) which is a comple e o hono mal basis in H. On he o he hand, le us conside ope a o s Bi,i= 1,· · · , N such ha hey a e diagonalizable in he same basis and a e bounded om abo e and below (see Kwiecinska [18]), i.e., he e exis cons an s dk i, k = 1, . . . , N such ha (Bkei, ej) = dk iδij, k = 1, . . . , N;i, j = 1,2, . . . The assump ion ha he ope a o s Bka e bounded om abo e and below is equi - alen o he condi ion ha he e exis posi i e cons an s mk, Mksuch ha 0< mk≤ |dk i| ≤ Mk, k = 1, . . . , N;i= 1,2, . . . Unde hese condi ions he ope a o s Bigene a e C0-g oups SBi( ) = eBi . Finally, conside a Lipschi z con inuous unc ion om Hin o Hde ined as (u)(x) = F(u(x)), x∈ O wi h F:R→RLipschi z con inuous wi h cons an L . Recall ha hAu, ui≤−ν1kuk2 o all u∈V, and he e o e he semig oup gene a ed by Asa is ies kS( )k ≤ e−ν1 o all ≥0. We now s udy he s ochas ic e olu ion equa ion (35) dX = (AX + (X))d + N X i=1 BiXdwi. 21 I ollows ha he cons an µin Theo em 3.3 sa is ies µ=−2ν1+ 2L +LB≤ −2ν1+ 2L + N X i=1 M2 i. I µ < 0, Theo em 3.3 implies he exis ence o a unique s a iona y solu ion o ou p oblem which is exponen ially s able in mean squa e. Also, hanks o Lemma 3.1 ii) and Theo em 3.5 he almos su e exponen ial s abili y o his s a iona y solu ion holds. Obse e ha µis nega i e only i he Lipschi z cons an s L and LBa e su icien ly small. Howe e , by using he echnique o andom dynamical sys ems we can p o e s abili y beha iou e en o la ge alues o he cons an LB. Indeed, we will be able o apply Theo em 4.4. To his end, we conside ou equa ion (35) in i s S a ono ich equi alen o m dX ="(A−1 2 N X i=1 B2 i)X+ (X)#d + N X i=1 BiX◦dwi. Deno e C=A−1 2PN i=1 B2 iwhich also gene a es a C0-semig oup SC( ) . I he ope a o s A, B1, ..., BNcommu e mu ually, hen his semig oup SC( ) is gi en as SC( ) = S( )e− 2PN i=1 B2 i. Now, by easy compu a ions (see [18]) we can deduce e− 2PN i=1 M2 i≤ ke− 2PN i=1 B2 ik ≤ e− 2PN i=1 m2 i and assump ion (28) in Theo em 4.4 becomes eµ=−ν1−1 2 N X i=1 m2 i+ N X i=1 MiλiE|z∗ i|+L N Y i=1 E(kSBi(−z∗ i)kkSBi(z∗ i)k)<0. Consequen ly, i he ope a o s Bia e such ha eµ < 0, we can apply Theo em 4.4 and ensu e ha he e exis s a unique exponen ially s able s a iona y solu ion gi en by a andom ixed poin . No ice ha his can p o ide be e s abili y esul s han he ones ob ained in Sec ion 3. Indeed, assume o ins ance ha Biu=miuwhe e mi∈R+ o i, ··· , N. This means ha mi=Mi, and i is easy o check ha eµ=−ν1−1 2 N X i=1 m2 i+ N X i=1 miλiE|z∗ i|+L , since now kSBi(−z∗ i)kkSBi(z∗ i)k= 1.Then, i −ν1−1 2 N X i=1 m2 i+L <0 (wha happens i , o example, he noise in ensi ies mia e la ge enough), we can choose s a iona y p ocess z∗ ico esponding o λisuch ha eµ < 0. Thus, some kind o s abiliza ion has been ob ained o he non- i ial s a iona y solu ion. Acknowledgemen s. This wo k was pa ially suppo ed by he DAAD (Ge - many), and he Minis e io de Ciencia y Tecnolog´ıa (Spain) unde he p ojec s HA2001-0075 and BFM2002-03068. 22 TOM´ AS CARABALLO, PETER E. KLOEDEN, AND BJ ¨ ORN SCHMALFUSS T. Ca aballo would like o hank B. Schmal uss and his amily o he kind hospi- ali y hey all o e ed him du ing his isi in No embe 2002. Re e ences [1] L. A nold. Random Dynamical Sys ems. Sp inge , New Yo k, 1998. [2] H. B ezis. Analyse Fonc ionnelle. Masson, Pa is, 1983. [3] T. Ca aballo and J. Langa. 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Kloeden: [email p o ec ed] E-mail add ess, Bj¨o n Schmal uß: [email p o ec ed] me sebu g.de (Tom´as Ca aballo) Dp o. Ecuaciones Di e enciales y An´ alisis Num´ e ico, Uni e sidad de Se illa, Apdo. de Co eos 1160, 41080-Se illa (Spain) (Pe e E. Kloeden) Fachbe eich Ma hema ik, Johann Wol gang Goe he-Uni e si ¨ a , D- 60054 F ank u am Main, Ge many (Bj¨o n Schmal uß) Depa men o Applied Sciences, Uni e si y o Technology and Ap- plied Sciences, Geusae S asse, D–06217 Me sebu g, Ge many,