Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters H ∈ (1/3, 1/2]

a Xi :1411.5237 1 [ma h.AP] 19 No 2014 LOCAL PATHWISE SOLUTIONS TO STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY FRACTIONAL BROWNIAN MOTIONS WITH HURST PARAMETERS H∈(1/3,1/2] MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Abs ac . In his a icle we a e conce ned wi h he s udy o he exis ence and uniqueness o pa hwise mild solu ions o e olu ions equa ions d i en by a H¨olde con inuous unc ion wi h H¨olde exponen in (1/3,1/2). Ou s ochas ic in eg al is a gene aliza ion o he well-known Young in eg al. To be mo e p ecise, he in eg al is de ined by using a ac ional in eg a ion by pa s o mula and i in ol es a enso o which we need o o mula e a new equa ion. F om his i u ns ou ha we ha e o sol e a sys em consis ing in a pa h and an a ea equa ions. In his pape we p o e he exis ence o a unique local solu ion o he sys em o equa ions. The esul s can be applied o s ochas ic e olu ion equa ions wi h a non-linea di usion coe icien d i en by a ac ional B ownian mo ion wi h Hu s pa ame e in (1/3,1/2], which is pa icula includes whi e noise. Sep embe 2, 2016 1. In oduc ion In his a icle, we shall ocus on he s udy o a local solu ion o he ollowing kind o s ochas ic e olu ion equa ions du( ) = Au( )d +G(u( ))dω( ), u(0) = u0, (1) in a Hilbe –space V, whe e he noise inpu ωis a H¨olde con inuous unc ion wi h H¨olde exponen in he in e al (1/3,1/2), Ais he in ini esimal gene a o o an analy ic semig oup S(·) on Vand Gis a nonlinea e m sa is ying ce ain assump ions which will be desc ibed in he nex sec ions. As a pa icula case o d i ing noises we can conside a ac ional B ownian mo ion BHwi h Hu s pa ame e H∈(1/3,1/2]. To be mo e p ecise, we will s udy (1) in he sense o mild solu ions gi en by (2) u( ) = S( )u0+Z 0 S( − )G(u( ))dω. Ou in e p e a ion o pa hwise is ha we ob ain a solu ion o hese s ochas ic equa- ions which does no p oduce excep ional se s depending on he ini ial condi ions. In he classical heo y o s ochas ic e olu ion equa ions, i.e., s ochas ic e olu ion 2000 Ma hema ics Subjec Classi ica ion. P ima y: 60H15; Seconda y: 60H05, 60G22, 26A33, 26A42. Key wo ds and ph ases. S ochas ic PDEs, Hilbe - alued ac ional B ownian mo ion, pa hwise solu ions. This wo k was pa ially suppo ed by MTM2011-22411, FEDER ounding (M.J. Ga ido-A ienza and B. Schmal uß), and by NSF0909400 (K. Lu). 1 2 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS equa ions (SEEs) d i en by B ownian mo ion B1/2, s ochas ic I o in eg als a e cons uc ed o be a limi in p obabili y o pa icula andom a iables de ined only almos su ely, whe e he excep ional se s may depend on he ini ial condi ions, which is in con adic ion wi h he cocycle p ope y needed o de ine a andom dy- namical sys em. Pa hwise esul s o ha classical heo y a e only a ailable o he whi e noise case (G= id) and a ew special cases when u7→ G(u) is linea . Du ing he las wo decades di e en in eg a ion heo ies ha e been de eloped o ea mo e gene al noise inpu s, and in pa icula , o ackling he ac ional B ow- nian mo ion BH. One o hese a emp s is gi en by he Rough Pa h Theo y, and we e e o Lyons and Qian [18] and F iz and Vic oi [8] o a comp ehensi e p e- sen a ion o his heo y. Some in e es ing pape s dealing wi h he s udy o SEEs by using he ough pa h heo y a e [2], [3], [7], [13], [15], [6] and [14] among o h- e s. In pa icula , in his las pape he au ho s p o ed he exis ence o local mild solu ions o s ochas ic SEEs d i en by ough pa hs o β-H¨olde –con inuous pa hs (β∈(1/3,1/2]) wi h a special quad a ic nonlinea i y. A di e en echnique called F ac ional Calculus was de eloped by Z¨ahle [24], who conside ed o a ac ional B ownian mo ion wi h H > 1/2 he well-known Young in eg al. In con as o he I o-o S a ono ich in eg al, ha in eg al can be de- ined in a pa hwise sense, gi en by ac ional de i a i es, which allows a pa hwise es ima e o he in eg als in e ms o he in eg and and he in eg a o using spe- cial no ms. In [20] i is shown he exis ence and uniqueness o he solu ion o a ini e-dimensional s ochas ic di e en ial equa ion d i en by a ac ional B ownian mo ion o H > 1/2. These esul s we e ex ended in [19] o show he exis ence o mild solu ions o SEEs d i en by ac ional B ownian mo ion o H > 1/2. Recen ly Hu and Nuala [16] ha e p o ed an exis ence and uniqueness esul o ini e-dimensional s ochas ic di e en ial equa ions ha ing coe icien s which a e su icien ly smoo h and d i en by a ac ional B ownian mo ion BHwi h H∈ (1/3,1/2], o which hey needed o o mula e a second equa ion o he so-called a ea in he space o enso s. In ou a icle we adap he echniques in [16] o ob ain a mild solu ion o (1). Howe e , he e a e signi ican di e ences be ween ou se ing and he one in [16], as o ins ance ha in o de o de ine he a ea equa ion in he in ini e-dimensional se ing we ha e o cons uc an a ea objec ω⊗Sω, depending on he noise pa h ωas well as on he semig oup S, sa is ying use ul p ope ies as he Chen–equali y. Unde gene al hypo hesis on he nonlinea i y Gwe de i e he exis ence and unique- ness o a local pa hwise mild solu ion u o (1). Howe e , global exis ence is missing in his gene al con ex . Unde some mo e es ic i e condi ions on G, in a o h- coming pape we will ob ain he exis ence and uniqueness o a global mild pa hwise solu ion, which in pa icula will gua an ee ha s ochas ic e olu ion equa ions like (1) and d i en by an Bm BHwi h H∈(1/3,1/2] gene a e andom dynamical sys ems, a challenging and a he open p oblem o he bes o ou knowledge. The a icle is o ganized as ollows. In Sec ion 2 we gi e he analy ical backg ound o p esen ou heo y. In Sec ion 3 we p esen he so called ac ional in eg a ion by pa s me hod. Using his echnique we can in oduce pa hwise s ochas ic in eg als allowing us o o mula e pa hwise s ochas ic di e en ial equa ions. In Sec ion 4 we in oduce mild pa h–a ea solu ions. In addi ion, we o mula e and sol e a ixed- poin equa ion ha ing wo componen s, a pa h- and an a ea-componen . The ole o he semig oup Sin he a ea equa ion will be gi en in e ms o a pa icula enso LOCAL PATHWISE SOLUTIONS TO SEES 3 objec ω⊗Sω. We also p esen an example o show a nonlinea i y G ha ma ches he abs ac heo y. The appendix sec ion con ains he p oo s o some echnical esul s. Finally, we wan o s ess ha wo di e en cons uc ions o he key enso objec ω⊗Sωby using an app oxima ion o he noise pa h by smoo h pa hs can be ound in [12]. One cons uc ion conside s as d i ing noise an in ini e-dimensional ac ional B ownian mo ion BHwi h H∈(1/3,1/2], while, in a less es ic i e se ing, he second one conside s a Hilbe - alued ace-class B ownian mo ion B1/2. Fu he mo e, we e e o [10] o a sho and ecen announcemen o ou esul s. 2. P elimina ies Le V= (V, (·,·),| ·|) be a sepa able Hilbe –space. On Vwe de ine A o be he nega i e and symme ic gene a o o an analy ic semig oup S. We suppose ha −A has a poin spec um 0 < λ1≤λ2≤ ··· ending o in ini e whe e he associa ed eigenelemen s (ei)i∈N o m a comple e o hono mal sys em on V.D((−A)κ) = Vκ deno es he domain o (−A)κ o κ∈R, and as usual, L(Vκ, Vζ) deno es he space o con inuous linea ope a o s om Vκin o Vζ, o κ, ζ ∈R. We hen ha e he ollowing es ima es o he semig oup S: kS( )kL(Vκ,Vγ)=k(−A)γS( )kL(Vκ,V )≤c κ−γ, o γ≥κ,(3) kS( )−idkL(Vσ,Vθ)≤c σ−θ, o σ−θ∈[0,1].(4) F om hese p ope ies we can de i e easily he ollowing esul : Lemma 1. Fo any ν, η, µ ∈[0,1],κ, γ, ρ ∈Rsuch ha κ≤γ+µ, he e exis s a cons an c > 0such ha o 0< q < < s < we ha e ha kS( − )−S( −q)kL(Vκ,Vγ)≤c( −q)µ( − )−µ−γ+κ, kS( − )−S(s− )−S( −q) + S(s−q)kL(Vρ,Vρ) ≤c( −s)η( −q)ν(s− )−(ν+η). Th oughou he whole pape we will w i e e y o en a cons an c. This cons an can change om line o line. Howe e his cons an is always chosen independen o ime pa ame e s con ained in a ixed in e al [0, T ]. Le V×Vand V⊗Vbe he ca esian p oduc and he enso p oduc o V, see [17]. The no m o V⊗Vis deno ed by k·k. Fo x, y ∈Vwe deno e by x⊗Vy he ank-one enso o V⊗V. Then (ei⊗Vej)i,j∈Nis a comple e o hono mal sys em o V⊗Vwhe e (ei)i∈Ncan be any comple e o hono mal sys em o V, al hough o he ollowing we choose he o hono mal sys em gi en a he beginning o his sec ion. Le ( ˆ V , | · |ˆ V,(·,·)ˆ V) be ano he sepa able Hilbe –space. By L2(V, ˆ V) (L2(V×V, ˆ V)) we deno e he Hilbe -Schmid ope a o s om V(V×V) o ˆ V. In pa icula G∈L2(V×V, ˆ V) i and only i X i,j |G(ei, ej)|2 ˆ V<∞. We no e ha G∈L2(V×V, ˆ V) can be ex ended o a linea ope a o ˆ Gde ined on V⊗Vsuch ha ˆ G∈L2(V⊗V, ˆ V), see [17] Chap e 2.6. Mo e p ecisely, we can cons uc a weak Hilbe -Schmid mapping p:V×V→V⊗Vwhe e 4 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS p(ei, ej) = ei⊗Vej o i, j ∈N. Then ˆ Gon V⊗Vis de e mined by ac o iza ion such ha G=ˆ Gp. In addi ion, we ha e kˆ Gk2 L2(V⊗V, ˆ V):= X i,j |ˆ G(ei⊗Vej)|2 ˆ V=X i,j |G(ei, ej)|2 ˆ V=kGk2 L2(V×V, ˆ V). In he ollowing we will w i e o ˆ Galso he symbol G. Le us now desc ibe he coe icien o he e olu ion equa ion ha we ha e in mind. Lemma 2. Le ˆ Vbe a subspace o V. Assume ha he mapping G:V→L2(V, ˆ V) is h ee imes con inuously F ´eche –di e en iable wi h bounded i s , second and hi d de i a i es DG(u),D2G(u)and D3G(u), o u∈V. Le us deno e, e- spec i ely, by cDG, cD2Gand cD3G he bounds o DG,D2Gand D3G, and le cG=kG(0)kL2(V, ˆ V). Then, o u1, u2, 1, 2∈V, we ha e • kG(u1)kL2(V, ˆ V)≤cG+cDG|u1|, • kG(u1)−G( 1)kL2(V, ˆ V)≤cDG|u1− 1|, • kDG(u1)−DG( 1)kL2(V×V, ˆ V)≤cD2G|u1− 1|, • kG(u1)−G(u2)−DG(u2)(u1−u2)kL2(V, ˆ V)≤cD2G|u1−u2|2, • kG(u1)−G( 1)−(G(u2)−G( 2))kL2(V, ˆ V)≤cDG|u1− 1−(u2− 2)| +cD2G|u1−u2|(|u1− 1|+|u2− 2|), • kDG(u1)−DG( 1)−(DG(u2)−DG( 2))kL2(V×V, ˆ V) ≤cD2G|u1− 1−(u2− 2)|+cD3G|u1−u2|(|u1− 1|+|u2− 2|). • kG(u1)−G(u2)−DG(u2)(u1−u2)−(G( 1)−G( 2)−DG( 2)( 1− 2))kL2(V, ˆ V) ≤cD2G(|u1−u2|+| 1− 2|)|u1− 1−(u2− 2)| +cD3G| 1− 2||u2− 2|(|u1−u2|+|u1− 1−(u2− 2)|). These es ima es ollow by he mean alue heo em; o a p oo o he las one see [20]. No ice ha , in pa icula , DG :V→L2(V, L2(V, ˆ V)) (o equi alen ly, DG :V→ L2(V×V, ˆ V)) is a bilinea map, ha can be ex ended o DG :V→L2(V⊗V, ˆ V), and D2G(u) is a ilinea map. Nex we in oduce some unc ion spaces. Le T > 0. Fo β∈(0,1], we conside he Banach–space o β–H¨olde –con inuous unc ions on [0, T ] wi h alues in V, deno ed by Cβ([0, T ]; V), wi h he semino m kukβ= sup 0≤ ≤T|u( )|+|||u|||β,|||u|||β= sup 0≤s< ≤T |u( )−u(s)| ( −s)β. I β= 1 we call hese unc ions Lipschi z–con inuous. Le Cβ,∼([0, T ]; V) be he space o unc ions on [0, T ] wi h alues in Vand wi h no m kukβ,∼= sup 0≤ ≤T|u( )|+ sup 0<s< ≤T sβ|u( )−u(s)| ( −s)β. LOCAL PATHWISE SOLUTIONS TO SEES 5 Lemma 3. Cβ,∼([0, T ], V )is a Banach–space. The p oo o his esul can be ound in Chen e al. [4]. Le ∆0,T be he iangle {(s, ) : 0 < s ≤ ≤T}. Fo β+β′<1, β ≤β′we in oduce he space Cβ+β′,∼(∆0,T , V ⊗V) o con inuous unc ions de ined on ∆0,T , which a e ze o o 0 < s = , such ha k kβ+β′,∼= sup 0<s< ≤T sβk (s, )k ( −s)β+β′<∞. These unc ions may no be de ined o s= 0 and can ha e a singula i y o (s, ), s = 0. Lemma 4. The space Cβ+β′,∼(∆0,T ;V⊗V)is a Banach–space. The p oo is simila o he p oo o Lemma 3 and he e o e we omi he e. Le us de ine ¯ ∆0,T ={(s, ) : 0 ≤s≤ ≤T}and conside he Banach–space Cβ+β′(¯ ∆0,T ;V⊗V) o con inuous unc ions de ined on ¯ ∆0,T , which a e ze o o s= , equipped wi h he no m k kβ+β′= sup 0≤s< ≤T k (s, )k ( −s)β+β′<∞. We o en use he ollowing in eg al o mula: o e e y s < ,µ, ν > −1 (5) Z s ( −s)µ( − )νd =c( −s)µ+ν+1 whe e conly depends on µ, ν. This p ope y ollows by he de ini ion o he Be a unc ion simply by pe o ming a sui able change o a iable. 3. F ac ional Calculus In his pape he main ins umen o ea (2) is ac ional calculus. In his sec ion we p esen he main ea u es o his heo y. We a e going o assume ha o some T > 0 we ha e ha ω∈Cβ′([0, T ]; V), u∈Cβ,∼([0, T ]; V) and ∈Cβ+β′,∼(∆0,T ;V⊗V) o 1/3< β < β′<1/2, and ha his iple o elemen s sa is ies he Chen–equali y gi en by (6) (s, ) + ( , ) + (u( )−u(s)) ⊗V(ω( )−ω( )) = (s, ) o 0 < s ≤ ≤ ≤T. We would like o emphasize ha when ωis smoo h an example o is gi en by (7) (u⊗ω)( , ) = Z (u(q)−u( )) ⊗Vdω(q). This enso a ea is clea ly well de ined and sa is ies, o 0 < < , k(u⊗ω)( , )k ≤ c βkukβ,∼kωkC1( − )1+β≤c β( − )β+β′, (8) and he e o e (u⊗ω)∈Cβ+β′,∼(∆0,T ;V⊗V). Mo eo e , he Chen–equali y easily ollows in his case. 6 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Le α∈(0,1). We de ine he igh hand side ac ional de i a i e o o de αo u and he le hand side ac ional de i a i e o o de 1 −αo ω −(·) := ω(·)−ω( ), gi en o 0 < s ≤ ≤ by he exp essions Dα s+u[ ] = 1 Γ(1 −α)u( ) ( −s)α+αZ s u( )−u(q) ( −q)1+αdq D1−α −ω −[ ] =(−1)1−α Γ(α)ω( )−ω( ) ( − )1−α+ (1 −α)Z ω( )−ω(q) (q− )2−αdq, whe e Γ(·) deno es he Gamma unc ion. Fo enso alued elemen s and o 0< < we de ine D1−α − [ ] =(−1)1−α Γ(α) ( , ) ( − )1−α+ (1 −α)Z ( , q) (q− )2−αdq. Lemma 5. Suppose ha β > α. Then he e exis s a cons an c > 0such ha o 0≤s < ≤ ≤T |Dα s+u(·)[ ]| ≤ ckukβ,∼ ( −s)α,|D1−α −ω −[ ]| ≤ c|||ω|||β′( − )α+β′−1, and o 0< < q < kD1−α − [q]−D1−α − [ ]k ≤ c β(kukβ,∼|||ω|||β′+k kβ+β′,∼)(q− )α+β+β′−1. (9) The p oo o he wo i s inequali ies ollows s aigh o wa dly, and he p oo o (9) is simila o he one o Lemma 6.3 in [16] wi h he di e ence ha in ha pape he au ho s wo k in di e en unc ion spaces. We e e he eade o Lemma 21 and Co olla y 29 in he Appendix sec ion. As an ex ension o he ac ional de i a i e o o de α, o he mapping G:V7→ L2(V, ˆ V) and s < we in oduce he so–called compensa ed ac ional de i a i e o o de αgi en by ˆ Dα s+G(u(·))[ ] = 1 Γ(1 −α)G(u( )) ( −s)α +αZ s G(u( )) −G(u(q)) −DG(u(q))(u( )−u(q)) ( −q)1+αdq. I is immedia e o p o e he ollowing esul : Lemma 6. Suppose ha α < 2βand Gsa is ies he assump ions o Lemma 2. Then he e exis s a posi i e cons an csuch ha o e e y 0≤s < ≤T |ˆ Dα s+G(u(·))[ ]| ≤ c(1 + kuk2 β,∼) ( −s)α. Le us assume o a while ha V, ˆ V=R. We i s ecall he ollowing use ul p ope y which is an in eg a ion by pa s o mula (−1)αZ s Dα s+u[ ]ω( )d =Z s u( )Dα −ω[ ]d ,(10) see Z¨ahle [24], o mula (21). Fo β > α and α+β′>1 he ac ional in eg al is gi en by Z s udω := (−1)αZ s Dα s+u[ ]D1−α −ω −[ ]d , LOCAL PATHWISE SOLUTIONS TO SEES 7 see again [24], which is a e sion o he Young in eg al. By Lemma 5 and he p ope y (5), o he abo e in eg al i is easy o de i e ha Z s udω≤ckukβ,∼|||ω|||β′( −s)β′. Fo H¨olde –con inuous uand ω his kind o in eg al was de ined by Young [23]. Howe e , ou unc ion uis no H¨olde –con inuous in he s ong sense bu u∈ Cβ,∼([0, T ]; R), in which case ha in eg al is also well de ined in he abo e sense since, acco ding o [24], wha we need is ha u∈Iα s+(Lp((s, ); R)), u(s+) bounded and ω −∈I1−α −(Lq((s, ); R)), wi h αp < 1, p−1+q−1≤1 ( o he de ini ion o hese spaces we e e o Samko e al. [22]). In pa icula , unde ou condi ions on α, β and β′, we know ha ω −∈I1−α −(Lq((s, ); R)) o any q > 1 and u∈Iα s+(Lp((s, ); R)) when αp < 1, see Theo em 13.2 o [22]. Nex we in oduce in eg als o ac ional ype wi h alues in a sepa able Hilbe – space. To do ha , we need a new sepa able Hilbe –space ( ˜ V , |·|˜ V,(·,·)˜ V). Lemma 7. Assume β > α and α+β′>1. Le ˆ V , ˜ Vbe wo sepa able Hilbe – spaces, being (˜ei)i∈Nand ( j)j∈Ncomple e o hono mal basis o ˜ Vand ˆ V esp., and le [s, ]∋ 7→ F( )∈L2(˜ V , ˆ V),[s, ]∋ 7→ ξ( )∈˜ V be measu able unc ions such ha F∈Cβ,∼([0, T ]; L2(˜ V , ˆ V)),ξ∈Cβ′([0, T ]; ˜ V) and 7→ kDα s+F[ ]kL2(˜ V , ˆ V)|D1−α −ξ[ ]|˜ Vis Lebesgue-in eg able. Then o 0≤s≤ ≤ ≤Twe can de ine Z s F( )dξ( ) := (−1)αX jX iZ s Dα s+( j, F(·)˜ei)ˆ V[ ]D1−α −(˜ei, ξ(·))˜ V[ ]d  j. Tha his exp ession is well de ined ollows by Z s F( )dξ( )ˆ V =X jX iZ s Dα s+( j, F(·)˜ei)ˆ V[ ]D1−α −(˜ei, ξ(·))˜ V[ ]d 21 2 ≤X jZ sX i (Dα s+( j, F(·)˜ei)ˆ V[ ])2X i (D1−α −(˜ei, ξ(·))˜ V[ ])21 2 d 21 2 ≤Z sX j,i (Dα s+( j, F(·)˜ei)ˆ V[ ])2X i (D1−α −(˜ei, ξ(·))˜ V[ ])21 2 d ≤Z sX j,i (Dα s+( j, F(·)˜ei)ˆ V[ ])21 2X i (D1−α −(˜ei, ξ(·))˜ V[ ])21 2 d =Z skDα s+F[ ]kL2(˜ V , ˆ V)|D1−α −ξ[ ]|˜ Vd < ∞.(11) 8 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Obse e ha his las in eg al is ini e since kDα s+F[ ]kL2(˜ V , ˆ V)=X j,i (Dα s+( j, F(·)˜ei)ˆ V[ ])21 2 =X j,i 1 Γ(1 −α)( j, F( )˜ei)ˆ V ( −s)α+αZ s ( j, F( )˜ei)ˆ V−( j, F(q)˜ei)ˆ V ( −q)1+αdq21 2 ≤√2c(Pj,i( j, F( )˜ei)2 ˆ V)1 2 ( −s)α+X j,i Z s ( j, F( )˜ei)ˆ V−( j, F(q)˜ei)ˆ V ( −q)1+αdq21 2 ≤√2ckF( )kL2(˜ V , ˆ V) ( −s)α+Z s kF( )−F(q)kL2(˜ V , ˆ V) ( −q)1+αdq ≤qc( −s)−αkFkCβ,∼([0,T ];L2(˜ V , ˆ V)), and |D1−α −ξ[ ]|˜ V≤c|||ξ|||β′( − )α+β′−1(see Lemma 5), so i su ices o apply (5). Now we can apply Lemma 7 o de ine in eg als o ac ional ype wi h alues in a sepa able Hilbe –space, as well as o conside in eg a o s wi h alues in Vo V⊗V. Fo example, conside an in eg and o he ype G(u( )) whe e u∈Cβ,∼([0, T ]; V), and β < α < 2β,α+β′>1, β + 1 >2α. No e ha i DG is bounded hen G(u)∈Cβ,∼([0, T ], V ), bu since we do no assume ha β > α, hen Dα s+G(u(·)) is no well-de ined. Howe e , we can apply Lemma 7 in he ollowing way Z s G(u(·))dω = (−1)αZ s Dα s+DG(u(·))(u(·)−u(s),·)[ ]D1−α −ω −[ ]d +(−1)αZ s Dα s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[ ]D1−α −ω −[ ]d . (12) Since u, ω, a e coupled by he Chen–equali y (6), we ge D1−α − (s, ·) −[ ] = (−1)1−α Γ(α) (s, )− (s, ) ( − )α+ (1 −α)Z (s, )− (s, q) (q− )2−αdq =(−1)1−α Γ(α)− ( , )−(u( )−u(s)) ⊗V(ω( )−ω( )) ( − )α + (1 −α)Z − ( , q)−(u( )−u(s)) ⊗V(ω(q)−ω( )) (q− )2−αdq =−D1−α − [ ] + (u( )−u(s)) ⊗VD1−α −ω −[ ]. (13) Simila ly, i is easy o de i e ha Dα s+(G(u(·)) −DG(u(·))(u(·)−u(s),·))[ ] =ˆ Dα s+G(u(·))[ ]−Dα s+DG(u(·))[ ](u( )−u(s),·), and hus, coming back o (12) we ob ain ha LOCAL PATHWISE SOLUTIONS TO SEES 9 Z s G(u(·))dω =(−1)αZ s ˆ Dα s+G(u(·))[ ]D1−α −ω −[ ]d −(−1)αZ s Dα s+DG(u(·))[ ]D1−α − (·, )[ ]d =(−1)αZ s ˆ Dα s+G(u(·))[ ]D1−α −ω −[ ]d −(−1)2α−1Z s D2α−1 s+DG(u(·))[ ]D1−α −D1−α − (·, )[ ]d , (14) whe e he las equali y is ue hanks o he ac ional in eg a ion by pa s o mula (10). The las in eg al on he igh hand side o (14) is well-de ined due o, o 0≤s≤ ≤ ≤T, (15) kD1−α −D1−α − [ ]k ≤ c(k kβ+β′,∼+kukβ,∼|||ω|||β′)( −s)−β( − )β+β′+2α−2 (see Lemma 21 abo e) and hence his in eg al can be de ined by using Lemma 7. Rema k 8. I may be checked ha he in eg al gi en in Lemma 7 can be also de ined o an in eg a o like F( ) = S( − )G(u( )), which is no H¨olde –con inuous a ze o. The eason is because he j-modes o his exp ession a e H¨olde con inuous. Fu he mo e, he in eg abili y condi ion (11) is sa is ied by he egula i y p ope ies o he analy ic semig oup Sand he egula i y o Gand u. Finally, on accoun o (14) and he es ima es o he di e en ac ional de i a i es ha ake pa in ha exp ession, we can es ablish he ollowing esul : Lemma 9. Suppose u, , ω sa is y he assump ions om he beginning o his sec- ion and Gsa is ies he assump ions o Lemma 2. Le β < α < 2β,α+β′> 1, β + 1 >2α. Then o 0≤s≤ ≤Twe ha e Z s G(u)dω≤c((1 + kuk2 β,∼)|||ω|||β′+kukβ,∼k kβ+β′,∼)( −s)β′. 4. Pa h-A ea-solu ions o s ochas ic e olu ion equa ions In his sec ion we gi e he de ini ion o a mild solu ion o (2) and es ablish a esul abou he local exis ence and uniqueness o such solu ion. In o de o unde s and he no ion o mild solu ion o (2), in a i s s ep we conside he case in which he d i ing noise is egula , o la e on conside he H¨olde case in which we a e in e es ed in. No e ha we could also add a nonlinea di usion e m Fon he igh -hand side o he equa ion in (1). Ne e heless, o simpli y he whole p esen a ion we ha e no conside ed i since he d -nonlinea i y is no he in e es ing p oblem o be ea ed in he pape . 4.1. Sys em (2) d i en by smoo h pa hs ω.In his si ua ion, since Ahas he p ope ies o Sec ion 2 and DG is bounded, we emind ha o any u0∈V he e exis s a unique solu ion u∈C([0, T ]; V) o (2) o any T > 0, see [21]. In addi ion, applying he p ope ies (3), (4) we ob ain ha his solu ion is in Cγ,∼([0, T ]; V) o any γ∈(0,1). Howe e , we do no ob ain ha he solu ion is in Cγ([0, T ]; V), due o he ac ha 7→ S( )u0is no H¨olde -con inuous in gene al bu in Cγ,∼([0, T ]; V). 16 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Lemma 17. Suppose ha he condi ions o Lemma 15 hold. Then he e exis s a c > 0depending on ωsuch ha o U1, U2∈ˆ W0,T ||T1(U1)−T1(U2)||β,∼≤cT β′(1 + ||U1||2 W+||U2||2 W)||U1−U2||W+c|u1 0−u2 0|. P oo . Le us deno e ∆u=u1−u2,∆U= (u1−u2, 1− 2). Then o ∈[0, T ] by Lemma 2 we ha e kG(u1( )) −G(u2( ))kL2(V, ˆ V)≤cDGk∆ukβ,∼ and kG(u1( )) −G(u1(q)) −DG(u1(q))(u1( )−u1(q)) −(G(u2( )) −G(u2(q)) −DG(u2(q))(u2( )−u2(q)))kL2(V, ˆ V) ≤cD2G(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼ ( −q)2β q2β +cD3Gku2kβ,∼ ( −q)β qβsup ∈[0,T ]|∆u( )|(2ku1kβ,∼+ku2kβ,∼)( −q)β qβ ≤c(ku1kβ,∼+ku2kβ,∼)k∆ukβ,∼ ( −q)2β q2β(1 + ku2kβ,∼). Now we can ollow he p oo o Lemma 15.  Up o now we ha e ob ained app op ia e es ima es o he i s componen T1o he ope a o Tgi en by (26). Now we aim a ge ing he co esponding es ima es o T2, he second componen o T. To his end, we need he wo ollowing Lemma a, dealing wi h es ima es o some o he e ms appea ing in he exp ession o T2. Lemma 18. Unde he Hypo hesis (H) he ollowing s a emen s hold: (i) Fo he mapping e∈V7→ ωS(s, )e= (−1)−αZ s (S(ξ−s)e)⊗Vdω(ξ) he ollowing p ope ies hold ue: o 0≤s≤ ≤ ≤T, e ∈Vand 1/3< β′< β′′ < H, kωS( , )e−ωS(s, )ek ≤ c( −s)β′(|||ω|||β′+|||ω|||β′′ )|e|, kωS(s, )ek ≤ c( −s)β′|||ω|||β′|e|, kωS(s, )(−A)β′ek ≤ c|||ω|||β′′ |e|. (ii) The mapping E∈L2(V, ˆ V)7→ Sω(s, )E=Z s S( − )Edω( ) is in L2(L2(V, ˆ V), V ), wi h no m bounded by c|||ω|||β′( −s)β′kEkL2(V, ˆ V). LOCAL PATHWISE SOLUTIONS TO SEES 17 P oo . Le β′< β′′ < H such ha o α < α′<1 we ha e β′+α′<1< β′′ +α. Then ωS( , )e−ωS(s, )e=(−1)αZ (S(ξ− )e−S(ξ−s)e)⊗Vdω(ξ) −(−1)αZ s S(ξ−s)e⊗Vdω(ξ). (29) In e p e ing hese in eg als in a ac ional sense and using Lemma 1 we ob ain |Dα +(S(·− )e−S(·−s)e)[ξ]| ≤ c( −s)β′ (ξ− )α+β′+αZξ ( −s)β′(ξ−q)α′ (ξ−q)1+α(q− )α′+β′dq|e| ≤c( −s)β′(ξ− )−α−β′|e|. Mo eo e , since β′′ < H, due o (H3) ω∈Cβ′′ ([0, T ]; V) (and in pa icula ω∈ Cβ′([0, T ], V )), hen by Lemma 5 |D1−α −ω −[ξ]| ≤ c|||ω|||β′′ ( −ξ)α+β′′−1. Hence, by applying (5), he i s in eg al on he igh -hand side o (29) is bounded in pa icula by c( −s)β′|||ω|||β′′ |e|. Fu he mo e, o he las e m in (29), hanks o (H1) and he egula i y p ope ies o he semig oup we ob ain Z skDα s+S(·−s)e[ξ]⊗VD1−α −ω −[ξ]kdξ ≤c|||ω|||β′( −s)β′|e|. The second s a emen o (i) ollows di ec ly om he i s one aking = . Fo he las conclusion o (i), o he pa ame e s chosen a he beginning o he p oo , |Dα s+(S(·−s)(−A)β′e)[ξ]| ≤ c|e| (ξ−s)α+β′ +Zξ s (ξ−q)α′|e| (ξ−q)1+α(q−s)α′+β′dq≤c|e|(ξ−s)−α−β′, and he e o e, since in pa icula β′+α < 1, Z skDα s+(S(·−s)(−A)β′e)[ξ]⊗VD1−α −ω −[ξ]kdξ ≤c|||ω|||β′′ |e|Z s (ξ−s)−α−β′( −ξ)α+β′′−1dξ ≤c|||ω|||β′′ ( −s)β′′−β′|e| ≤ c|||ω|||β′′ |e|. Now we p o e (ii). Fi s o all, no e ha Sωis well-de ined since, hanks o Rema k 8, i su ices he H¨olde –con inui y o 7→ S( − )Eon any in e al [ǫ, ]. In addi ion by he embedding ˆ V⊂V he mapping L2(V, ˆ V)∋E7→ ED1−α −ω −[ ]∈V is in L2(L2(V, ˆ V), V ), and he no m can be es ima ed cV, ˆ V|D1−α −ω −[ ]|, see Lemma 28 (ii). Fu he mo e, he in eg and Dα s+S( −·)[ ]·D1−α −ω −[ ] is weakly measu - able wi h espec o L2(L2(V, ˆ V), V ) such ha by Pe is’ heo em he in eg and is 18 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS measu able. Because o D1−α −Eω −[ ] = ED1−α −ω −[ ], we ge kSω(s, )·kL2(L2(V, ˆ V),V )=   Z s S( − )·dω( )   L2(L2(V, ˆ V),V ) ≤Z skDα s+S( −·)[ ]kL(V)k·D1−α −ω −[ ]kL2(L2(V, ˆ V),V )d ≤c|||ω|||β′( −s)β′.  Co olla y 19. Le (ωn)n∈Nbe a sequence con e ging o ωin Cβ′([0, T ]; V). Then lim n→∞ sup 0≤s < ≤T |e|= 1 k(ω−ωn)S(s, )ek ( −s)β′= 0, lim n→∞ sup 0≤s< ≤T kSω−ωn(s, )·kL2(L2(V, ˆ V),V ) ( −s)β′= 0. F om he p e ious esul we also ob ain ha Dα s+(S(·−s)(Sω(q, s)·))[ξ]⊗VD1−α −ω −[ξ] is weakly L2(L2(V, ˆ V), V ⊗V)–measu able. Lemma 20. Suppose ha he hypo hesis (H) holds. Fo 0< s ≤q≤ ≤T, ˜ E∈L2(V⊗V, ˆ V)and U= (u, )∈ˆ W0,T conside he mapping w( , s, q) : L2(V⊗V, ˆ V)∋˜ E7→ ˜ Ew( , s, q)∈V⊗V gi en by (27) and (28). Then i is well-de ined and sa is ies, o β′< β′′ he es ima e kw( , s, q)kL2(L2(V⊗V, ˆ V),V ⊗V)≤c||U||Ws−β(q−s)β+β′( −s)β′, whe e he cons an cdepends on |||ω|||β′′ and k(ω⊗Sω)k2β′. In pa icula c∼(1 + |||ω|||2 β′′ )k(ω⊗Sω)k2β′. Fo he p oo , see he Appendix. We would like o poin ou ha wsa is ies he gene alized Chen–equali y ˜ Ew( , s, ) + ˜ Ew( , , q)−˜ E(u( )−u(s),·)(ω⊗Sω)( , q) =˜ Ew( , s, q) + ωS(q, )Sω(q, )˜ E(u( )−u(s),·), (30) o 0 < s ≤ ≤q≤ ≤T, and he Chen–equali y ˜ Ew( , s, ) + ˜ Ew( , , )−˜ E(u( )−u(s),·)(ω⊗Sω)( , ) = ˜ Ew( , s, )(31) whe e he la e one is ob ained om (30) aking q= . In o de o p o e hese wo p ope ies we only need o ollow an app oxima ion a gumen , conside ing (ωn,(ωn⊗Sωn)) sa is ying (H3), and he e o e con e ging o (ω, (ω⊗Sω)) in Cβ′([0, T ], V )×C2β′(¯ ∆0,T ;L2(L2(V, ˆ V), V ⊗V)) and ake in o accoun ha he app oxima ing elemen s wn:= (u⊗(ωn⊗Sωn)) gi en in Lemma 10 sa is y he Chen–equali ies (23) and (24). In pa icula , he e ms (ωn S⊗ωn) con e ge o he LOCAL PATHWISE SOLUTIONS TO SEES 19 co esponding e m (ωS⊗ω), see he p oo o Lemma 20 in he Appendix sec ion. The ollowing esul gi es es ima es o he ac ional de i a i es o and w. Lemma 21. Le U= (u, )∈ˆ W0,T . Then o 0< < ≤Twe ha e kD1−α −D1−α − [ ]kβ+β′,∼≤c||U||W −β( − )2α+β+β′−2, kD1−α −D1−α −w( , ·,·)[ ]k ≤ c||U||W −β( − )2α+β+2β′−2, whe e he i s cons an cdepends on |||ω|||β′′ , and he second one on |||ω|||β′′ and k(ω⊗Sω)k2β′. We ha e also shi ed he p oo o his esul o he Appendix sec ion. Lemma 22. Assume ha u0∈V. The e exis s posi i e cons an s ˜cand csuch ha o U∈ˆ W0,T kT2(U, ω,(ω⊗Sω), u0)kβ+β′,∼≤˜c|u0|+cT β′(1 + kUk2 W) and, in addi ion, o wo elemen s U1, U2∈ˆ W0,T : kT2(U1, ω, (ω⊗Sω), u0)−T2(U2, ω, (ω⊗Sω), u0)kβ+β′,∼ ≤˜c|u1 0−u2 0|+cT β′(1 + kU1k2 W+kU2k2 W)kU1−U2kW. The cons an cdepends on |||ω|||β′′ ,|||ω|||β′and k(ω⊗Sω)k2β′, while ˜con |||ω|||β′. P oo . Le us deno e T2(U)(s, ) =: B1(s, ) + B2(s, ) + B3(s, ), co esponding o he h ee di e en addends o T2. Fo B1we can conside he ollowing spli ing: B1(s, ) = Z s (S(ξ−s)−id)u(s)⊗Vdω(ξ) =Z s (S(ξ)−S(s))u0⊗Vdω(ξ) +Z s (S(ξ−s)−id)Zs 0 S(s− )G(u( ))dω( )⊗Vdω(ξ) =:B11(s, ) + B12(s, ). B1can be in e p e ed in he ac ional sense hanks o he egula i y o i s in eg and, which means ha B11(s, ) = (−1)αZ s Dα s+((S(·)−S(s))u0)[ξ]⊗VD1−α −ω −[ξ]dξ. Fo α < α′whe e α′is su icien ly close o αand s > 0, applying (3) and (4), |Dα s+((S(·)−S(s))u0)[ξ]| ≤ c|(S(ξ)−S(s))u0| (ξ−s)α+Zξ s |(S(ξ)−S(q))u0| (ξ−q)1+αdq ≤c(ξ−s)β−α sβ+Zξ s (ξ−q)α′q−β (ξ−q)1+αqα′−βdq|u0| ≤c(ξ−s)β−α sβ+1 sβZξ s (ξ−q)α′ (ξ−q)1+α(q−s)α′−βdq|u0| ≤ c(ξ−s)β−α sβ|u0|, 20 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS hence, o s > 0, |B11(s, )| ≤ c|||ω|||β′|u0| sβZ s (ξ−s)β−α( −ξ)β′+α−1dξ ≤c|||ω|||β′|u0|s−β( −s)β′+β, which implies kB11kβ+β′,∼≤˜c|u0|. Mo eo e , no e ha  Dα s+(S(·−s)−id) Zs 0 S(s− )G(u( ))dω( )[ξ] ≤c|S(ξ−s)−id) Rs 0S(s− )G(u( ))dω( )| (ξ−s)α +Zξ s |Rs 0S(ξ− )−S(q− ))G(u( ))dω( )| (ξ−q)1+αdq. To deal wi h B12 on accoun o Co olla y 16 and hanks o he ac ha β′> β we ha e Zξ s |Rs 0(S(ξ−q)−id)S(q− )G(u( ))dω( )| (ξ−q)1+αdq ≤Zξ s (ξ−q)α′ (q−s)α′−β(ξ−q)1+α (−A)βZs 0 S(s− )G(u( ))dω( ) dq ≤c(1 + |||ω|||β′)(1 + kUk2 W)(ξ−s)β−αsβ′−β and by (5) kB12kβ+β′,∼≤cT β′|||ω|||2 β′(1 + kUk2 W). Finally, a simila es ima e ollows o B2and B3. In o de o see his, no e ha B2and B3can be conside ed in a simila way o C11 and C12 in he p oo o Lemma 15, wi h he di e ence ha now we ha e o es ima e D1−α −(ω⊗Sω)(·, ) −[ ] and D1−α −D1−α −w( , ·,·)[ ], o which we use he 2β′-H¨olde con inui y o (ω⊗Sω) oge he wi h Lemma 21, a i ing a kB2kβ+β′,∼≤cT β′k(ω⊗Sω)k2β′(1 + kUk2 W), kB3kβ+β′,∼≤cT β′(1 + |||ω|||2 β′′ )k(ω⊗Sω)k2β′(1 + kUk2 W). The second pa o his lemma can be p o en simila ly and hus we omi i he e.  Now we app oxima e T(U, ω, (ω⊗Sω), u0) by piecewise linea noise. Lemma 23. Le U∈ˆ W0,T and assume ha (ωn,(ωn⊗Sωn)) sa is ies (H3). Then lim n→∞ kT(U, ωn,(ωn⊗Sωn), u0)−T(U, ω, (ω⊗Sω), u0)kW= 0. In addi ion, T(U, ω, (ω⊗Sω), u0)∈ˆ W0,T . P oo . No e ha i a e m o Tcon ains ωo (ω⊗Sω) hen his e m depends on hese exp essions linea ly o bi-linea ly (see he de ini ion o T1,T2 oge he wi h (27)) which jus gi es he con e gence conclusion. To see he second pa o he s a emen we ake U= (u, )∈ˆ W0,T . Fo his elemen we choose an app oxima ing sequence (un,(un⊗ωn)) om (25). We no e ha T1((un,(un⊗ ωn)), ωn, un 0)∈Cγ([0, T ], V ) and T1((un,(un⊗ωn)), ωn, un 0)(0) ∈D((−A)) o any γ∈(0,1), see Pazy [21] Theo em 4.3.1 and (3)-(4). The e o e T2((un,(un⊗ ωn)), ωn,(ωn⊗Sωn), un 0) can be de ined as (T1((un,(un, ωn)), ωn, un 0)⊗ωn) gi en LOCAL PATHWISE SOLUTIONS TO SEES 21 by (7). By de ini ion o he space ˆ W0,T , by Lemmas 17, 22, and he i s pa o his lemma we ha e ha ha T(U, ω, (ω⊗Sω), u0)∈ˆ W0,T . Le us now p o e he uniqueness o he pa h-a ea-solu ion in ˆ W0,T i such a solu ion exis s. Theo em 24. Suppose ha U1= (u1, 1), U2= (u2, 2)∈ˆ W0,T a e wo pa h-a ea solu ions ela ed o he ini ial condi ion u0∈V. Then we ha e U1=U2. P oo . Le T1∈[0, T ) such ha [0, T1] ep esen s he maximal in e al o unique- ness. He e T1= 0 means ha wo di e en solu ions a e jus b anching om 0. No e ha he es ic ions ˜ Ui= (ui [T1,˜ T], i ∆[T1,˜ T]) a e pa h-a ea solu ions oo wi h espec o he in e al [T1,˜ T],˜ T≤Twi h ini ial condi ion u(T1). This ol- lows because o T1we can apply Lemma 12. On he o he hand, T2 es ic ed o T1< s < ≤˜ Tkeeps he same s uc u e as he o iginal T2. Then, i he cons an C is an es ima e o k˜ Uik2 WT1,˜ T, simila o he es ima es ha we ob ained in Lemmas 17 and 22, we ob ain 06=k˜ U1−˜ U2kWT1,˜ T≤c(˜ T−T1)β′(1 + 2C)k˜ U1−˜ U2kWT1,˜ T o equi alen ly 1≤c(˜ T−T1)β′(1 + 2C) o any ˜ T > T1, which is a con adic ion i ˜ T−T1is su icien ly small.  Now we p esen he main heo em o he pape . Theo em 25. Suppose ha he s anding condi ions (H) a e sa is ied and suppose ha T > 0is chosen su icien ly small depending on he pa ame e s o he p oblem. Then Thas a ixed poin in ˆ W0,T ⊂W0,T ha de ines a mild pa h-a ea solu ion o (2) which is unique. P oo . Lemmas 15, 22 and 23 p o e ha Tmaps a closed cen e ed ball om ˆ W0,T in o i sel i T > 0 is su icien ly small and by Lemma 17 and Lemma 22 we ob ain ha his mapping is a con ac ion i T > 0 is chosen su icien ly small.  Co olla y 26. Le U∈ˆ W0,T be a mild pa h-a ea solu ion gi en by Theo em 25 and le Unbe he pa h-a ea solu ion co esponding o he equa ions (2) and (21) and d i en by a smoo h noise ωn. Then we ha e lim n→∞ kUn−UkW= 0. In ac , o la ge enough nwe can ind a ball in ˆ W0,T which is mapped in o i sel by T(·, ωn,(ωn⊗Sωn), u0) and T(·, ω, (ω⊗Sω), u0), and whe e in addi ion hese mappings a e con ac ions wi h uni o m con ac ion condi ion. Then he conclusion ollows by he pa ame e e sion o he Banach- ixed poin Theo em oge he wi h Lemma 23. Rema k 27. An applica ion o he main esul s o his a icle is o p o e ha he (pa hwise) mild pa h–a ea solu ions a e global, see he o hcoming pape [11]. In addi ion, he esul s p esen ed in his pape a e he basemen o p o e ha SEEs wi h non- i ial di usion coe icien s Gand d i en by ac ional B ownian–mo ions BHwi h H∈(1/3,1/2] gene a e a andom dynamical sys em, see also [11]. Tha p ope y has been es ablished ecen ly by he same au ho s when dealing wi h a mo e egula Bm, namely BHwi h H∈(1/2,1), see [9]. 22 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS 5. Example Le V=l2be he space o squa e addi i e sequences wi h alues in R. In addi ion, le Abe a nega i e symme ic ope a o de ined on D(−A)⊂l2wi h compac in e se. In pa icula , we can assume ha −Ahas a disc e e spec um 0 < λ1≤ λ2≤ ··· ≤ λi≤ ··· → ∞ whe e he associa ed eigenelemen s (ei)i∈N o m a comple e o hono mal sys em in l2. The spaces D((−A)ν) = Vνa e hen de ined by {u= (ui)i∈N∈l2:X i λ2ν iu2 i=: |u|2 Vν<∞}. Assume ha he e exis s κ > 0 such ha he Hilbe -Schmid embedding Vκ⊂V holds ue, and ake ˆ V=Vκ. Conside a sequence o unc ions (gij )i,j∈N, wi h gij :V→R, and de ine G(u) o u∈Vby G(u) =X j gij(u) ji∈N o all ∈V. We assume ha kG(u)k2 L2(V,Vκ)=X j|G(u)ej|2 Vκ=X jX i λ2κ i(G(u)ej)2 i=X i,j λ2κ ig2 ij(u)≤c uni o mly wi h espec o u∈V. In addi ion, assume ha gij a e ou imes di e en iable and hei de i a i es a e uni o mly bounded in he ollowing way |Dgij(u)(ek)| ≤ cijk g,1,|D2gij(u)(ek, h1)| ≤ cijk g,2|h1|,|D3gij(u)(ek, h1, h2)| ≤ cijk g,3|h1||h2|, |D4gij(u)(ek, h1, h2, h3)| ≤ cijk g,4|h1||h2||h3| o any u∈V, and hese bounds sa is y X ijk λ2κ i(cijk g,1)2<∞,X ijk λ2κ i(cijk g,2)2<∞,X ijk λ2κ i(cijk g,3)2<∞,X ijk λ2κ i(cijk g,4)2<∞. To see o ins ance ha DG exis s, no e ha by Taylo expansion |gij(u+h)−gij(u)−Dgij(u)(h)|2≤1 2|D2gij(u+ηh)(h, h)|2≤(cijk g,2)2|h|4 whe e u, h ∈Vand η∈[0,1]. In pa icula , we also no e ha X jk  DG(u)(ek, ej) 2 Vκ =X ijk λ2κ i|Dgi,j(u)(ek)|2≤X ijk λ2κ i(cijk g,1)2=: c2 DG <∞. This condi ion ensu es he Lipschi z con inui y o Gas well as he Hilbe -Schmid p ope y o DG. Simila ly, we ob ain ha DG is also Lipschi z wi h espec o he Hilbe -Schmid no m. We also ob ain he exis ence o he second and hi d de i a i e. Hence he condi ions on Gin Hypo hesis (H) hold. The de eloped heo y can be also applied o o he examples o G, like ke nel in e- g als, see [11]. LOCAL PATHWISE SOLUTIONS TO SEES 23 6. Appendix In he appendix we p o e some echnical es ima es ela ed o w= (u⊗(ω⊗Sω)). P oo o Lemma 10 P oo . Fo ˜ E∈L2(V⊗V, ˆ V) we de ine ˜ E:L2(V, V ⊗V)×V→L2(V, V ⊗V) gi en by ˜ E(Q, u) = Q(˜ E(u, ·)). F om (17) o smoo h ωwe ha e ha ˜ E(u⊗(ω⊗Sω)( ))(s, q) = −(−1)αZq s ωS( , )˜ E(u( )−u(s), ω′( ))d =−(−1)αZq s ˜ E(ωS( , ), u( )−u(s))ω′( )d . Following Theo em 3.3 in [16], we ha e Zq s ˜ E(ωS( , ), u( )−u(s))ω′( )d =(−1)αZq s ˆ Dα s+ ˜ E(ωS(·, ), u(·)−u(s))[ ]D1−α q−ωq−[ ]d −(−1)αZq s (ωS( , )−ωS(s, ))Dα s+˜ E(u(·)−u(s),·)[ ]D1−α q−ωq−[ ]d −(−1)αZq s Dα s+ωS(·, )[ ]˜ E(u( )−u(s),·)D1−α q−ωq−[ ]d +Zq s D ˜ E(ωS( , ), u( )−u(s))(ωS( , )−ωS(s, ), u( )−u(s))ω′( )d . (32) Now we calcula e he de i a i e o ˜ E: D ˜ E(ωS( , ), u( )−u(s))(ωS( , )−ωS(s, ), u( )−u(s))ω′( ) =(ωS( , )−ωS(s, )) ˜ E(u( )−u(s), ω′( )) + ωS( , )˜ E(u( )−u(s), ω′( )) =˜ E(u( )−u(s),·)D2(ωS( )⊗ω)(s, ) + ωS( , )˜ ED2(u⊗ω)(s, ). Subs i u ing he abo e exp ession in (32), a e applying ac ional in eg a ion (10) o he las wo e ms, we ha e o calcula e ED1−α q−(ωS( )⊗ω)q−(s, ·)[ ], o E∈ L2(V, ˆ V), and ˜ ED1−α q−(u⊗ω)( )(s, ·)q−[ ]. Fi s , we ha e ED1−α q−(ωS( )⊗ω)q−(s, ·)[ ] = D1−α q−E(ωS( )⊗ω)q−(s, ·)[ ] =(−1)1−α Γ(α)E(ωS( )⊗ω)(s, )−E(ωS( )⊗ω)(s, q) (q− )1−α + (1 −α)Zq E(ωS( )⊗ω)(s, )−E(ωS( )⊗ω)(s, θ) (θ− )2−αdθ =−(−1)1−α Γ(α)E(ωS( )⊗ω)( , q) + (ωS( , )−ωS(s, ))E(ω(q)−ω( )) (q− )1−α + (1 −α)Zq E(ωS( )⊗ω)( , θ) + (ωS( , )−ωS(s, ))E(ω(θ)−ω( )) (θ− )2−αdθ =−ED1−α q−(ωS( )⊗ω))[ ] + (ωS( , )−ωS(s, ))ED1−α q−ωq−( ). 24 MAR´ IA J. GARRIDO-ATIENZA, KENING LU, AND BJ ¨ ORN SCHMALFUSS Secondly, simila o (13) we ob ain ha ˜ ED1−α q−(u⊗ω)( )(s, ·)q−[ ] = −˜ ED1−α q−(u⊗ω)( )[ ] + ˜ E(u( )−u(s), D1−α q−ωq−( )), and subs i u ing he las wo exp essions in o (32) we ob ain he conclusion.  In he ollowing we wan o p o e ha wis well–de ined o U∈ˆ W0,T . To his end we will conside a mapping w( , s, q) = w(U, ω, (ω⊗Sω))( , s, q) which coincides o smoo h ωwi h he exp ession in oduced in Lemma 10. In o de o p o e he egula i y o ws a ed in Lemma 20 we need se e al p ope ies ha we collec and p o e in he ollowing esul . Lemma 28. (i) Le ˜ E∈L2(V⊗V, ˆ V)and le ∈V⊗Vbe ixed. Then he mapping ˜ E7→ ˜ E is in L2(L2(V⊗V, ˆ V), V ). (ii) Le E∈L2(V, ˆ V)and le u∈Vbe ixed. Then he mapping E7→ Eu is in L2(L2(V, ˆ V), V ). (iii) Le ˜ E∈L2(V⊗V, ˆ V)and u∈V. Then k˜ E(u, ·)kL2(V, ˆ V)=k˜ E(u⊗V·)kL2(V, ˆ V)≤ |u|k˜ EkL2(V⊗V, ˆ V). P oo . Conside he sepa able Hilbe –space L2(V⊗V, ˆ V) equipped wi h he com- ple e o hono mal basis ( ˜ Eijk)i,j,k∈Ngi en by ˜ Eijk(el⊗Vem) = ˜ Eijk(el, em) = 0 : j6=lo k6=m i:j=land k=m. We emind ha (ei)i∈Nand ( i)i∈Na e, espec i ely, comple e o hono mal basis o Vand ˆ V. Then o ∈V⊗V, sX ijk (˜ Eijk )2=sX i| i|2X jk 2 jk =cV, ˆ Vk k whe e jk is he mode o wi h espec o ej⊗Vek. This comple es (i). The second s a emen can be p o en simila ly and he e o e we omi i s p oo . Finally, we ha e k˜ E(u, ·)k2 L2(V, ˆ V)=X i|˜ E(u, ei)|2 ˆ V =X i|X k uk˜ E(ek, ei)|2 ˆ V≤X iX k|uk||˜ E(ek, ei)|ˆ V2 ≤X iX k|uk|21 2X k|˜ E(ek, ei)|2 ˆ V1 22 =|u|2k˜ Ek2 L2(V⊗V, ˆ V).  LOCAL PATHWISE SOLUTIONS TO SEES 25 In wha ollows we abb e ia e he no a ion in he ollowing way: le us deno e L2,⊗=L2(L2(V, ˆ V), V ⊗V), L2,⊗,⊗=L2(L2(V⊗V, ˆ V), V ⊗V). P oo o Lemma 20. P oo . Le us conside sepa a ely he h ee e ms o w( , s, q) gi en by (27). P e- cisely, we s a es ima ing he hi d e m I3(˜ E) := Zq s D2α−1 s+ωS(·, )[ ]D1−α q−D1−α q−˜ E [ ]d . As we ha e seen in Lemma 28 (i), o a ixed ∈V⊗V he mapping L2(V⊗V, ˆ V)∋ ˜ E7→ ˜ E is in L2(L2(V⊗V, ˆ V), V ) whe e an es ima e o he no m o his ope a o is gi en by cV, ˆ Vk k. Then, since Lemma 18 (i) in pa icula implies ha D2α−1 s+ωS( , ) is in L(V, V ⊗V), he mapping ˜ E7→ I3(˜ E) is in L2,⊗,⊗. We ha e kI3(·)kL2,⊗,⊗≤Zq skD2α−1 s+ωS(·, )[ ]·D1−α q−D1−α q− [ ]kL2,⊗,⊗d ≤Zq skD2α−1 s+ωS(·, )[ ]kL(V,V ⊗V)k·D1−α q−D1−α q− [ ]kL2(L2(V⊗V, ˆ V),V )d . In o de o es ima e he second ac o in he in eg and o I3, no e ha due o Lemma 21, o ∈(s, q) and U∈ˆ W0,T , we ob ain kD1−α q−D1−α q− [ ]k ≤ ckUkW −β(q− )β+β′+2α−2 and as we ha e said a he beginning o his p oo kD1−α q−D1−α q−· [ ]kL2(L2(V⊗V, ˆ V),V )≤ccV, ˆ VkUkW −β(q− )β+β′+2α−2. On he o he hand, by Lemma 18 (i), kD2α−1 s+ωS(·, )[ ]kL(V,V ⊗V)≤c( − )β′ ( −s)2α−1+Z s ( −ξ)β′ ( −ξ)2αdξ(|||ω|||β′+|||ω|||β′′ ). Combining he p e ious es ima es we can conclude kI3(·)kL2,⊗,⊗≤ckUkWs−β( −s)β′(q−s)β+β′, whe e cdepends on |||ω|||β′′ and |||ω|||β′. Nex we deal wi h I2(˜ E) := Zq s ˆ Dα s+ωS(·, )˜ E(u(·)−u(s),·)[ ]D1−α q−ωq−[ ]d . Obse e ha kˆ Dα s+ωS(·, )˜ E(u(·)−u(s),·)[ ]D1−α q−ωq−[ ]kL2,⊗,⊗ ≤kωS( , )˜ E(u( )−u(s), D1−α q−ωq−[ ])kL2,⊗,⊗ ( −s)α +cZ s k(ωS( , )−ωS(θ, )) ˜ E(u( )−u(θ), D1−α q−ωq−[ ])kL2,⊗,⊗ ( −θ)1+αdθ ≤ccV, ˆ V(q− )α+β′−1s−βk˜ EkL2(V⊗V, ˆ V)kukβ,∼|||ω|||β′ ×(|||ω|||β′+|||ω|||β′′ )(( − )β′( −s)β−α+ ( −s)β′+β−α),