Tracking Properties of Trajectories On Random Attracting Sets

acking p ope ies o ajec o ies on andom a ac ing se s Tom´as Ca aballo and Jos´e A. Langa 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) e-mails: ca aball@nume .us.es, langa@nume .us.es Abs ac The heo y o andom a ac ing se s highligh s in e es ing p ope - ies o he asymp o ic beha iou o some s ochas ic di e en ial equa- ions. In his pape some esul s on he ela ion be ween he dynamics on andom a ac o s and s ochas ic ine ial mani olds, and he dy- namics in he associa ed andom dynamical sys em a e s udied. In pa icula , some acking p ope ies o ajec o ies on andom a ac- o s and a gene al esul on he asymp o ic comple eness o s ochas ic ine ial mani olds a e shown. 1 INTRODUCTION One o he main concep s o he s udy o he asymp o ic beha iou o dis- sipa i e dynamical sys ems is he global a ac o (see Cons an in e al. [4], Hale [18], Temam [26] and he e e ences he ein). This is a compac , in- a ian se a ac ing uni o mly e e y ajec o y s a ing in a bounded se o he phase space. The unde s anding o he dynamics on he global a - ac o gi es us ele an in o ma ion abou he asymp o ic beha iou o he dynamical sys em. One o he mos impo an esul s in he heo y o global a ac o s claims ha he ac al, and so he Hausdo , dimension o his se is ini e, e en i he dynamical sys em is posed in an in ini e-dimensional phase space. Tha is, al hough he ajec o ies depend on an in ini e numbe o deg ees o ee- dom, he ini e dimensionali y o he a ac o s seems o be desc ibing he asymp o ic beha iou o he dynamical sys em wi h a ini e numbe o ime- dependen coo dina es. This makes e en mo e in e es ing he s udy o he dynamics on he global a ac o , as we expec ha his dynamics can be desc ibed by a sys em o o dina y di e en ial equa ions (see Eden e al. [12], chap e 10, and Robinson [23],[24]). In ac , his is wha we ob ain in he heo y o ine ial mani olds (smoo h in a ian mani olds which a ac s e e y ajec o y exponen ially as , see Foias e al. [16]), ha is, he dynamics on he ine ial mani old, go e ned by a sys em o o dina y di e en ial equa ions, is de e mining he asymp o ic beha iou o he dynamical sys em. F om a geome ical poin o iew, his ac can be seen when i is p o ed ha e e y ajec o y o he sys em can be ollowed a bi a ily close, as ime g ows o in ini e, by ano he ajec o y mo ing on he ine ial mani old. We say in his case ha he ine ial mani old sa is ies a acking p ope y (Foias e al. [17]) o ha i is asymp o ically comple e (Robinson [22]). In ecen yea s, C auel and Flandoli [7] (see also Schmal uss [25] and C auel e al. [6]) ha e in oduced a concep o he a ac o o some s ochas- ic di e en ial equa ions. The gene aliza ion o his si ua ion is no i ial, as in his case he sys em is non au onomous and he new o cing e m may ha e e y la ge luc ua ions which make ha solu ions a e pushed ou om any bounded ball in he phase space. Howe e , i is possible o de ine in hese cases a gene alized concep o global a ac o as a mo ing (in he pa ame e s ime and omega o he andom e m) compac se , in a ian wi h espec o he shi associa ed o he andom dynamical sys em (see sec ion 2) and a - ac ing, backwa ds in ime, all he ajec o ies s a ing in any bounded se . This se is called andom a ac o and, al hough he con e gence p ope y is p o ed, oughly speaking, om −∞, i is easy o show ha we also ha e con e gence in p obabili y, o wa d in ime, o he andom a ac o . On he o he hand, he e a e some esul s on he ini e dimensionali y (wi h p obabili y one) o he andom a ac o (see Debussche [10], [11]), which mo i a es again he s udy o he dynamics on hese andom a ac ing se s o know in wha sense i is de e mining he asymp o ic beha iou o hese s ochas ic di e en ial equa ions. In his di ec ion, Flandoli and Langa [14] ha e p o ed a esul on de e mining modes o andom dynamical sys ems which gene alizes he esul s o Foias and P odi [15] o he s ochas ic case. In sec ion 3, we p esen a gene aliza ion o a esul al eady known in he de e minis ic case (see Langa and Robinson [21]). The e a e also some pape s ex ending he concep o ine ial mani old o he s ochas ic case (see Bensoussan and Flandoli [3], Chuesho and Gi ya [8] o Da P a o and Debussche [9]). A s ochas ic ine ial mani old is a andom in a ian Lipschi z mani old which a ac s e e y ajec o y exponen ially as . As in he de e minis ic case, he ine ial mani old is gi en as he g aph o some ( andom) Lipschi z unc ion. In sec ion 4, a esul ha shows he asymp o ic comple eness p ope y o hese s ochas ic ine ial mani olds is p o ed. In ac , i is e en mo e gene al han classical esul s on asymp o ic comple eness on ine ial mani olds in he de e minis ic case, since i is s ill ue o gene al in a ian exponen ially a ac ing se s and no necessa ily smoo h mani olds gi en as g aphs o some unc ions. The esul can be applied o he p oblems s udied in [3] and [8]. Finally, some conclusions and possible gene aliza ions a e p esen ed in las sec ion. 2 RANDOM DYNAMICAL SYSTEMS AND ATTRACTORS Le (Ω,F, P) be a p obabili y space and {θ : Ω →Ω, ∈} a amily o mea- su e p ese ing ans o ma ions such ha ( , ω)7→ θ ωis measu able, θ0= id, θ +s=θ θs, o all s, ∈. The low θ oge he wi h he p obabili y space (Ω,F, P, (θ ) ∈) is called a (measu able) dynamical sys em. Fu he mo e, we suppose ha he shi θ is e godic. A andom dynamical sys em (RDS) on a Polish space (X, d) wi h Bo el σ-algeb a Bo e θ on (Ω,F, P) is a measu able map ϕ:+×Ω×X→X ( , ω, x)7→ ϕ( , ω)x such ha P−a.s. i) ϕ(0, ω) = id (on X) ii) ϕ( +s, ω) = ϕ( , θsω)◦ϕ(s, ω),∀ , s ∈+(cocycle p ope y). A RDS is con inuous o di e en iable i ϕ( , ω) : X→Xis con inuous o di - e en iable. The heo y o andom dynamical sys ems co e s all sys ems wi h andomness, in pa icula andom and s ochas ic di e ence and di e en ial equa ions (see A nold and C auel [2]). A andom se K(ω) is said o abso b he se B⊂Xi P−a.s. he e exis s B(ω) such ha o all ≥ B(ω) ϕ( , θ− ω)B⊂K(ω). Finally, a andom se A(ω) is a andom a ac o associa ed o he RDS ϕi P−a.s. i) A(ω) is a andom compac se , ii) ϕ( , ω)A(ω) = A(θ ω),∀ ≥0 (in a iance) and iii) o all B⊂Xbounded (and non andom) lim →+∞dis (ϕ( , θ− ω)B, A(ω)) = 0, whe e dis ( . , . ) deno es he Hausdo semidis ance dis (A, B) = sup a∈A in b∈Bd(a, b), A, B ⊂X. In his si ua ion, we ha e he ollowing heo em abou exis ence o andom a ac o s due o C auel and Flandoli ([7], heo em 3.11). Theo em 1 Suppose he e exis s a compac se D(ω)abso bing e e y bounded non andom se B⊂X. Then, he se A(ω) = [ B⊂X ΛB(ω) is a andom a ac o o ϕ, whe e he union is aken o e all B⊂Xbounded, and ΛB(ω)deno es he omega-limi se o Bwhich is gi en by ΛB(ω) = n≥0[ ≥n ϕ( , θ− ω)B. 2 Mo eo e , in C auel [5] i is shown ha andom a ac o s a e unique and, unde he e godici y assump ion on θ , he e exis s a compac se K⊂X such ha P−a.s. he andom a ac o is he omega limi se o K, ha is, A(ω) = n≥0[ ≥n ϕ( , θ− ω)K. 3 A TRACKING PROPERTY ON RANDOM ATTRACTORS In his sec ion we shall p o e a p ope y on andom a ac o s al eady known o (de e minis ic) global a ac o s (see Langa and Robinson [21]). I says, oughly speaking, ha gi en a andom dynamical sys em o which he e ex- is s a andom a ac o , e e y andom ajec o y can be ollowed a bi a ily closely by skipping om one solu ion o ano he on he andom a ac o . Le us conside he ollowing pa ial di e en ial equa ion on he Hilbe space H(no m |.|) pe u bed by an addi i e H− alued whi e noise p ocess on he p obabili y space (Ω,F, P) wi h shi θ on Ω du =−Aud + (u)d +dW( ),(1) whe e Ais a linea ope a o , sel adjoin , posi i e and wi h compac in e se, whose domain is D(A), and is he nonlinea e m. In hese condi ions, he e exis an inc easing sequence o eigen alues o A,λn%+∞, and he co esponding sequence o eigen unc ions {en}∞ n=1, which o ms an o hogonal basis in H. Le ϕ:+×Ω×H→Hbe a andom dynamical sys em associa ed o p oblem (1) and assume he ollowing p ope y o con inui y wi h espec o ini ial condi ions: he e exis s L > 0 such ha , gi en s≥0, i ollows ha , o all u, ∈H, o all ≥0, and P−a.s. |ϕ( , θsω)u−ϕ( , θsω) | ≤ eL |u− |.(2) In addi ion o he p eceden hypo heses, suppose he ones o he ex- is ence o a andom a ac o a e sa is ied, so ha he e exis s a andom compac se A(ω), in a ian , and such ha , o all B⊂Hbounded and P−a.s. lim →+∞dis (ϕ( , θ− ω)B, A(ω)) = 0. Since he shi θ is measu e p ese ing, i is known (see C auel and Flandoli [7]) ha his las con e gence implies con e gence in p obabili y, ha is, o all ² > 0, lim →+∞P(dis (ϕ( , ω)B, A(θ ω)) < ²) = 1.(3) ¿F om (2) and (3) he ollowing esul holds: P oposi ion 1 W i ing a ajec o y o p oblem (1) as ϕ( , ω)u0,u0∈H, and gi en 0< ² < 1,0< δ < 1, and T > 0, he e exis s a ime τ(², δ, T)>0 such ha , o all τ≥τ(², δ, T), he e exis s e Ωwi h P(e Ω) >1−δsuch ha , i ω∈e Ω, he e exis s a poin τ∈ A(θτω)sa is ying |ϕ( +τ, ω)u0−ϕ( , θτω) τ|< ² , o all 0≤ ≤T. P oo . Gi en 0 < ², δ < 1, T > 0, om (3) we deduce ha he e exis s τ=τ(², δ, T)>0 such ha , ∀ ≥τ(²), P(dis (ϕ( , ω)u0,A(θ ω)) < ²e−LT )≥1−δ. In pa icula , o =τ, he e exis s Ωτ,P(Ωτ)≥1−δsuch ha , o all ω∈Ωτ, dis (ϕ(τ, ω)u0,A(θτω)) < ²e−LT . As A(θτω) is compac , he e exis s τ∈ A(θτω) such ha |ϕ(τ, ω)u0− τ| ≤ ²e−LT , and, by (2), |ϕ( , θτω)ϕ(τ, ω)u0−ϕ( , θτω) τ| ≤ |ϕ(τ, ω)u0− τ|eL ≤², o all ∈[0, T], and his gi es he p oposi ion by he cocycle p ope y. 2 As a consequence o his esul we ob ain he ollowing acking p ope y o ajec o ies o p oblem (1). Co olla y 1 Gi en 0< δ < 1,T > 0, and {²n}∞ n=1,²n>0,²n&0, he e exis a sequence o imes {τn}∞ n=1,τn%+∞, τn+1 > τn∀n∈N, τn+1 −τn→ ∞ as n→ ∞, and a subse e Ω⊂Ωwi h P(e Ω) >1−δ, such ha , o each ω∈e Ω, he e exis s τn∈ A(θτnω)sa is ying |ϕ( +τn, ω)u0−ϕ( , θτnω) τn|< ²n,0≤ ≤nT, ∀n∈ P oo . Gi en 0 < δ < 1, we choose a dec easing sequence o {δn}∞ n=1 such ha ∞ X n=1 δn< δ ( o ins ance, δn=δ 2n+1 ).(4) Then, applying P oposi ion 1 o ²1,δ1and T, he e exis s τ1=τ(δ1, ²1, T) such ha , o all τ≥τ(δ1, ²1, T), he e exis s Ωτ⊂Ω wi h P(Ωτ)>1−δ1 sa is ying ha , o each ω∈Ωτ, he e exis s τ∈ A(θτω) wi h |ϕ( +τ, ω)u0−ϕ( , θτω) τ|< ²1, o all 0 ≤ ≤T. Now, again by P oposi ion 1 o ²2,δ2and 2T, we ge ha he e exis s τ2=τ(δ2, ²2, T)≥τ1such ha , o all τ≥τ2, he e exis s Ωτ⊂Ω wi h P(Ωτ)>1−δ2such ha , o each ω∈Ωτ, he e exis s τ∈ A(θτω) wi h |ϕ( +τ, ω)u0−ϕ( , θτω) τ|< ²2, o all 0 ≤ ≤2T. In gene al, gi en ²n,δnand nT, he e exis s τn=τ(δn, ²n, T)≥τn−1such ha , o all τ≥τn, he e exis s Ωτ⊂Ω wi h P(Ωτ)>1−δnsuch ha , o each ω∈Ωτ, he e exis s τ∈ A(θτω) wi h |ϕ( +τ, ω)u0−ϕ( , θτω) τ|< ²n, o all 0 ≤ ≤nT. Le us call Ωn⊂Ω he se wi h he p ope y ha , o all ω∈Ωn, he e exis s τn∈ A(θτnω) sa is ying |ϕ( +τn, ω)u0−ϕ( , θτnω) τn|< ²n,0≤ ≤nT. We ha e ha P(Ωn)≥1−δn.Deno ing e Ω = ∩∞ n=1Ωn, i is clea om (4) ha P(e Ω) >1−δ. Now, aking ω∈e Ω we ha e ha he e exis s τn∈ A(θτnω) wi h |ϕ( +τn, ω)u0−ϕ( , θτnω) τn|< ²n,0≤ ≤nT, o all n∈,and hus P(|ϕ( +τn, ω)u0−ϕ( , θτnω) τn|< ²n,0≤ ≤nT, ∀n∈)>1−δ. 2 3.1 Applica ion. A eac ion-di usion equa ion wi h addi i e noise Le D⊂nbe an open bounded se wi h egula bounda y and (u) = 2p−1 X k=0 akuk, a2p−1<0. We conside he ollowing pa ial di e en ial equa ion o eac ion-di usion ype in Dwi h an addi i e whi e noise p ocess:          du = ∆ud + (u)d +Pd i=1 φidWi in D u= 0 on ∂D u(0) = u0 (5) whe e Wi : Ω →, ∈, a e independen one dimensional wo-sided Wiene p ocesses on a p obabili y space (Ω,F, P). As i is well known, (5) can be exp essed as a di e en ial equa ion in H= L2(D),    du =Aud +F(u)d +Pd i=1 φidWi in H u(0) = u0 (6) whe e A:D(A)⊂H→H,Au = ∆u,F:Z→Z0,Z=L2p(D) and Z0=L(2p)0(D),wi h (2p)0= (2p−1)/2p, and is de ined as F(u) = (u). We ake φi∈D(A). We can de ine a andom dynamical sys em ϕ( , ω) : H→Hon (Ω,F, P, (θ ) ( ∈), wi h he shi θ e godic, o which he exis ence o a andom a ac o A(ω) has al eady been p o ed ( o he s udy o he andom a ac o o his p oblem, see C auel and Flandoli [7], C auel e al. [6], and Debussche [10]). Due o he condi ion on he nonlinea e m , and unde s anda d compu- a ions (see, o ins ance, Debussche [10]), i can be shown ha P−a.s. |u( , ω;u0)−u( , ω; 0)| ≤ ek |u0− 0|, so ha he con inui y p ope y wi h espec o ini ial condi ions (2) is sa is- ied and, consequen ly, he esul s in his sec ion a e ue o his p oblem. Thus, e u ning o (11) |ϕ( −(s+h), θs+hω) s+h−ϕ( −s, θsω) s| ≤D(θsω)D(θs+hω)e−γ( −s)eγhC(ω)e−νs(e−νh +eLh), and hus, o all h≤h0,(p e iously chosen), ≤D(θsω)D(θs+hω)e−γ e−(ν−γ)sC(ω)K. (12) ¿F om (12) we can conclude ha (10) con e ges uni o mly on bounded in- e als o [0,+∞) since, o any τ > T, |ϕ( −τ, θτω) τ−ϕ( −T, θTω) T| ≤Ke−γ C(ω)e−(ν−γ)T ∞ X n=0 D(θT+nhω)D(θT+(n+1)hω)e−(ν−γ)nh and, by he condi ion on D(ω), he se ies abo e is con e gen , so ha he las exp ession is =K0e−γ e−(ν−γ)T which ends o ze o uni o mly o ∈[0, 0], o all 0>0,as T→+∞. The e o e, he limi in (10) exis s and sa is ies he equa ion o he di e en ial equa ion, since i is he uni o m limi o solu ions o he p oblem. Mo eo e , i is now clea ha ∞( , ω) sa is ies he acking p ope y o ϕ( , ω)u0,since | ∞( , ω)−ϕ( , ω)u0| ≤ | ∞( , ω)− (ω)|+| (ω)−ϕ( , ω)u0| ≤ | lim T→∞ ϕ( −T, θTω) T−ϕ( − , θ ω) |+| (ω)−ϕ( , ω)u0| ≤KC(ω)e−γ ∞ X n=0 D(θ +nhω)D(θ +(n+1)hω)e−(ν−γ)( +nh)+C(ω)e−ν ≤K0C(ω)e−γ e−(ν−γ) +C(ω)e−ν ≤C(ω)˜ K0e−ν . 2 Rema k. No e ha in he p oo we ha e no made any e e ence o he ine ial mani old gi en as a g aph o some Lipschi z unc ion, so ha he heo em is ue o a gene al in a ian exponen ially a ac ing closed andom se which sa is ies he low no mally hype bolic p ope y. 4.1 Applica ion. A Semilinea S ochas ic Di e en ial Equa ion wi h Addi i e Noise In Bensoussan and Flandoli [3] (see also Chuesho and Gi ya [8]) i is p o ed he exis ence o a s ochas ic ine ial mani old o he ollowing di e en ial equa ion wi h addi i e noise du( ) + Au( )d =R(u( ))d +dW( ) u(0) = u0, whe e Ais a sel adjoin posi i e linea ope a o wi h a disc e e spec um and compac in e se, so ha he e exis s a sequence o eigen alues 0< λ1≤λ2≤ · · · ≤ λn≤ · · · whose co esponding eigen ec o s o m an o hogonal basis o H. R is he nonlinea e m which is Lipschi z con inuous wi h cons an LR. Unde he hypo heses on R(u), i is clea ha condi ion (2) is sa is ied, and ha he andom a iable D(ω) in (8) is independen o ω. Fu he mo e, in [3] i is p o ed ha unde he spec al gap condi ion λn+1 −λn>4LR(13) he e exis s a s ochas ic ine ial mani old M(ω) gi en as he g aph o some andom unc ion. In his case γ= (1 + M)LR+λn. Indeed, a bound on he sepa a ion o ajec o ies on M(ω) is gi en by a bound on he sepa a ion o ajec o ies o he ODE dp +Apd =PmR(p+φ (ω)p)d +dPmW , and he Lipschi z cons an o Ap +PmR(p+φ (ω)p) is (1 + M)LR+λn. On he o he pa , we ha e ha he a e o a ac ion ν=λn+1 −LR(1 + M),whe e M=Lφ, and i can be chosen o be less o equal o one (see [3]). Thus, om (13) we ob ain γ= (1 + M)LR+λn<2LR+λn< λn+1 −2LR< λn+1 −LR(1 + M) = ν, so ha , as γ < ν,M(ω) is low no mally hype bolic and, by heo em 2, i is asymp o ically comple e. CONCLUSIONS Some esul s on he ela ion be ween he dynamics on andom a ac ing se s and dissipa i e andom dynamical sys ems ha e been s udied. On he o he hand, i has been p o ed a gene al esul o he asymp o ic comple eness o in a ian exponen ially a ac ing andom se s, which can be success ully applied o some in e es ing p oblems in he li e a u e o which he exis ence o s ochas ic ine ial mani olds has been p o ed. The applica ion o hese esul s o o he possible examples wi h andom a ac o s o s ochas ic ine ial mani olds is an in e es ing p oblem which could lead us o mo e gene al esul s on acking p ope ies o ajec o ies o andom a ac ing se s. This would imp o e he unde s anding o he asymp o ic beha iou o in e es ing s ochas ic sys ems. ACKNOWLEDGMENTS The au ho s would like o hank P o . James Robinson o many in e es ing and help ul discussions on he opic o his pape . This wo k has been pa ially suppo ed by D.G.I.C.Y.T. (Spain) P oyec o No. PB95–1242 Re e ences [1] L. A nold,S ochas ic Di e en ial Equa ions. Theo y and Applica- ions. Wiley, New Yo k (1974). [2] L. A nold and H. C auel, Random dynamical sys ems, Lya- puno Exponen s, P oceedings, Obe wol ach, Lec . No es in Ma h. 1486 (1990), 1-22. [3] A. Bensoussan and F. Flandoli, S ochas ic ine ial mani old, S ochas ic and S ochas ic Repo s, 53 (1995), 13-39. [4] P. Cons an in, C. Foias and R. Temam,A ac o s ep esen ing u bulen lows, Mem. Ame . 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