A CLASS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL NOISE

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1 A CLASS OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS DRIVEN BY A FRACTIONAL NOISE M. Sanz-Solé, U. of Barcelona, Span Isaac Newton Insttute for Mathematcal Scences May 13, 1 sanz

2 Plan of the lecture 1. Descrpton of the SPDE. A stochastc ntegral sutable to formulate the SPDE 3. Two notons of solutons -varatonal and mld- and ther relatonshp 4. Propertes of the sample paths of the solutons 5. Dscusson on unqueness Based on jont work wth P. Vullermot, J. Evol. Eq. 9.

3 1. The SPDE D R d bounded doman, D of class C +β, β (, 1), T >, du(x, t) = dv (k(x, t) u(x, t)) dt + g(u(x, t))dt + h(u(x, t))w H (x, dt), u(x, ) = ϕ(x), x D, u(x, t) n(k) (x, t) D (, T ]. Related wth =, (x, t) D (, T ], (1) Stochastc heat equaton (Walsh, 1986;...). Extensons of heat equaton (Dawson-Gorostza; 1987, Krylov,...).

4 Assumptons g, h : R R Lpschtz, k : D [, T ] R d R d 1. symmetrc: k,j (.) = k j, (.),. strctly ellptc: (k(x, t)q, q) R d c q, for all q R d, (x, t) D [, T ]. L = dv(k(, t) ) s self-adjont and postve. 3. regular, jontly n (x, t). ϕ C +β (D). S.D. Edelman, N.V. Zhtarashu, 1998.

5 The nose: nfnte dmensonal fractonal Brownan moton Set C : L (D) L (D), lnear, self-adjont, postve, non-degenerate, trace-class. (e ) Z+ ONB of L (D), sup Z+ e <, Ce = λ e. (λ ) Z+, λ >, λ 1 <. (t), t }, Z +, sequence of ndependent fbm. {B H W H (, t) = λ 1 e ( )B H Nose fractonal n tme, colored n space. (t).

6 Fractonal Brownan Moton (fbm) Fx H (, 1) (Hurst parameter). A Fractonal Brownan moton {B H t, t } s a Gaussan process wth E(B H t ) =, E(B H t B H s ) = 1 ( s H + t H t s H). H = 1 : Brownan moton, H > 1 : Increments are postvely correlated (aggregaton, cluster phenomena), H < 1 : Increments are negatvely correlated (sequences wth ntermttency), P. Lévy ( s), Mandelbrot and Van Ness (1968),

7 Some propertes The sample paths of B H are a-hölder contnuous wth a (, H). Indeed, E ( B H t B H s ) = t s H. Thus E ( B H t B H s p) C p t s ph, and we can apply Kolmogorov s crteron. For H 1, B H does not have ndependent ncrements, B H s not a semmartngale. Itô Calculus does not apply.

8 References on related work Same settng as ours, H = 1 M. S-S, P. Vullermot, Ann. I. H. Poncaré-PR, 3. More general operators and coeffcents, f-d Brownan mot. L. Dens, L. Stoca, Electronc J. of Probablty, 4. On UMD Banach spaces, type, H = 1 M. C. Veraar, J. of Evoluton Equ., 1. Autonomous operators, fbm H > 1 B. Maslowsk, D. Nualart, J. of Funct. Analyss, 3. Same settng as ours D. Nualart, P. Vullermot, J. of Funct. Analyss, 6.

9 Calculus wth fractonal Brownan moton Pathwse approach 1. Fractonal Calculus: Zähle, Ppras-Taqqu, Hu-Nualart... (H > 1, but not only). Rough Path Analyss: Coutn-Qan, Feyel-La Pradelle, Frz-Vctor, Gubnell, Lyons, Lejay,... (H > 1 4 ) Applcatons 1. Stochastc Dfferental Equatons: Zähle, Nualart-Rascanu, Rough Path Famly,.... Stochastc Partal Dfferental Equatons: Duncan, Hu-Øksendal, Frz-Oberhauser, Gubnell, Lejay, Maslowsk-Nualart, Tndel, Tudor, Vens, Quer-Sardanyons,... Mallavn Calculus Alòs-Mazet-Nualart, 1. Applcatons not extensvely developed. Motvaton: Models n engneerng, bophyscs, mathematcal fnance, etc.

10 An ntegral wth respect to fbm, H > 1 Remann-Louvlle ntegrals and Marchaud dervatves f L 1 (a, b), α >, x (a, b): I α a+f (x) = 1 Γ(α) x Ib α ( 1) α f (x) = Γ(α) (x y) α 1 f (y)dy, a b x (y x) α 1 f (y)dy. Fx p 1. For f I α a+(l p ) (f I α b (Lp )), α (, 1): D α a+f (x) = 1 ( f (x) Γ(1 α) Db α ( 1) α f (x) = Γ(1 α) (x a) α + α ( f (x) (b x) α + α x a b x f (x) f (y) (x y) α+1 dy f (x) f (y) dy (y x) α+1 ) 1 (a,b) (x), ) 1 (a,b) (x).

11 Generalzed Steltjes ntegral: M. Zähle, 1998 Defnton Let p, q 1, 1 p + 1 q 1, αp < 1, f I a+(l α p ). Set g b (x) = g(x) g(b ); assume g b I 1 α b (Lq ). Then, b a fdg := ( 1) α b a Da+f α (x)d 1 α b g b (x) dx.

12 Fx α (, 1 ) and f, g : [, T ] R such that f α,1 = T f (s) T s s α + f (s) f (y) dyds <, (s y) α+1 g 1 α,,t = sup <s<t<t ( g(t) g(s) t (t s) 1 α + s ) g(y) g(s) dy <. (y s) α Then, t fdg, t [, T ] s well defned and (Zähle; Rascanu-Nualart) t fdg C α f α,1 g 1 α,,t. Extenson to Hlbert-valued functons.

13 Fact: For a fbm wth H ( 1, 1), where α ( 1 H, 1 ). ) E ( B H 1 α,,t <, Let f : Ω [, T ] L (D), f (t) = f (t)e, 1. Assume sup f α,1 <, a.s. Then, a.s. λ 1 t f (τ)b H (dτ) λ 1 B H 1 α,,t f α,1 <.

14 Set t f (t)w H (dt) := λ 1 t f (τ)b H (dτ). (Remember: W H (, t) = λ 1 e ( )B H (t).) Ths s a well defned random varable a.s. n L (D): t f (t)w H (dt) rα H sup f α,1, a.s.

15 Space for the solutons of the SPDE H > 1 α (1 H, 1) B α, (, T ; L (D)) space of measurable mappngs u : [, T ] L (D) such that u α, = < +. Remark: α,1 C α,. T ( t sup u(t) + dt dτ u(t) u(τ) ) t [,T ] (t τ) α+1

16 Two notons of soluton to equaton (1) I. Varatonal Soluton (Nualart-Vullermot, 6) u V L (, T ; H 1 (D)) B α, (, T ; L (D)) a.s., a.s., for every v H 1 (D (, T )) and t [, T ], dx v(x, t)u V (x, t) D t = dx v(x, )ϕ(x) + dτ dx v τ (x, τ)u V (x, τ) D D t dτ dx ( v(x, τ), k(x, τ) u V (x, τ)) R d D t + dτ dx v(x, τ)g(u V (x, τ)) + + =1 D λ 1 t (v(., τ), h(u V (., τ))e ) B H (dτ).

17 II. Mld Soluton u M L (, T ; H 1 (D)) B α, (, T ; L (D)) a.s., a.s., for every t [, T ] n L (D): u M (., t) = dy G(., t; y, )ϕ(y) D t + dτ dy G(., t; y, τ)g (u M (y, τ)) + + =1 D λ 1 t ( D ) dy G(., t; y, τ)h (u M (y, τ)) e (y) B H (dτ).

18 G s the Green functon assocated wth the operator L = t dv(k(x, t) ), on D (, T ) wth Neumann boundary condtons. Gaussan property: G : D [, T ] D [, T ] {s, t [, T ] : s t} R, [ ] x µ t ν G(x, t; y, s) c(t s) d+ µ +ν x y exp c t s µ = (µ 1,..., µ d ) N d, ν N and µ + ν, µ = d j=1 µ j.

19 If u B α,, all terms n the mld formulaton make sense For example, + =1 + = λ 1 t ( ) dy G(x, t; y, τ)h (u(y, τ)) e (y) B H D =1 λ 1 t f,t (., τ)b H (dτ). Remember: + λ 1 t f,t (u)(., τ)b H (dτ) =1 ( t C α sup dτ f,t(u)(., τ) τ α + α t τ (dτ) dσ f,t(u)(., τ) f,t (u)(., σ) (τ σ) α+1 ), a.s.

20 1. Gaussan property of G and lnear growth of h mples t. Lpschtz contnuty mples dτ f,t(u)(., τ) τ α C(1 + u(., τ) ). f,t (u)(x, τ) f,t (u)(x, σ) { C dy G(x, t; y, τ) u(y, τ) u(y, σ) D } + C dy G(x, t; y, τ) G(x, t; y, σ) (1 + u(y, σ) ). D Then, estmates related wth the Gaussan property of G yeld t τ dσ f,t(u)(., τ) f,t (u)(., σ) (τ σ) α+1 C(1 + u α, ).

21 Results Assumptons: g, h : R R Lpschtz, h s γ Hölder contnuous, γ (, 1]. Assumptons on L gven before. Theorem 1. Fx H ( 1 γ+1, 1 ) and then α ( 1 H, ) γ γ+1. Equaton (1) possesses a varatonal soluton u V (Nualart-Vullermot). Every such u V s a mld soluton u M. Indeed, u V (., t) = u M (., t), a.s. n L (D), for every t [, T ].

22 Theorem (cont.) ( ). Fx H 1 γ+1 d+1 d+, 1 and then α ( 1 H, ) γ γ+1 1 d+. If h s an affne functon, u V s the unque varatonal soluton to (1), whle u M s ts unque mld soluton. 3. For all s, t [, T ] we have θ u M (., t) u M (., s) C H t s θ ( 1 + u M α, ), (, ( ) 1 α) β, C H < + a.s. Hence {u M (t), t [, T ]} s a L (D) valued process wth θ Hölder contnuous sample paths.

23 Remarks For affne h there s ndstngushablty of varatonal and mld solutons. For constant h one can prove a result on exstence n L (, T ; L (D)). ( Moreover, ) the Hölder exponent of the soluton s θ, β.

24 About the Proof 1. Exstence of u V and unqueness for affne h s establshed n Nualart, Vullermot, 6 usng a Galerkn scheme: sequence of varatonal solutons on an ncreasng sequence of fnte dmensonal subspaces of L (D).. Every varatonal soluton s a mld soluton. Ths avods usng fxed pont arguments, (vald only for affne h?). 3. Unqueness s proved usng a fxed pont argument on a sutable space. 4. Hölderanty s establshed usng accurate estmates on ncrements of the Green functon. 5. Sample path regularty can also be proved by usng the factorsaton method, but the results are n general weaker.

25 . Every varatonal soluton s a mld soluton Let v C (D). Consder the test functon { v t v(x), f s = t, (x, s) = dy G(y, t; x, s)v(y), f t > s, D The defnton of varatonal soluton, the self-adjont property of the dfferental operator yeld: u V (., t) dy G(., t; y, )ϕ(y) D t dτ dy G(., t; y, τ)g (u V (y, τ)) D t dy G(., t; y, τ)h (u V (y, τ)) W H (y, dτ), D s a.s. orthogonal for every t [, T ] to C (D), then to L (D).

26 3. Unqueness when h s affne We modfy the norm α, by ntroducng dependence on tme: u α,,t t ( τ := sup u(τ) + dτ dσ u(τ) u(σ) ) τ [,t] (τ σ) α+1, t [, T ] ( α, = α,,t ). Let u M and ũ M be two mld solutons, z M = u M ũ M. We prove z M α,,t R t dτ z M α,,τ, a.s. for any t, wth R < a.s. By Gronwall s lemma, ths mples unqueness of the mld soluton.

27 Step 1: We obtan the estmate ( t z M (., t) R 1 dτ sup t ( τ + R dτ ρ [,τ] z M (., ρ) ) dσ z M(., τ) z M (., σ) (τ σ) α+1 a.s. for every t [, T ], for some fnte and postve random varables R 1, R. The left-hand sde can be replaced by z M (., t) α,,t. Ths holds for affne h (extenson?). ),

28 Step : Work further on ( τ to see ths term replaced by dσ z ) M(., τ) z M (., σ) (τ σ) α+1, τ ( σ sup z M (., ρ) + dσ dρ z ) M(., σ) z M (., ρ) ρ [,τ] (σ ρ) α+1. Obtan estmates of z M (., τ) z M (., σ), keepng track of powers of the factor τ σ. Use that h s affne to deal wth second order ncrements of u.

29 Usng the equaton satsfed by z M and estmates on ncrements of the Green functon, we obtan: ( ) t 6 z M α,,t R dτ z M (., ρ) + T k (t), a.s. Two type of terms: T 1 (t) = T 4 (t) = t = t ( τ dτ sup ρ [,τ] σ k=1 ( τ ( τ ) 1 ) dτ dσ(τ σ) 1 α dρ z M (., ρ), ( dσ τ (τ σ) α+1 dρ σ Sngulartes can be ntegrated. ρ σ dξ z )) M(., ρ) z M (., ξ) (ρ ξ) α+1,

30 4. Hölder contnuty The ntegral w.r.t. fbm C(u M )(., t) C(u M )(., s) = where f,t (u M )(., τ) = D + + =1 + =1 λ 1 t f,t (u M )(., τ)b H s λ 1 s f,t,s(u M )(., τ)b H (dτ) dy G(x, t; y, τ)h (u M (y, τ)) e (y), f,t,s(u M )(., τ) = f,t (u M )(., τ) f,s (u M )(., τ). (dτ), Second term n the RHS: Requres, n partcular, the sharp sze of G(x, t; y, τ) G(x, s; y, τ) G(x, t; y, σ) + G(x, s; y, σ) n terms of powers of t s, s τ and τ σ.

31 Factorzaton method (Da Prato, Kwapen, Zabczyk, 1987) C(u M )(., t) = + =1 λ 1 t = sn(επ) π where ε (, 1 ) and Y ε (u M )(., t) = t + λ 1 t =1 f,t (u M )(., τ)b H dτ(t τ) ε 1 (dτ) D dyg(, t; y, τ)y ε (u M )(y, τ), (t τ) ε f,t (u M )(., τ)b H (dτ). ()

32 Facts: sup t [,T ] Y ε (u M )(., t) R(1 + u M α,,t ). C(u M )(., t) C(u M )(., s) c(t 1 + T ), t T 1 = dτ(t τ) ε 1 dyg(., y; y, τ)y ε (u M )(, τ) s D s T = dτ dyy ε (u M )(y, τ) D ( (t τ) ε 1 G(., t; y, τ) (s τ) ε 1 G(., s; y, τ) ) ( ) Upper Bound: t s α (1 + u α,,t ), α, d+ 1.

33 Open problems, further drectons If h s not affne, we have not succeed yet to control two-dmensonal ncrements of h(u M (., τ)) to obtan smlar estmates. Fractonal dfferentaton? Free terms g(u(s, t), u(x, t), t), h(u(s, t), u(x, t), t). End