Asymptotic exponential stability for diffusion processes driven by stochastic differential equations in duals of nuclear spaces

ASYMPTOTIC EXPONENTIAL STABILITY FOR DIFFUSION PROCESSES DRIVEN BY STOCHASTIC DIFFERENTIAL EQUATIONS IN DUALS OF NUCLEAR SPACES 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-Se illa, SPAIN KAI LIU* Depa men o P obabili y and S a is ics The Uni e si y o She ield The Hicks Building, Houns ield Road She ield, S3 7RH, UK ABSTRACT The main objec i e o his pape is o in es iga e he asymp o ic s abili y o di usion p ocesses d i en by a class o I ˆo s ochas ic di e en ial equa ions in duals o nuclea spaces. A coe ci i y condi ion imposed on his so o equa ion plays he ole o an exponen ial s abili y c i e ion. An example is s udied o illus a e ou heo y. AMS Classi ica ion: p ima y 93E03; seconda y 60H10. Keywo ds: Almos su e exponen ial s abili y; L2-exponen ial s abili y; S ochas ic di u- sion p ocesses in duals o nuclea spaces. * Au ho o Co espondence. 1 1. In oduc ion In he pape we shall s udy he exponen ial s abili y o s ochas ic di usion equa ions in duals o nuclea spaces. These equa ions na u ally a ise in he esea ch o chemical eac ion-di usion equa ions, neu ophysiology and u bulence, especially, in he ecen i e pollu ion model esea ches (see [11], [15] and [16]). Roughly speaking, we shall conside he ollowing s ochas ic di usion equa ion: X =X0+Z 0 A(s, Xs)ds +Z 0 B(s, Xs)dWs(1.1) whe e A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) a e wo measu able mappings and W is a Φ0- alued Wiene p ocess. He e Φ0is he dual space o a ce ain coun ably Hilbe ian nuclea space and L(Φ0,Φ0) is he space o all bounded linea ope a o s om Φ0in o i sel . Di usion equa ions o he ype (1.1) ha e been s udied by a numbe o au ho s, o ins ance, G. Kallianpu and R.L. Wolpe [10], G. Kallianpu and J. Xiong [11], H. Tuckwell [15] and J.B. Walsh [16] among o he s. The eade is e e ed o G. Kallianpu and J. Xiong [11] o u he de ails conce ning ce ain p ope ies o he solu ions (1.1) and some ela ed opics. In he pape , we a e pa icula ly in e es ed in he c i e ia o exponen ial s abili y in he sense o mean squa e and pa hwise wi h p obabili y one o he s ong solu ions o he equa ions (1.1). I is a long his o y o he in es iga ion o he exponen ial s abili y o s ochas ic di - e en ial equa ions in ini e dimensional spaces and, mo e ecen ly, o s ochas ic e olu ion equa ions in Hilbe spaces. Fo in ini e dimensional case, we should men ion U.G. Hauss- mann [6] (linea case) and A. Ichikawa [7] (semilinea case) o hei undamen al wo k on his aspec . Ne e heless, o nuclea space- alued s ochas ic di e en ial equa ion si - ua ions, o he bes o ou knowledge i seems ha nobody e e ca ied ou he s udy o exponen ial s abili y ei he in he sense o mean squa e o pa hwise wi h p obabili y one. This is he main ask in his pape o ill his gap. I is pa icula ly wo h poin ing ou ha ou app oaches, which a e de o ed o he conside a ion o he s ochas ic di e en- ial equa ions in duals o nuclea spaces (1.1), could e en be used o ex end he esul s o [6][7] o co e gene al non-au onomous Hilbe space- alued s ochas ic di e en ial sys ems. Fi s ly, we shall gi e su icien condi ions o he exponen ial s abili y in mean squa e o he s ong solu ions o he equa ions (1.1). Nex , we ob ain exponen ial s abili y o pa hs wi h p obabili y one. Ou a gumen is based on a coe ci i y condi ion which plays a key ole o he exis ence and uniqueness o he equa ions (1.1). As a consequence, we will obse e how a sui able coe ci i y condi ion may be ega ded as an exponen ial s abili y c i e ion. The exposi ion is as ollows. In Sec ion 2, we shall b ie ly collec some no ions and no a ions which a e essen ial o ou s abili y analysis. Sec ion 3 is de o ed o he in es- iga ion o exponen ially asymp o ic s abili y o s ong solu ions. Finally, in Sec ion 4 we will illus a e he heo ems de i ed in he las sec ion by s udying an example. 2 2. P elimina ies In his sec ion we a e going o s a e some basic no ions and no a ions in a sui able way. In pa icula , he eade is s ongly e e ed o G. Kallianpu and J. Xiong [11] o a sys ema ic and de ailed s a emen conce ning he ma e ial in his sec ion. Le Φ be a sepa able F ´eche space which is a coun ably Hilbe ian space, ha is, i s opology is gi en by an inc easing sepa able k·kn,n≥0, o compa ible Hilbe ian no ms. In pa icula , h oughou his pape we suppose Φ is nuclea , p ecisely, o each n≥0 he e exis s m > n such ha he canonical injec ion om Φmin o Φnis Hilbe -Schmid . He e Φnis he comple ion o Φ wi h espec o k·kn. Le Φ0be he collec ion o all con inuous linea maps om Φ o R, i.e., he dual space o Φ. We could show ha {Φn}n≥0is a sequence o dec easing Hilbe ian spaces and Φ = ∩∞ n=0Φn. Iden i ying Φ0 0wi h Φ0by Riesz’s ep esen a ion heo em, we deno e Φ0 nby Φ−nwi h no ms k·k−n,n≥0. Then {Φ−n}n≥0is a sequence o inc easing Hilbe ian spaces, Φ0is sequen ially comple e and Φ0=∪∞ n=0Φ−n.In he la e case, we shall deno e by {φp j} ⊂ Φ a comple e o hono mal sys em, o simply, CONS o Φpand {φ−p j} he CONS o Φ−pconjuga e o {φp j} o p≥0. Le θpbe he isome y om Φ−p o Φpsuch ha θpφ−p j=φp j,∀j≥1. A class o impo an examples o coun ably Hilbe ian spaces can be desc ibed app o- p ia ely as ollows. Le Hbe a eal sepa able Hilbe space and A=−La closed densely de ined sel -adjoin ope a o on Hsuch ha <−Lφ, φ >≤0 o φ∈Dom(L), he domain o L. Le {T }be he semig oups on Hde e mined by A. Fu he assume ha some powe o he esol en o Lis a Hilbe -Schmid ope a o , i.e., ∃ 1such ha (λI +L)− 1is Hilbe -Schmid .(2.1) This condi ion enables us o p o e ha he e exis 0 ≤λ1≤λ2≤ ··· and {φj} ⊂ H, a CONS o H, such ha Lφj=λjφj, o any j≥1. De ine Φ =nφ∈H:k(I+L) φk2 H<∞,∀ ∈Ro =½φ∈H: ∞ X j=1 (1 + λj)2 < φ, φj>2 H<∞,∀ ∈R¾, and he inne p oduc <·,·> on Φ by < φ, ψ > = ∞ X j=1 (1 + λj)2 < φ, φj>H< ψ, φj>H and kφk2 =< φ, φ > . Le Φ be he k·k -comple ion o Φ. We hen ha e Φ = Φ ,Φ0=[ Φ 3 and o ≤s,φ∈Φ, kφk ≤ kφksand u he mo e Φs⊂Φ wi h Φ0=H. Condi ion (2.1) implies ha he injec ion om Φqin o Φpis Hilbe -Schmid o q≥p+ 1and he e o e Φ is a coun ably Hilbe ian nuclea space, simply, CHNS. As usual, we also call he compa ible amily (Φ, H, T ) o (Φ, H, L) a special compa ible amily. We assume h oughou ha (Ω,F,{F } ≥0, P ) is a comple e p obabili y space wi h a igh con inuous il a ion {F } ≥0. A map X: Ω →Φ0is a Φ0- alued andom a iable i i is F/B(Φ0)-measu able, whe e B(Φ0) is he Bo el ield o he opological space Φ0(in he sense o s ong opology). A amily {X ; ∈R+}o Φ0- alued andom a iables is called a Φ0-p ocess. In he es o his pape , we shall conce n wi h Φ0- alued ma ingales. In pa icula , we ha e he ollowing: De ini ion 2.1. A Φ0- alued p ocess M={M } ≥0is a Φ0-ma ingale wi h espec o {F } ≥0i o each φ∈Φ, M [φ] is a ma ingale wi h espec o {F }. I is called a Φ0-squa e-in eg able-ma ingale i , in addi ion, E³M [φ]2´<∞,∀φ∈Φ, ≥0.(2.2) We le M(Φ0) ( esp. M2(Φ0)) deno e he collec ion o all Φ0-ma ingales ( esp. Φ0-squa e- in eg able-ma ingales). We also le M2,c(Φ0) = nM∈ M2(Φ0) : M [φ] has a con inuous e sion o each φ∈Φo. De ini ion 2.2. A con inuous (in he sense o s ong opology) Φ0- alued s ochas ic p ocess W= (W ) ≥0on (Ω,F, P) is called a cen e ed Φ0-Wiene p ocess wi h Q(·,·) i W sa is ies he ollowing h ee condi ions: a). W0= 0 a.s.; b). Whas independen inc emen s, i.e., he andom a iables W 1[φ1],(W 2−W 1)[φ2],···,(W n−W n−1)[φn] (2.3) a e independen o any φ1,φ2,···,φn∈Φ, 0 ≤ 1≤ ··· ≤ n,n≥1; c). Fo each ≥0 and φ∈Φ E³eiW [φ]´=e− Q(φ,φ)/2(2.4) whe e Qis a co a iance unc ional, i.e., a posi i e de ini e symme ic con inuous bilinea o m on Φ ×Φ. Clea ly, W∈ M2,c(Φ0), {W [φ] : φ∈Φ, ≥0}is a cen e ed Gaussian sys em and E³W [ψ]Ws[φ]´= (s∧ )Q(ψ, φ), ψ, φ ∈Φ, s, ≥0.(2.5) 4 De ini ion 2.3. Le Hbe a sepa able Hilbe space wi h no m k·kH. A amily {B (h) : ≥0, h ∈H}o eal- alued andom a iables is called a cyclind ical B ownian mo ion (c.B.m) on Hwi h co a iance Σ i Σ is a con inuous sel -adjoin posi i e de ini e ope a o on Hsuch ha he ollowing condi ions hold: i). Fo each h∈Hsuch ha h6= 0, <Σh, h >−1/2 HB (h) is a one-dimensional s anda d Wiene p ocess; ii). Fo each ≥0, α1,α2∈Rand 1, 2∈H B (α1 1+α2 2) = α1B ( 1) + α2B ( 2)a.s.; iii). Fo each h∈H,{B (h)}is an FB -ma ingale, whe e FB =σ{Bs(h) : s≤ , h ∈H}. {B (h) : ≥0, h ∈H}is called a s anda d H-c.B.m o simply, H-c.B.m. i i is a H-c.B.m. wi h co a iance Σ = I. Fo each φ∈Φ, le ıφ := Q(φ, ·). Then ıis an injec i e linea ope a o om Φ on o a linea subspace R(ı) o Φ0. In pa icula , o a bi a y 1, 2∈ R(ı), le HQ:= Q(ı−1 1, ı−1 2). Then <·,·>HQis an inne p oduc on R(ı). Le k·kHQbe he no m on R(ı) de e mined by he inne p oduc <·,·>HQand le HQbe he comple ion o R(ı) wi h espec o k · kHQ. Then HQis a sepa able Hilbe space and HQ⊂Φ0. I could also be shown ha he e exis s a one- o-one co espondence be ween a Φ0- alued Wiene p ocess Wwi h co a iance Qand an HQ-c.B.m. B: W = ∞ X j=1 B ( j) j(2.6) whe e { j}is a CONS o HQ; B ( ) = lim n→∞ W [ı−1 n],∀ ∈HQ(2.7) whe e { n} ⊂ R(ı) con e ges o in HQ. Conside he ollowing s ochas ic di usion equa ion (see [11] o u he de ails on s ochas ic in eg al and ela ed p ope ies) X =X0+Z 0 A(s, Xs)ds +Z 0 B(s, Xs)dWs(2.8) whe e A:R+×Φ0→Φ0,B:R+×Φ0→ L(Φ0,Φ0) a e wo measu able mappings and W is a Φ0- alued Wiene p ocess. He e L(Φ0,Φ0) deno es he collec ion o all con inuous linea mappings om Φ0in o Φ0. De ini ion 2.4. Le (Ω,F,{F } ≥0, P) be he s ochas ic basis and W a Φ0- alued Wiene p ocess wi h co a iance unc ion Q. Suppose ha X0is a Φ−p- alued andom 5 a iable such ha EkX0k2 −p<∞. Then by a Φ−p- alued s ong solu ion on Ω o he SDE (2.8) o ∈[0, T] we mean a p ocess X de ined on Ω such ha (a). X is a Φ−p- alued F -measu able andom a iable; (b). X ∈C([0, T],Φ−p), a.s.; (c). The e exis s a sequence (σn) o bounded s opping imes on Ω inc easing o in ini y such ha ∀n≥1 EZT∧σn 0kA(s, Xs)k−qds < ∞,(2.9) and EZT∧σn 0kB(s, Xs)k2 L(2)(HQ,Φ−p)ds < ∞.(2.10) He e L(2)(HQ,Φ−p) deno es he class o all Hilbe -Schmid ope a o s om HQin o Φ−p and qwill be in oduced in he ollowing assump ion (H1); (d). The SDE (2.8) is sa is ied o all ∈[0, T] and almos all ω∈Ω. I Tis eplaced by ∞, we call X a global s ong solu ion o (2.8). As we a e mainly in e es ed in he s abili y analysis, one always assumes ha he equa ion (2.8) has a unique global s ong solu ion. In pa icula , o his pu pose we shall make he ollowing assump ion (H1) [11]: The e exis s an index p0>0 such ha , ∀p≥p0,∃q≥pand a cons an K=K(p, q)> 0 such ha (D1). (Con inui y) ∀ ∈R+, he maps ∈Φ−p→A( , )∈Φ−qand ∈Φ−p→ B( , )∈L(2)(HQ,Φ−p) a e con inuous; (D2). (Coe ci i y) ∀ ∈R+and ∈Φ−p, we ha e 2A( , )[θp ] + kB( , )k2 L(2)(HQ,Φ−p)≤K(1 + k k2 −p); (2.11) (D3). (G ow h) ∀ ∈R+and ∈Φ−p, we ha e kA( , )k2 −q≤K(1 + k k2 −p); (2.12) (D4). (Lipschi z) ∀ ∈R+, 1, 2∈Φ−p, we ha e kA( , 1)−A( , 2)k−q≤Kk 1− 2k−p(2.13) and kB( , 1)−B( , 2)kL(2)(HQ,Φ−p)≤Kk 1− 2k−p.(2.14) 6 3. The Main Resul s In his sec ion, we shall de o e ou sel es o he in es iga ion o exponen ial s abili y o he equa ion (2.8). Fo simplici y, h oughou his sec ion we ake he special compa ible amily (Φ, H, L) desc ibed in Sec ion 1 as ou basic CHNS. In pa icula , o ou end we shall make he ollowing addi ional assump ion (H2): ∀ ∈R+, ∈Φ−p,p≥p0, he e exis posi i e cons an s ν > 0, µ > 0, p≤ ≤qand posi i e unc ion γ( ), ∈R+, such ha 2A( , )[θq ] + kB( , )k2 L(2)(HQ,Φ− )≤ −νk k2 − +γ( )e−µ (3.1) whe e p0,qa e in oduced as in he assump ion (H1) and γ( ) sa is ies ha o a bi a y δ > 0, γ( ) = o(eδ ), as → ∞, i.e., lim →∞ γ( )/eδ = 0. Be o e p oceeding o ou s abili y a gumen s, le us i s make he ollowing commen s on he condi ion (H2): Rema k 1. As is well known, he coe ci i y condi ion (2.11) plays an essen ial ole in he es ablishmen o he exis ence and uniqueness o he equa ion (2.8). The u he es ic i e coe ci i y condi ion (3.1) will play he ole o an exponen ial s abili y c i e ion as desc ibed below. Rema k 2. The exponen ial decay e m appea ing on he igh hand side o (3.1) is o he essence o ou s abili y pu poses. In ac , o see his, le us simply conside he ollowing one dimensional linea I ˆo equa ion: Example 3.1. Assume X sa is ies he ollowing dX =−pX d + (1 + )−qdW , ≥0 wi h ini ial da a X0= 0, whe e p,q > 0 a e wo posi i e cons an s and W is a one- dimensional s anda d B ownian mo ion. Clea ly, he le -hand side o he coe ci i y ype condi ion (3.1) now u ns ou o be 2<−p , > +h(1 + )−qi2=−2p 2+ (1 + )−2q.(3.2) whe e <·,·>deno es he s anda d inne p oduc in R. Howe e , since he las e m (1 + )−2qis no exponen ially dec easing, he solu ion is exponen ially uns able. Indeed, i is easy o ob ain he explici solu ion X =e−p Z 0 eps ·(1 + s)−qdWs=: e−p M , ≥0, which immedia ely implies ha o a bi a ily gi en q > 0 Lyapuno exponen lim →∞ log E|X |2 = 0. 7 In he mean ime, no icing he law o he i e a ed loga i hm lim sup →∞ M √2 log log = 1 a.s. and lim sup →∞ log ³R 0e2ps(1 + s)−2qds´ = 2p, we he e o e ge Lyapuno exponen lim sup →∞ 1 log |X |= 0 a.s. Tha is, in spi e o he ypical s abili y o an o dina y di e en ial equa ion dX =−pX d , he polynomial ype decay o he noise e m is no su icien o ensu e he exponen ial s abili y o i s s ochas ically pe u bed sys em. Now we a e in a posi ion o ob ain ou main esul s in he pape . Theo em 3.2. Suppose X is a solu ion o he equa ion (2.8) sa is ying (H1). Fu - he mo e we assume he coe ci i y condi ion (3.1) holds. Then he e exis cons an s τ > 0, C > 0such ha EkX k2 − ≤C·e−τ ,∀ ≥0.(3.3) Tha is, he s ong solu ion is exponen ially s able in mean squa e. In pa icula , cons an τ > 0can be aken as ollows: τ < µ, i µ≤νand τ=ν, i µ > ν. P oo . Fo a bi a y φ∈Φ, we ha e X [φ] = Z 0 A(s, Xs)[φ]ds +X jZ 0 < B(s, Xs)0φ, j>HQdWs[ı−1 j],(3.4) whe e { j} ⊂ R(ı) is a CONS o HQand ıis de ined as in Sec ion 2. He e B(s, ·)0deno es he dual ope a o o B(s, ·)∈ L(HQ,Φ− ), s≥0. I ollows om I ˆo’s o mula and De ini ion 2.4 ha o a bi a y δ > 0 wi h µ−δ > 0, we ha e e(µ−δ) ∧σnX ∧σn[φ]2−X0[φ]2 =(µ−δ)Z ∧σn 0 e(µ−δ)sXs[φ]2ds + 2 Z ∧σn 0 e(µ−δ)sXs[φ]A(s, Xs)[φ]ds + 2 X jZ ∧σn 0 e(µ−δ)sXs[φ]HQdWs[ı−1 j] +Z ∧σn 0 e(µ−δ)sQ(B(s, Xs)0φ, B(s, Xs)0φ)ds 8 whe e (σn) is he sequence o s opping imes de ined as in De ini ion 2.4. Now, since R ∧σn 0e(µ−δ)sXs[φ]HQdW [ı−1 j], ∈R+, is a con inuous ma ingale, i ollows ha E³Z ∧σn 0 e(µ−δ)sXs[φ]HQdWs[ı−1 j]´= 0, ∈R+. The e o e, le ing φ=φ k,n→ ∞,k∈Nand hen adding on index k∈N, we can deduce by Fa ou’s lemma and he condi ion (3.1) Ee(µ−δ) kX k2 − ≤EkX0k2 − + (µ−δ−ν)Z 0 e(µ−δ)sEkXsk2 − ds +Z 0 γ(s)e−δsds. (3.5) I µ−ν≤0, we he e o e deduce Ee(µ−δ) kX k2 − ≤EkX0k2 − +Z 0 γ(s)e−δsds, ha is, le ing k(δ) = R∞ 0γ(s)e−δsds, we ha e EkX k2 − ≤³EkX0k2 − +k(δ)´e−(µ−δ) . On he o he hand, i µ−ν > 0, i is always possible o choose a sui able δ > 0 such ha µ−ν−δ > 0. Then, by i ue o G onwall’s lemma we easily de i e om (3.5) ha Ee(µ−δ) kX k2 − ≤³EkX0k2 − +Z 0 γ(s)e−δsds´e(µ−δ−ν) . Hence, le ing δ > 0 small enough immedia ely yields ha he e exis s a cons an k(δ)>0 such ha EkX k2 − ≤³EkX0k2 − +k(δ)´e−ν . Combining he a gumen s abo e, we hus ob ain ou conclusion. Theo em 3.3. Assume he assump ions in Theo em 3.2 hold. Then he e exis posi i e cons an s M,βand a subse Ω0⊂Ωwi h P(Ω0)=0such ha , o each ω6∈ Ω0, he e exis s a posi i e andom numbe T(ω)such ha he ollowing holds: kX k2 − ≤M·e−β ,∀ ≥T(ω).(3.6) Tha is, he s ong solu ion is almos su ely s able. P oo . Ou p oo s a e di ided in o he ollowing se e al s eps. S ep 1. We i s ly claim ha he e exis s a cons an C > 0, independen o ∈R+, such ha Z s EkB(u, Xu)k2 L(2)(HQ,Φ− )du ≤C < ∞,0≤s≤ . (3.7) 9