On fractional Brownian motions and random dynamical systems

BoI. Soc. Esp. Ma . ApI. n051(2010), 71-79 ON FRACTIONAL BROWNIAN MOTIONS AND RANDOM DYNAMICAL SYSTEMS MARIA J. GARRIDO-ATIENZA* AND BJORN SCHMALFUg *Dp o. EDAN, Uni e si y o Se illa Ap do. 1160 41080 Se illa, Spain lns i u u Ma hema ik Fakul a ElM, Uni e si a Pade bo n, Va bu ge S asse 100, 33098, Pade bo n, Ge many [email p o ec ed] [email p o ec ed] Abs ac In his pape we conside a class o nonlinea s ochas ic pa ial di e en ial equa ions (SPDEs) d i en by a ac ional B ownian mo ion wi h he Hu s pa ame e bigge han 1/2. We show ha hese SPDEs gene a e andom dynamical sys ems. Key wo ds: F ac ional B ownian mo ions, Random dynamical sys ems, S ochas ic di e en ial equa ions. AMS subjec classi ica ions: 60H15, 37Hl0, 60H05. 1 In oduc ion A cen al ma hema ical objec in S ochas ics and S ochas ic P ocesses is he I o in eg al. I plays an impo an ole in many a eas o pu e and applied ma hema ics including ma hema ical inance, popula ion dynamics, luid dynamics, s a is ics, signal p ocessing, con ol, pa icle sys ems, o name a ew. The in eg a o o such an in eg al is o en chosen o be he B ownian mo ion ( he Wiene p ocess) o i s semima ingale gene aliza ions. These andom unc ions a e o unbounded o al a ia ion, so ha hei S iel jes in eg als do no exis . Special p ope ies o he in eg a o s and he in eg ands a e necessa y o gene alize he de ini ion o he S iel jes in eg al o he I o in eg al, and enable he de ini ion o solu ions o di e en ial equa ions d i en by B ownian mo ion. A p ope y o pa amoun impo ance o his e ec o B ownian mo ion is he independence o i s inc emen s. To mo e beyond in eg als and p ocesses cons uc ed using his p ope y is one o he mos impo an asks in he heo y o S ochas ics. We a e mos in e es ed in using he ac ional B ownian mo ion ( E n) p ocess BH whe e H E (0,1) is ixed. I is a ype o s ochas ic p ocess which de ia es signi ican ly om B ownian mo ion and semima ingales. 71 72 M.J. Ga ido-A ienza, B. Schmal uR As a cen e ed Gaussian p ocess, i is cha ac e ized by he s a iona i y o i s inc emen s and a medium- o long-memo y p ope y which is in sha p con as wi h ma ingales and Ma ko p ocesses. I also exhibi s powe scaling and pa h egula i y p ope ies wi h Holde pa ame e H, which a e e y dis inc om B ownian mo ion (no e ha he B ownian mo ion is included in his amily o models when conside ing H = 1/2). F ac ional B ownian mo ion has become a popula choice o la e o applica ions whe e classical p ocesses canno model hese non- i ial p ope ies; o ins ance long memo y, which is also known as pe sis ence, and co esponds o he case H E (1/2,1), is o undamen al impo ance o inancial da a and in in e ne a ic, see [12], [16] . F ac ional B ownian mo ion is also a good candida e o model andom long ime in luences in clima e sys ems, see [15]. E e since he pionee ing wo ks o Ziihle [17], Dec euse ond and Us iinel [5], and Lyons [11], he main h us has been o unde s and how o pe o m s ochas ic in eg a ion wi h espec o Bm in a way which is consis en wi h some p ope ies o he classical I o heo y o B ownian mo ion. In he case o highe egula i y (H > 1/2), simple ajec o ial me hods, labelled as pa hwise, can be used which make i easy o ansla e one in eg a ion heo y in o ano he , as ac ional de i a i es allow a pa hwise es ima e o he in eg als in e ms o in eg and and in eg a o using special no ms. Pa hwise in eg als his o ically ga e he i s cases whe e adequa e solu ions o s ochas ic di e en ial equa ions (SDEs) we e es ablished, e.g. Nuala and Rascanu [14]; in ini e-dimensional equa ions ha e been ea ed wi h he same success as ini e-dimensional ones, e.g. Nuala and Maslowski [13], Ga ido-A ienza e al. [6]. In his pape , we aim o in es iga e he equa ions' asymp o ics. The e a e wo heo ies dealing wi h he asymp o ic quali a i e beha io o gene al SDEs: he heo y o andom dynamical sys ems (RDS) and he heo y o exis ence and uniqueness o in a ian measu es o he associa ed Ma ko semig oup. Howe e , simila ly o Bm i sel , equa ions d i en by Bm do no gene a e a Ma ko p ocess; his p ecludes he s udy o in a ian measu es using classical ools o Bm-d i en sys ems. This mo i a es ou plan o concen a e on he s udy o Bm-d i en SDEs as RDS. The heo y o RDS, de eloped by L. A nold and cowo ke s, see [1], can be used o desc ibe he asymp o ical and quali a i e beha io o sys ems o andom and s ochas ic di e en ial/di e ence equa ion in e ms o s abili y, Lyapuno exponen s, in a ian mani olds, and a ac o s. As we ha e said, conside ing Bm ins ead o B ownian mo ion has some ad an ages because o he nice p ope ies ha he Bm enjoys and he B ownian mo ion does no . Ano he c ucial ad an age is he ollowing: o many B ownian-d i en SPDEs wi h non- i ial di usion coe icien s, i is no known i hese equa ions gene a e a RDS. The eason is ha usually s ochas ic di e en ial equa ions a e only de ined almos su ely whe e he excep ional se may depend on w since his excep ional se is ela ed o he de ini ion o an I o in eg al which is de ined as a limi o andom a iables in p obabili y. And such a amily o excep ional se s does no allow o use he heo y o RDS. Bu we can o e come such excep ional se s dealing wi h SPDEs d i en by a Bm wi h H > 1/2, On F ac ional B ownian mo ions and andom dynamical sys ems 73 p o ided he s ochas ic in eg als a e in e p e ed in he pa hwise sense. 2 P elimina ies on andom dynamical sys ems In his sec ion we e iew some basic concep s and esul s on andom dynamical sys ems ha will be used la e . In he nex de ini ion, we in oduce a sys em ha models he e olu ion o a noise. De ini ion 1 A me ic dynamical sys em (0, F, lP', {e hE1 ) wi h wo-sided ime 'll' (which is JR. in he con inuous case and Z in he disc e e one) consis s o a p obabili y space (0, F, lP') and a amily o ans o ma ions {ed E1 such ha : 1. I is a one-pa ame e g oup, i.e. 2. ( , w) E 'll' x 0 ---7 e w is measu able, 3. lP' is in a ian wi h espec o e, i.e., e lP' = lP', o all E 'll', which means ha lP'(e A) = lP'(A), o all A E F and all E 'll'. 4. lP' is e godic wi h espec o e, i.e, o any {e hE1 -in a ian se BE F, which means ha e B = B o all E 'll', we ha e ei he lP'(B) = 0 o lP'(B) = 1. We now in oduce a couple o examples o me ic dynamical sys ems. Le V = (V, 11·11, e, .)) be a sepa able Hilbe space. Conside i s he B ownian mo ion. We choose o 0 he se o con inuous unc ions Cd' = Co (JR., V) on JR. wi h alues in V which a e ze o a ze o. On his se we in oduce he compac open opology gi en by he uni o m con e gence on compac in e als in R The Bo el-o--algeb a o e his space is deno ed by B (Cd'). lP'l is he Wiene measu e. The exis ence o such a canonical p ocess 2 (Cd', B( Cn, lP'l) ollows by Kolmogo o 's heo em abou he exis ence o a 2 con inuous modi ica ion o a p ocess, see Baue [2]. The low e is gi en by e we) = we + ) - w( ), wEO (1) which is called he Wiene shi . The Wiene shi is measu able, see A nold [1] Page 544, because Cd' is sepa able and ( , w) -7 e w is con inuous. We emphasize ha his me ic dynamical sys em is e godic, see Boxle [3]. Now le us in oduce he ac ional B ownian mo ion. Gi en H E (0,1), a con inuous cen e ed Gaussian p ocess 3H ( ), E JR., wi h he co a iance unc ion , s E JR. is called a wo-sided one-dimensional ac ional B ownian mo ion ( E n), and H is he Hu s pa ame e . Assume ha Q is a bounded and symme ic linea ope a o on V which is o ace class, i.e., he e exis a comple e o hono mal basis {ediEN in V and a 74 M.J. Ga ido-A ienza, B. Schmal uR sequence o nonnega i e numbe s {AdiEN such ha Q = L:~l Ai < (X) and Qei = Aiei, i E N. A con inuous V- alued ac ional B ownian mo ion BH wi h inc emen al co a iance ope a o Q and Hu s pa ame e H is de ined by co BH ( ) = L V>:;ei/3 ( ), i=l whe e {;3 ( )}iEN is a sequence o s ochas ically independen one-dimensional Bm. No ice ha he abo e se ies is con e gen in L2(0, F, JID) since L:~l Ai < (X) and 1E(;3 ( ))2 = I l 2H o E R Rema k 1 Bl/2 is he B ownian mo ion. Using he de ini ion o BH, Kolmogo o 's heo em ensu es ha BH has a con inuous e sion. Thus we can conside he canonical in e p e a ion o an Bm: le 0 = Co(lR, V), equipped again wi h he compac open opology. Le F be he associa ed Bo el-o--algeb a and JID H he dis ibu ion o he Bm BH, and {e hER be he low o Wiene shi s de ined by (1). Then he quad uple (0, F, JID, e) is a me ic dynamical sys em which is e godic, see [9]. Fu he mo e, We now in oduce he concep o andom dynamical sys ems ha is used o desc ibe he dynamics o sys ems unde he in luence o a noise. De ini ion 2 A andom dynamical sys em (RDS) wi h one-sided ime ']['+ and phase space V is a pai consis ing o he me ic dynamical sys em (0, F, JID, e) and a mapping cp : ']['+ x 0 x V ---7 V which is (3(']['+) ® F ® 3(V), 3(V))- measu able and sa is ies he cocycle p ope y cp ( , e T W, .) 0 cp ( T, W, .) = cp ( + T, W, .), o , T E ']['+, W EO, cp(O,w,·) = id . A ypical example o co cycle mapping is he solu ion ope a o o ini e o in ini e dimensional di e en ial equa ions wi h andom coe icien s sa is ying pa icula egula i y assump ions. Ano he example is he solu ion ope a o o ini e dimensional I o-equa ions. As we announced in he In oduc ion, o in ini e dimensional I o-equa ions wi h non- i ial di usion coe icien s his p oblem is a he unsol ed. No ice ha he cocycle p ope y is he gene aliza ion o he semig oup p ope y; in ac , i we dele ed all w-dependence in he co cycle p ope y we would jus ge he semig oup p ope y. We wan o s ess ha we ha e equi ed he MDS o be de ined on wo- sided ime '][', while he RDS is only equi ed o be de ined on one-sided ime ']['+. The eason is ha we canno expec he mapping cp o be de ined on '][', since i is gi en, o ins ance, by he solu ion ope a o o a SPDE, which is no in e ible in gene al. Howe e , we can conside exp essions o he ollowing On F ac ional B ownian mo ions and andom dynamical sys ems 75 ype: cp( ,e_ w,x), o x E V, w E 0, E 11'+, exp essions ha playa c ucial ole when analyzing he exis ence o andom ixed poin s o andom a ac o s associa ed o he RDS cp, see [8]. As we ha e men ioned, he pu pose o his pape is o show ha an in ini e dimensional s ochas ic di e en ial equa ion d i en by an Bm wi h gene al di usion coe icien s gene a es a andom dynamical sys em. 3 Main esul s In his sec ion we i s in oduce some basic concep s and esul s on ac ional calculus and s ochas ic in eg als wi h espec o he Bm 3H and BH. Fo T > 0, le Wa,l(O, T; V) be he space o measu able unc ions : [0, T] ---7 V such ha I I = iT (1I (s)11 + is Il (s) - (()11 dl") ds < (X) a a (1")00+1" , o S 0 s-" whe e 1- H < a < ~ is ixed, so we need o conside om now on H E (1/2,1). Following Ziihle [17], o E wa,l (0, T; V) we de ine he s ochas ic in eg al as he gene alized S iel jes in eg al !aT d 3H = (-l)a!aT Do+ (s)D~-=-a 3! -(s)ds, (3) i d 3H = !aT 1(s, )d 3H, o 0::; s < ::; T, whe e, in gene al, o ° ::; a < b ::; T, 3 !- (s) := 3H (s) - 3H (b), and o a < < b he Weyl de i a i es a e gi en by a 1 ( ( ) j ( ) - (() ) Da+ ( ) = (l _ a) ( _ a)a + a a ( _ ()a+1 d( , D1- a 3H ( ) = (_1)1-00 ( 3H( ) - 3H(b) +(l-a)l b 3H( ) - 3 H (()d() b- b- (a) (b - )1-a (( - )2-a ' whe e deno es he Gamma unc ion. I can be p o ed (see, o ins ance, Nuala and Ra§canu [14], Dec euse ond and Us iinel [5], Ziihle [17]) ha he s ochas ic in eg al (3) exis s. Now we de ine he s ochas ic in eg al wi h espec o he in ini e dimensional Bm BH. Le L(V) deno e he space o linea bounded ope a o s on V and le G : 0 x [0, T] ---7 L(V) be an ope a o such ha G(w, ·)ei E wa,l(o, T; V) o each i E Nand w E o. We de ine whe e he con e gence o he sums in (4) is unde s ood in V. 76 M.J. Ga ido-A ienza, B. Schmal uR The ollowing esul es ablish ha when making a change o a iable in he s ochas ic in eg al, we no only ha e o shi he in eg a ion in e al and he a iable bu also he pa h o he Bm ( o he p oo , see [6]). Lemma 1 Fo a, b, E JR., assuming ha bo h in eg als a e well-de ined, Ib Ib- a G(s)dw(s) = a- G(s + )de w(s). Conside now he ollowing s ochas ic e olu ion equa ion in V { du( ) = (Au( ) + F(u( )))d + G(u( ))dw( ), u(O) = Uo E V whe e w deno es he in ini e dimensional Bm BH (see (2)). (5) Assume ha A is he in ini esimal gene a o o an analy ic semig oup Se), and ha F : V --7 V is Lipschi z con inuous wi h Lipschi z cons an LF, and G : V --7 L(V) and G' : V --7 L(V, L(V)) a e Lipschi z con inuous in he ollowing senses: sup IIG( dei -G( 2)eill ::; Lcll l - 211, iEN sup IIG'( dei -G'( 2)eiIIL(V) ::; L~ll l - 211, iEN (6) (7) whe e {ediEN is he comple e o hono mal basis in V in oduced in Sec ion 2. The solu ion o (5) on [0, T] is a V- alued p ocess u whose pa hs a e o e e y wE 0 elemen s o Woo,l(O, T; V), o an a E (1 -H, ~), and u( ) = S( )uo+ !a S( -s)F(u(s))ds+ !a S( -s)G(u(s))dw, E [O,T], (8) whe e he s ochas ic in eg al has o be unde s ood acco ding o (4). Fo such an a E (1 - H, ~ ), deno e by w ~(X) (0, T; V) he Banach space o measu able unc ions x : [0, T] --7 V such ha (I ( ~ i Ilx( ) - x( )11 ) IlxIIOO,~,(I = sup e- Ilx( )11 + ( )1+ 00 d < 00 E[O,T] ° - o (J ?: 1, and ~ E [a, 1 - a). The ole o he ac o ~ is c ucial when p o ing he ollowing exis ence heo em, which p oo can be ound in [6]. Theo em 2 Le a E (1 -H, ~), (J ?: 1 and ~ E [a,l - a). Assume F is Lipschi z con inuous, and ha G and G' sa is y (6) and (7). Then, o each ini ial poin Uo E V he e exis s a unique solu ion o equa ion (8) wi h i s pa hs in w ~(X)(O, T; V). In addi ion, he mapping <I> : V --7 w ~(X)(O, T; V) gi en by <I> : Uo -7 u is con inuous o w E o. On F ac ional B ownian mo ions and andom dynamical sys ems 77 Theo em 3 The solu ion u o (8) de ines a andom dynamical sys em cp : lR+ x [2 X V ---7 V, gi en by cp( , w, uo) = S( )uo + a S( -s)F(u(s))ds + a S( -s)G(u(s))dw. P oo . The measu abili y ollows by [4] Lemma 111.14. T i ially cp(O, w, x) = Uo. Le us check hen he cocycle p ope y: o , T E lR+, wE [2 and Uo E V, we ha e + T cp( +T,W,UO)=S( +T)UO+ Jo S( +T-s)F(u(s))ds + T + Jo S( +T-s)G(u(s))dw(s) = S( ) (S(T)UO + aT S(T -s)F(u(s))ds + aT S(T -s)G(U(S))dW(S)) I +T I +T + T S( + T -s)F(u(s))ds + T S( + T -s)G(u(s))dw(s). Making he change o a iable s -T = , applying Lemma 1, I +T T S( + T -s)G(u(s))dw(s) = Jo S( - )G(u( + T))deTw( ), and hen, se ing y(s) = u(s + T), o s E [0, l, cp( + T, w, uo) = S( )y(O) + a S( - )F(y( ))d + a S( - )G(y( ))deTw( ) = cp( , eTw,') 0 cp(T, W, uo). D P o ing ha ou s ochas ic equa ion (8) gene a es a RDS is he s a ing poin o analyze i s asymp o ic beha io . One possibili y, which is a key concep desc ibing he dynamics o RDS gene a ed by Bm-d i en SDEs, is he so-called global a ac o , which is an in a ian compac andom se a ac ing o he bounded andom se s. The essen ial dynamics ake place in a neighbo hood o he a ac o (see [8]). Ano he op ion o discuss he s abili y o Bm-d i en SDEs is o s udy he exis ence o s able and uns able mani olds and Lyapuno exponen s, see [10] and [7]. Such smoo h mani olds a e in a ian unde he dynamics o he sys ems, and on hem, he s a es a e a ac ed o epelled by a s eady s a e. Re e ences [1] L. A nold, Random Dynamical Sys ems, Sp inge Monog aphs in Ma hema ics, Sp inge -Ve lag, Be lin 1998. 78 M.J. Ga ido-A ienza, B. Schmal uR [2] H. Baue , P obabili y Theo y, de G uy e S udies in Ma hema ics. Wal e de G uy e & Co., Be lin 1996. [3] P. Boxle , S ochas ische Zen umsmannig al igkei en. Ph.D. hesis, Ins i u U Dynamische Sys eme, Uni e si ii B emen, 1988. [4] C. Cas aing, M. 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