A Stochastic Pitchfork Bifurcation in a Reaction-Diffusion Equation

A s ochas ic pi ch o k bi u ca ion in a eac ion-di usion equa ion By Tom´ as Ca aballo, Jos´ e A. Langa, & James C. Robinson† Depa amen o de 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 We s udy in some de ail he s uc u e o he andom a ac o o he Cha ee-In an e eac ion-di usion equa ion pe u bed by a mul iplica i e whi e noise, du= (−Au +βu −u3) d +σu ◦dW x∈D⊂Rm. Fi s we p o e, o m≤5, a lowe bound on he dimension o he andom a ac o which is o he same o de in βas he uppe bound we de i ed in an ea lie pape , and is he same as ha ob ained in he de e minis ic case. Then we show, o m= 1, ha as βpasses h ough λ1( he i s eigen alue o he nega i e Laplacian) om below, he sys em unde goes a s ochas ic bi u ca ion o pi ch o k ype. We belie e ha his is he i s example o such a s ochas ic bi u ca ion in an in ini e- dimensional se ing. Cen al o ou app oach is he exis ence o a andom uns able mani old. Keywo ds: Random a ac o s, s ochas ic bi u ca ion, Hausdo dimension 1. In oduc ion A ac o s o in ini e-dimensional dynamical sys ems ha e p o en e y use ul ool in he s udy o he asymp o ic beha iou o many pa ial di e en ial equa ions (see o example Hale (1988), Ladyzhenskaya (1991), Temam (1988, 2nd ed. 1997)). Much a en ion has been ocused on he dimension o hese a ac o s, since in many cases one can p o e ha his is ini e, and deduce ha he long- ime beha iou o he sys em depends on only a ini e numbe o deg ees o eedom. The de ini ion o an a ac o has been gene alized o he s ochas ic case by C auel & Flandoli (1994) and Schmal uß (1992), and has once again been ui ul in u he ing ou unde s anding o he associa ed andom and s ochas ic di e en ial equa ions, in pa icula in ini e dimensions (A nold 1998). Roughly speaking, a andom a ac o is a amily o compac andom se s, which a e in a ian o he s ochas ic low and a ac all solu ions ‘ om =−∞’. Once again, i is possible o show ha ce ain o hese a ac o s a e ini e-dimensional, al hough he numbe o examples is much mo e limi ed han in he de e minis ic case. Debussche (1997) has adap ed he mos powe ul de e minis ic echnique o †Pe manen add ess: Ma hema ics Ins i u e, Uni e si y o Wa wick, Co en y CV4 7AL. UK. A icle submi ed o Royal Socie y T EX Pape 2T. Ca aballo, J.A. Langa, & J.C. Robinson ea he s ochas ic case, and in a p e ious pape (Ca aballo e al. 2000) we used his me hod o ob ain an uppe bound on he Hausdo dimension o he andom a ac o o he Cha ee-In an e eac ion-di usion equa ion pe u bed by a mul i- plica i e noise in he sense o S a ono ich, du= (−Au +βu −u3) d +σu ◦dW x∈D⊂Rm.(1.1) Di e en echniques a e equi ed, o bo h s ochas ic and de e minis ic sys ems, o ob ain lowe bounds on he dimension o hese a ac o s. Such bounds a e gene ally based on inding some in a ian mani olds which mus be subse s o he a ac o (Babin & Vishik 1983). He e we adop his app oach, using some ideas due o DaP a o & Debussche (1996) o show (in §4) ha he e is an n-dimensional uns able mani old nea he ze o solu ion when λn< β < λn+1 ( he λja e he eigen alues o he nega i e Laplacian, see §3). This gi es a lowe bound on he a ac o dimension which is o he same o de as ou p e ious uppe bound. Rema kably hese bounds do no depend on he le el o he noise (σin (1.1)), and a e o he same o de as he bounds in he de e minis ic case. In he las decade he e has been some esea ch in s ochas ic bi u ca ion heo y, al hough i s ill seems o be unclea how his heo y can be se up in gene al (A nold 1998, 2000). Howe e , o ou knowledge he e a e as ye no s udies o bi u ca ions in in ini e-dimensional s ochas ic di e en ial equa ions. We adop he e he dynamical concep o a s ochas ic bi u ca ion, which is unde s ood as a quali a i e change in he in a ian se s (o in a ian measu es) o he sys em (see §5). Fo he one-dimensional case m= 1 we use he mani old s uc u e, which ga e us he lowe bound on he dimension, o in es iga e in mo e de ail wha happens o he andom a ac o as βpasses h ough λ1 om below. Fo β < λ1we showed p e iously ha he a ac o is jus he poin {0}, and ou lowe bound show ha o β > λ1 he Hausdo dimension o he a ac o is a leas 1. We show ha o λ1< β < λ2 he uns able mani old is in ac angen o he space spanned by he i s eigen unc ion o he Laplacian, and hus ac ually has one b anch in he cone o posi i e solu ions and he o he one in he cone o nega i e solu ions (§5). I is only a sho s ep om he e o he equi ed pi ch o k bi u ca ion, using he heo y o mono one andom dynamical sys ems de eloped by A nold & Chuesho (1998a & b). We end wi h some conclusions and open p oblems. 2. Fo mula ion o he p oblem and de e minis ic esul s Le D⊂Rm,m≤5, be an open bounded se wi h egula bounda y. We conside he ollowing Cha ee-In an e eac ion-di usion equa ion in Dpe u bed by a linea mul iplica i e whi e noise du= (∆u+βu −u3)d +σu ◦dW ,(2.1) wi h u(x, ) = 0 o x∈∂D, and whe e W·(ω) : Ω →C0(R,R) is a one-dimensional Wiene p ocess on he p obabili y space (Ω,F,P). We ew i e (2.1) as he ollowing di e en ial equa ion on L2(D), du= (−Au +βu −u3) d +σu ◦dW (2.2) A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 3 whe e A=−∆ on Dwi h he app op ia e (Di ichle ) bounda y condi ions. The ope a o Ais posi i e, linea , sel -adjoin and wi h compac in e se. Thus he e exis s a sequence {λj}o posi i e eigen alues, whose associa ed eigen unc ions wj (wi h Awj=λjwj) o m an o hono mal basis o H(e.g. Rena dy & Roge s 1992). We o de hese so ha λn+1 ≥λn. (a)The de e minis ic case (σ= 0) When σ= 0, exis ence and uniqueness esul s a e p o ed o he equa ion in Ma ion (1987), Temam (1988), and Robinson (2001). The e exis s a unique weak solu ion u( ;u0)∈L2(0, T;H1 0(D)) ∩L4((0, T)×D)∩C([0, T]; L2(D)), so ha in pa icula we can use he solu ions o de ine a semig oup S( ) on L2(D), ia S( )u0=u( ;u0). S( ) sa is ies he usual semig oup p ope ies, S(0) = id, S( )S(s) = S( +s),and S( )u0con inuous in and u0.(2.3) I is shown in all h ee o he abo e e e ences ha he equa ion also enjoys he exis ence o a global a ac o A, ha is, a compac in a ian se which a ac s he o bi s o all bounded se s, i.e. S( )A=A o all ∈R(2.4) and dis (S( )B, A)→0 as → ∞,(2.5) whe e Bis any bounded subse o L2(D) and dis (A, B) is he Hausdo semidis- ance be ween Aand B, dis (A, B) = sup a∈A in b∈B|a−b|.(2.6) Es ima es o he Hausdo dimension o his a ac o in e ms o βcan also be ob ained, namely dH(A)≤Cβm/2. (Recall ha D⊂Rm.) By showing he exis ence o an uns able mani old nea he o igin, Babin & Vishik (1983) show ha a simila lowe bound holds, so ha in ac he e exis s a cons an csuch ha cβm/2≤dH(A)≤Cβm/2 (see also Temam 1988). We w i e his mo e compac ly as dH(A) = O(βm/2).(2.7) We will ob ain, below, he same beha iou in he s ochas ic case. A icle submi ed o Royal Socie y 4T. Ca aballo, J.A. Langa, & J.C. Robinson In he pa icula case m= 1 he s uc u e o his a ac o is ex emely well un- de s ood: he a ac o consis s o a collec ion o s a iona y poin s, which bi u ca e om he o igin as βpasses h ough each eigen alue λj, and he uns able mani olds joining hem (Hale 1988; Hen y 1984). In pa icula , as βpasses h ough λ1 he a ac o changes om a single s able ixed poin a u≡0 o a se , homeomo - phic o an in e al, which consis s o he one-dimensional uns able mani old o he o igin. This has wo dis inc componen s, one o which lies in he cone o posi i e solu ions and on which all solu ions app oach a new posi i e ixed poin , and one which lies in he cone o nega i e solu ions and on which all solu ions app oach a new nega i e ixed poin . I is his pi ch o k bi u ca ion ha we will seek o mi o in he s ochas ic case. 3. Random a ac o s We now discuss b ie ly he de ini ion o a andom dynamical sys em and a andom a ac o , unsu p isingly using ou equa ion (2.2) as an illus a i e example. (a)Random dynamical sys ems Le (Ω,F,P) be a p obabili y space and {θ : Ω →Ω, ∈R}a amily o measu e p ese ing ans o ma ions such ha ( , ω)7→ θ ωis measu able, θ0= id, and θ +s=θ θs o all s, ∈R. The low θ oge he wi h he co esponding p obabili y space, (Ω,F,P,(θ ) ∈R) is called a (measu able) dynamical sys em. In ou case we ake (Ω,F,P) o be he p obabili y space which gene a es he one-dimensional Wiene p ocess dW . The shi θ ac s on Ω so ha W (θsω) = W +s(ω)−Ws(ω),(3.1) he addi ional sub ac ed e m ensu ing ha W·(θsω) is s ill a B ownian mo ion. Fo his example i also ollows ha he shi θ is e godic (A nold 1998). A con inuous andom dynamical sys em (RDS) on a Polish space (X, d) wi h Bo el σ-algeb a Bo e θon (Ω,F,P) is a measu able map ϕ:R+×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) o all , s ∈R+(cocycle p ope y) iii) ϕ( , ω) : X→Xis con inuous. When σ6= 0 i is known (Pa doux 1975) ha o each u0∈L2(D) and T > 0, he e exis s a unique solu ion u( ;x0) o (2.2), wi h u( ;x0)∈L2(Ω ×(0, T); H1 0(D)) ∩L4(Ω ×(0, T)×D)∩L2(Ω; C(0, T;L2(D)). A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 5 In pa icula , i ollows ha he solu ions o (2.2) can be used o gene a e a andom dynamical sys em i we de ine ϕ( , ω)u0=u( ;ω, u0), whe e u( ;ω, u0) is he solu ion o (2.2) wi h noise ωand ini ial condi ion u(0) = u0. (b)Random a ac o s A andom se A(ω) is said o be a andom a ac o o he RDS ϕi i) A(ω) is a andom compac se , ha is, P−a.s.,A(ω) is compac and o all x∈X, and he map ω7→ dis (x, A(ω)) is measu able wi h espec o F. ii) P−a.s. ϕ( , ω)A(ω) = A(θ ω) o all ≥0 (in a iance) and iii) o all B⊂Xbounded (and non- andom), P−a.s., lim →∞ dis (ϕ( , θ− ω)B, A(ω)) = 0, whe e dis ( . , . ) deno es he Hausdo semidis ance in X(c . (2.6)). Since ϕ( , θ− ω)u0can be in e p e ed as he posi ion a = 0 o he ajec- o y which was a u0a ime − , his pullback con e gence p ope y is essen ially a ac ion ‘ om =−∞’. In Ca aballo e al. (2000) we p o ed he exis ence o a andom a ac o o ou equa ion, using a heo em due o C auel & Flandoli (1994). We also showed ha i β < λ1 hen A(ω) = {0}, and mo e gene ally (using a esul o Debussche 1998) ha i β < 1 d d X j=1 λj hen dH(A(ω)) < d. Since one can bound Pd j=1 λj≤Cd(m+2)/m, his implies he uppe bound dH(A(ω)) ≤c1βm/2.(3.2) No e ha his is o he same o de as he de e minis ic bound in (2.7), and o cou se is ex emely sugges i e o he ac ha he a ac o becomes mo e complica ed as βinc eases h ough λ1. 4. A lowe bound on he a ac o dimension We now u n o p o ing he p omised lowe bound on he dimension o he andom a ac o A(ω). We make use o an idea om he de e minis ic heo y, which is o show ha he a ac o mus con ain an uns able mani old o a ce ain dimension. Howe e , his app oach usually equi es a ious di e en iabili y p ope ies o he low, and such echnicali ies a e by no means s aigh o wa d in he s ochas ic case. Fu he mo e, he e is cu en ly no well-de eloped heo y o uns able mani olds o gene al s ochas ic PDEs. A icle submi ed o Royal Socie y 6T. Ca aballo, J.A. Langa, & J.C. Robinson So we p oceed in a manne which we belie e is sligh ly unusual, and could p o e use ul no only o o he s ochas ic examples bu also in he de e minis ic case, by p o ing no he exis ence o a C1uns able mani old, bu o a Lipschi z mani old in a neighbou hood o he o igin. The heo y o ine ial mani olds, in oduced by Foias e al. (1988) in he de e minis ic case, and de eloped by a ious au ho s o s ochas ic equa ions (Bensoussan & Flandoli 1994; Chuesho & Gi ya 1994), is well sui ed o his, and we ollow some ideas om a pape o DaP a o & Debussche (1996) in ou p oo . In his way, we p o e he ollowing heo em. Theo em 4.1. I λn< β < λn+1 hen he e exis s a se Mδ(ω), a (locally) in- a ian n-dimensional Lipschi z mani old, which is pa o he uns able se o he o igin, i.e. lim →∞ dis (ϕ(− , ω)u, 0) = 0 (4.1) o all u∈ M(ω). In (4.1), ϕ(− , ω)uis he poin ∈ M(θ− ω) such ha ϕ( , θ− ω) =u( he ac ha M(ω) is a ini e-dimensional in a ian mani old ensu es ha such a poin exis s – see p oposi ion 4.6 o mo e de ails). We now appeal o a esul o C auel (1999), which gua an ees ha he uns able se o he o igin (de ined p ecisely as in (4.1)) mus be a subse o he andom a ac o , and so ob ain he lowe bound on he a ac o dimension gi en in he ollowing heo em. Theo em 4.2. P o ided ha m≤5, i λn< β < λn+1 hen dH(A(ω)) ≥n. In pa icula , he dimension o he a ac o sa is ies dH(A(ω)) = O(βm/2). (a)T unca ing he equa ion We p o e he exis ence o Mδ(ω) in a se ies o p oposi ions. We apply a e sion o he heo y in DaP a o & Debussche (1996) de eloped o p o e he exis ence o ine ial mani olds o s ochas ic PDEs wi h a gene al mul iplica i e noise e m. Howe e , he e we do no look o an ine ial mani old, bu in ac an uns able mani old in a neighbou hood o he o igin. The i s ask is o unca e he equa ion in a neighbou hood o he o igin, o ensu e ha he nonlinea e m is globally Lipschi z, wi h a small Lipschi z cons an . [Ou app oach is, in ac , simila o ha o Boxle (1991) o cen e mani olds, al hough he e we can unca e ou equa ion in a manne which is independen o ω.] Al hough his is a s anda d app oach in he de e minis ic heo y, in which one has a compac abso bing ball, i is pe haps he main weakness o he heo y o ine ial mani olds o s ochas ic equa ions, whe e in gene al he adius o he abso bing ball will depend on ω, and as ye he e is no p oo which allows he Lipschi z cons an o he nonlinea i y o a y in his way. Howe e , since we equi e only a local uns able mani old, we will be able o unca e he equa ion in a consis en ashion. We es ic o he case m≤5, since hen we can use he embedding H2α⊂L6 wi h 3/4< α < 1 o show ha F(u) = u3is Lipschi z om H2αin o L2(we supp ess he Din all ou unc ion spaces om now on), |u3− 3|L2≤C(kuk2 H2α+k k2 H2α)ku− kH2α.(4.2) A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 7 Indeed, we ha e |u3− 3|2 L2=ZD (u3(x)− 3(x))2dx =ZDµZu(x) (x) 3s2ds¶2 dx ≤9ZD |u(x)− (x)|2(u(x)2+ (x)2)2dx ≤9µZD |u(x)− (x)|2pdx¶1/pµZD |u(x)2+ (x)2|2qdx¶1/q ≤9|u− |2 L2p(|u|L4q+| |L4q)4 ≤9|u− |2 L6(|u|L6+| |L6)4, aking (p, q) = (3,3/2). Fo a C1cu -o unc ion θ:R+→[0,1], such ha θ( ) = ½1 ≤1 0 ≥2 wi h |θ0( )| ≤ 2, i is s aigh o wa d o e i y (c . Temam, 1988) ha F(u) = −θµkukH2α R¶u3(4.3) is globally bounded, |F(u)| ≤ M0 o all u∈H2α, and globally Lipschi z, |F(u)−F( )| ≤ ˜ L ku− kH2α o all u, ∈H2α.(4.4) wi h ˜ L ≤c0R2 o some cons an c0. No e in pa icula ha F(u) = −u3in BH2α(0, R), and so he wo equa ions ag ee in a small H2α-neighbou hood o he o igin. We will ind i mo e con enien o use he no m equi alence be ween he H2α no m and he no m in D(Aα). W i ing |Aαu|=|u|α, whe e Ais he nega i e Laplacian on Dwi h Di ichle bounda y condi ions, we ha e kukH2α≤c|u|α.(4.5) We can e-w i e (4.4) as |F(u)−F( )| ≤ L |u− |α o all u, ∈D(Aα),(4.6) whe e L =c˜ L (wi h cas in (4.5)). No e ha we can make L as small as we wish p o ided ha Rin (4.3) is chosen small enough. A icle submi ed o Royal Socie y 8T. Ca aballo, J.A. Langa, & J.C. Robinson (b)An in a ian mani old o he unca ed equa ion We now wo k wi h he unca ed equa ion du= (−Au +βu +F(u)) d +σu ◦dW ,(4.7) and p o e he exis ence o an in a ian mani old o dimension nwhen λn< β < λn+1. In he s a emen o he heo em we w i e Pn o he o hogonal p ojec ion on o he i s neigen unc ions o A, o de ed so ha he co esponding eigen alues a e non-dec easing in n, and Qn o i s o hogonal complemen , Qn=I−Pn. Theo em 4.3. I λn< β < λn+1, and Rin (4.3) is chosen small enough ha L sa is ies β−λn>4L λα n(4.8) and λn+1 −β > (2KL )1/(1−α),(4.9) hen (4.7) has an n-dimensional in a ian mani old M(ω), ϕ(λ, ω)M(ω) = M(θλω), λ ≥0. Fu he mo e, M(ω)is gi en as he g aph o a Lipschi z unc ion Φω:PnH→ QnH∩D(Aα), |Φω(p1)−Φω(p2)|α≤2|p1−p2|α.(4.10) We analyse he equa ion ω-by-ω. To his end we conside no equa ion (2.2) o u, bu , se ing = e−σW uand obse ing ha d = e−σW du−e−σW σu ◦dW , we can ea he non-au onomous equa ion o , d d =−A +β + e−σW F(eσW ).(4.11) We w i e his as d d =−A +β +Fσ( ), whe e Fσ( ) = e−σW F(eσW ). No e ha |Fσ(u)−Fσ( )| ≤ e−σW |F(eσW u)−F(eσW | ≤e−σW L |eσW u−eσW |α =L |u− |α,(4.12) A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 9 so ha he Lipschi z p ope y o Fin (4.6) is ans e ed o Fσ. Fu he mo e i ollows, since he suppo o Fis con ained in a bounded se in D(Aα), and F(0) = 0, ha |Fσ(u)| ≤ M1 o some cons an M1. The solu ion ope a o co esponding o he ans o med equa ion we w i e as ψ( , ω). Clea ly, ψ( , ω) = e−σW (ω)ϕ( , ω).(4.13) Fo a unc ion y: (−∞,0] →PnH(4.14) we de ine a no m kykE= sup ∈(−∞,0] |y( )|α,(4.15) and simila ly o z: (−∞,0] →QnD(Aα).(4.16) Fo a pai (y, z), we de ine k(y, z)kE= max(kykE,kzkE).(4.17) The space o all unc ions (y, z) as in (4.14) and (4.16) wi h ini e Eno m ( he no m in (4.17)) we deno e by E. We will need he ollowing s anda d bound on he ope a o no m o Aαe−A Qn om L2in o L2(see Temam 1988, o example), kAαe−A Qnkop ≤˜ K( −α+λα n+1)e−λn+1 o all ≥0.(4.18) We also se K=˜ KZ∞ 0 ( −α+ 1)e− d . (4.19) ¿F om now on we w i e P=Pnand Q=Qn. Roughly speaking, we ollow he p oo in DaP a o & Debussche (1996). Howe e , he e he a gumen is somewha di e en , since ou pa icula choice o mul iplica i e noise allows us o conside he ans o med equa ion (4.11) ω-by-ω, and we do no ha e o deal wi h he p oblems which a ise when ying o sol e a mo e gene al s ochas ic equa ion backwa ds in ime. We also need o ailo he analysis o make su e ha he dependence on β emains explici . P oposi ion 4.4. Fo each ω∈Ω, he e exis s a unique solu ion (y, z)o he coupled equa ions dy d =−Ay +βy +PFσ(y+z)y(0) = y0 dz d =−Az +βz +QFσ(y+z)z( )→0 as → −∞,(4.20) on he in e al (−∞,0]. A icle submi ed o Royal Socie y 16 T. Ca aballo, J.A. Langa, & J.C. Robinson (b)Di e en iabili y o M(ω)a 0 Fi s we p o e he di e en iabili y p ope y o he mani old. In pa icula , he cubic bound (5.1) will be used o show ha a po ion o M(ω) lies in K+. P oposi ion 5.2. The mani old M(ω)is angen o PnHa he o igin, and mo e- o e |Φω(p)|α≤Cω|p|3 α.(5.1) P oo . Fi s , obse e ha Fσ(u) = −e−σW θµ|ueσW |α R¶(ueσW )3 is essen ially cubic, so ha |Fσ(u)| ≤ e2σW µZu6dx¶1/2 ≤e2σW kuk3 L6 ≤Ce2σW |u|3 α. Now, Φω(y0) is z(0) om he coupled equa ions in p oposi ion 4.4, so we know (c . (4.22)) ha z(0) = Z0 −∞ e(A−βI)sQFσ(y(s) + z(s)) ds, and hence |z(0)|α≤˜ KZ −∞ ((−s)−α+ (λn+1 −β)α)e(λn+1−β)sCe2σWs|u|3 αds. Since |u|3 α≤(|y|α+|z|α)3≤8(|y|3 α+|z|3 α), we ha e, using (4.29), |z(0)|α≤Kω[kyk3 E+kzk3 E].(5.2) Now, we know ha z( ) = Z −∞ e−(A−βI)( −s)QFσ(y+z) ds, and we can es ima e |z( )|αas in he p oo o p oposi ion 4.4, using he ac ha Fσ(0) = 0, |z( )|α≤˜ KL Z −∞ (( −s)α+ (λn+1 −β)α)e−(λn+1−β)( −s)|y+z|αds ≤µ˜ KL Z −∞ (( −s)α+ (λn+1 −β)α))e−(λn+1−β)( −s)ds¶ A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 17 ×(kykE+kzkE). Now, we saw in p oposi ion 4.4 ha he in eg al exp ession he e is bounded by 1 2, and so we ha e kzkE≤1 2(kykE+kzkE), i.e. kzkE≤ kykE. Equa ion (5.2) now becomes |z(0)|α≤2Kωkyk3 E,(5.3) so i only emains o es ima e kykE. Howe e , his is he solu ion o he go e ning equa ion which has y(0) = y0and z(0) = Φω(y0), and so which lies on M(ω). The inequali y (4.28) om p oposi ion 4.6 shows ha in ac |y( )|α≤ |y0|α o all ≤0, and so kykE≤ |y0|α. Combining his wi h (5.3) gi es he equi ed bound. (c)The bi u ca ion heo em We now p o e ou second main esul . Theo em 5.3. Le m= 1. Then (2.2) unde goes a s ochas ic pi ch o k bi u ca ion a β=λ1. In pa icula , o β > λ1 he e exis posi i e and nega i e andom ixed poin s ±a(ω), and a(ω)→0as β↓λ1. No e ha one could easily ecas his esul in e ms o he appea ance o wo new in a ian measu es, namely he andom Di ac measu es concen a ed a ±a(ω). P oo . Choose and ix β > λ1. Since he i s eigen unc ion o he Laplacian is posi i e, we use he cubic es ima e o show ha Mδ(ω) mus con ain a posi i e unc ion. Mo e p ecisely, we le K+={u∈L2(D) : u(x)≥0 almos e e ywhe e}, and K−={u∈L2(D) : u(x)≤0 almos e e ywhe e}. The main idea is o show ha one po ion o he uns able mani old mus be a subse o K+, and ano he a subse o K−. On a one-dimensional domain [0, L] we know ha he eigen unc ions o he Laplacian wi h Di ichle bounda y condi ions a e p opo ional o wn(x) = sin(nπx/L). Since he mani old Mδ(ω) is gi en as a g aph o e he space spanned by he i s eigen unc ion, o each ixed ω he mani old consis s o a amily o unc ions (he e pa ame ised by ²) gi en in he o m u(x, ²) = ²sin(πx/L) + ∞ X j=2 cjsin(jπx/L), whe e cj=cj(², ω). A icle submi ed o Royal Socie y 18 T. Ca aballo, J.A. Langa, & J.C. Robinson Since ¯¯¯¯ sin(jπx/L) sin(πx/L)¯¯¯¯ ≤j (simply ew i e he sine e ms using complex exponen ials) we ha e u(x, ²)≥sin(πx/L)·²− ∞ X j=2 |cj|j¸. Using he cubic bound on Φω, we know ha ¯¯¯¯ ∞ X j=2 cjsin(jπx/L)¯¯¯¯α =Cµ∞ X j=2 |cj|2j4α¶1/2 ≤Cω¯¯²sin(πx/L)|3 α=Kω²3. Since α > 3/4 we can now w i e ∞ X j=2 |cj|j≤µ∞ X j=2 j4α|cj|2¶1/2µ∞ X j=2 j2(1−2α)¶1/2 ≤C²3, and so u(x, ²)≥sin(πx/L)[²−C²3]. Thus o ² > 0 small enough, u(x, ²)≥0 on [0, L]; simila ly o ² < 0 and small enough, u(x, ²)≤0 on [0, L]. In o he wo ds, he e a e po ions o Mδ(ω) nea he o igin which in e sec non- i ially wi h K+and K−. Now, Ko elenez (1992) shows ha he cocycle gene a ed by he equa ion is o de -p ese ing on L2(D), so ha i u0≥ 0almos e e ywhe e hen ϕ( , ω)u0≥ϕ( , ω) 0. In pa icula , since ze o is a ixed poin o he equa ion, K±a e in a ian subspaces o ϕ. (These wo ac s a e easy o check i u0and 0a e con inuous unc ions by applying he s anda d de e minis ic heo y (essen ially he maximum p inciple, see Smi h 1995) o he equa ion o ( ) = e−σW u( ).) By ollowing he analysis in Ca aballo e al. (2000) one can show ha he low in each o hese subspaces has a posi i ely in a ian compac abso bing se , namely a bounded se in H1(D) wi h adius (ω), whe e (ω) is a empe ed andom a iable. In pa icula he e o e, using he heo y in Flandoli & Schmal uss (1996) he low in each o hese subspaces has i s own andom a ac o , A+(ω) and A−(ω), and hese a ac all empe ed andom se s. Since Mδ(ω)∩K± o ms pa o he uns able se o he o igin, i mus be a subse o A±(ω), and so in pa icula A±(ω) is non- i ial. In hei 1998 pape A nold & Chuesho de elop an ex ensi e heo y o o de - p ese ing andom dynamical sys ems, and in pa icula one o hei esul s ( he- o em 2) p o ides he exis ence o new s ochas ic ixed poin s in his example (es- sen ially we ollow Chuesho , 2000): i ϕis an o de -p ese ing andom dynamical sys em o which he e exis s a andom in e al J(ω) = {u:b(ω)≤u≤c(ω)} which is a ac ed o and con ains A(ω), hen he e exis andom ixed poin s u−(ω)≤u+(ω) such ha u−(ω)≤u≤u+(ω) o all u∈ A(ω). A icle submi ed o Royal Socie y A s ochas ic pi ch o k bi u ca ion 19 Since A(ω) lies wi hin a andom ball in H1(D) o adius (ω), and since H1(D)⊂ C0(D) wi h kuk∞≤C(D)kuk1 H, i ollows in pa icula ha 0≤u≤c(D) (ω) o all u∈ A+(ω). Using he abo e esul we can deduce ha he e exis wo andom ixed poin s u−(ω) and u+(ω) such ha u−(ω)≤u≤u+(ω) o all u∈ A+(ω). Clea ly u−(ω) = 0, bu since A+(ω) is non- i ial his p o es he exis ence o a new andom (posi i e) ixed poin a(ω) = u+(ω). By symme y he e is also a new andom (nega i e) ixed poin −a(ω). (Fo a simila a gumen see Chuesho (2000).) Tha a(ω)→0 as β↓λ1 ollows om a sligh a ian o he gene al esul on he uppe -semicon inui y o andom a ac o s o be ound in Ca aballo & Langa (2001, heo em 3), which gua an ees unde condi ions which a e easily e i ied in ou case (con e gence o he co esponding cocyles; con e gence o he compac abso bing se s; and posi i e in a iance o he compac abso bing se o he limi equa ion (β=λ1)) ha P-a.s. lim β↓λ1 dis (A+ β(ω), Aλ1(ω)) = 0. Since a(ω)⊂ A+(ω) and Aλ1(ω) = {0}we ha e a(ω)→0. In ac , we know a li le mo e han is s a ed in he heo em. Fo β > λ1 he a ac o con ains he wo dis inc , non- i ial, in a ian subse s A+(ω) and A−(ω), which a ac all ini ial condi ions in K+and K− espec i ely. One can hink o his, loosely, as a ans e o s abili y o he o igin o A+(ω) and A−(ω). Fo a mo e conc e e s abili y esul , heo em 2 in A nold & Chuesho (1998) gua an ees a limi ed ype o s abili y o he new ixed poin s. In pa icula , a(ω) is s able om abo e, in ha i u(ω)≥a(ω) (no e ha his “ini ial condi ion” canno in gene al be aken o be de e minis ic) hen lim →∞ ϕ( , θ− ω)u(θ− ω) = a(ω) (s abili y “ om below” holds o −a(ω)). Since ajec o ies nea ze o in K+mo e away om he o igin, i seems easonable o expec ha a(ω) is a ac ing in K+. I one could show ha solu ions e en ually en e K+o K− his would gi e he s abili y o he ixed poin s. (A gene al esul gua an eeing he exis ence o s able ixed poin s is gi en by Schmal uß (1996), bu we ha e no been able o apply his in ou case.) 6. Conclusion We ha e shown ha he s uc u e o he andom a ac o o ou example, he Cha ee-In an e equa ion wi h a mul iplica i e noise e m, is su p isingly close o ha o he de e minis ic equa ion. In pa icula , we ha e shown ha he dimen- sion o he a ac o , asymp o ically in β, exhibi s he same beha iou as in he de e minis ic case, and is independen o he le el o noise (σ). A icle submi ed o Royal Socie y 20 T. Ca aballo, J.A. Langa, & J.C. Robinson Analysing u he he s uc u e o his a ac o we ha e shown ha as βpasses h ough λ1 he a ac o g ows and wo new andom ixed poin s appea . No ing ha a±(ω)∈ A±(ω), wha we expec is ha in ac , o λ1< β < λ2, A(ω) = A+(ω)∪A−(ω), wi h A±(ω) consis ing o a one-dimensional mani old ( he image o Mδ(ω)∩ K± unde he low) joining he o igin o ±a(ω). This would ensu e ha he a ac o ’s s uc u e is exac ly ha in he de e minis ic case. Some suppo o his pic u e, a leas o small σ, is gi en by he uppe semi- con inui y esul in Ca aballo e al. (1998). We p o ed he e ha as σ→0, o each ω, dis (Aσ(ω),A)→0. Thus he andom a ac o , a se which is a leas one-dimensional, mus lie wi hin a small neighbou hood o A, which is i sel homeomo phic o a line when λ1< β < λ2. Finally, we commen ha we would no expec simila ideas o wo k in he case o an addi i e noise e m (e.g. +²φ dW , o some φ∈H). Wi hou he ixed poin {0} he e is no clea way o show ha an in a ian mani old o a unca ed e sion o he equa ion is a subse o he andom a ac o . Indeed, he one-dimensional ex- ample in C auel & Flandoli (1998) sugges s ha wi h an addi i e noise he a ac o will be e y much simple han in he de e minis ic case. We hope o in es iga e his u he in a u u e pape . Many hanks o Igo Chuesho , Hans C auel, F anco Flandoli, and An onio Su´a ez o some in e es ing and help ul con e sa ions. James Robinson is cu en ly a Royal Socie y Uni e si y Resea ch Fellow, and would like o hank he Socie y o hei suppo . He would also like o hank EDAN o hei hospi ali y, and Ibe d ola o hei inancial assis ance du ing his isi . Tom´as Ca aballo and Jos´e Langa ha e been pa ially suppo ed by DGICYT P ojec PB98-1134. Re e ences A nold, L. 1998 Random dynamical sys ems. New Yo k: Sp inge . A nold, L. 1998 Recen p og ess in s ochas ic bi u ca ion heo y. Repo no. 439, Uni e - si y o B emen. A nold, L. & Boxle , P. 1992 S ochas ic bi u ca ion: ins uc i e examples in dimension one. In Di usion p ocesses and ela ed p oblems in analysis, Vol II (eds. Pinsky & Wihs u z), pp. 241–256. S u ga : Bi kh¨ause . A nold, L. & Chuesho , I. 1998 O de -p ese ing andom dynamical sys ems: Equilib ia, a ac o s, applica ions. Dyn. S ab. Sys. 13, 265–280. Babin, A. V. & Vishik, M. 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