David Nualart

Pierre-A. Vuillermot

Comptes Rendus Mathématique

1/lariationalboundedivee solutionéfinies dansit d’unC. R. Acad. Sci. Paris, Ser. I 340 (2005) 281–286http://france.elsevier.com/direct/CRASSPartial Differential Equations/Probability TheoryVariational solutions for a class of fractional stochastic partiadifferential equationsDavid Nualarta, Pierre-A. Vuillermotba Facultat de Matemàtiques, Universitat de Barcelona, Gran Via, 585, 08007 Barcelona, Spainb I.E.C.N., université Henri-Poincaré, Nancy 1, BP 239, 54506 Vandoeuvre-lès-Nancy cedex, FranceReceived 18 October 2004; accepted 4 January 2005Presented by Pierre-Louis LionsAbstractIn this Note we present new results regarding the existence, the uniqueness and the equivalence of two notions of vsolution related to a class of non autonomous, semilinear, stochastic partial differential equations defined on an opendomainD ⊂ Rd . The equations we consider are driven by an infinite-dimensional noise derived from anL2(D)-valued frac-tional Wiener processWH with Hurst parameter H∈ ( 1γ+1,1), whereγ ∈ (0,1] denotes the Hölder exponent of the derivatof the nonlinearity that appears in the stochastic term.To cite this article: D. Nualart, P.-A. Vuillermot, C. R. Acad. Sci. Paris,Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméSolutions variationnelles pour une classe d’équations aux dérivées partielles stochastiques fractionnaires.Dans cetteNote nous présentons des résultats nouveaux concernant l’existence, l’unicité et l’équivalence de deux notions dvariationnelle relatives à une classe d’équations aux dérivées partielles stochastiques semilinéaires non autonomes dun ouvert bornéD ⊂ Rd . Les équations que nous considérons sont dirigées par un bruit en dimension infinie déduprocessus de Wiener fractionnaireWH à valeurs dansL2(D) de paramètre de Hurst H∈ ( 1γ+1,1), oùγ ∈ (0,1] est l’exposantde Hölder de la dérivée de la nonlinéarité apparaissant dans le terme stochastique.Pour citer cet article : D. Nualart, P.-A.Vuillermot, C. R. Acad. Sci. Paris, Ser. I 340 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.E-mail addresses:dnualart@ub.edu (D. Nualart), vuillerm@iecn.u-nancy.fr (P.-A. Vuillermot).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.01.006282 D. Nualart, P.-A. Vuillermot / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 281–286(Hnotionsnvaria-s ensuiteopriétésapproxi-ns dude typede ce typeéquationsien que) lorsqueivé entreématiques ou aux8–10,13]complèteall for an by ais Notefferentialnctionaland theirfcallytyVersion française abrégéeLa numérotation, la notation et la terminologie que nous utilisons ici ainsi que les hypothèses (C), (K),γ ) et(I) se réfèrent directement aux formules, à la notation, à la terminologie et aux hypothèses (C), (K), (Hγ ) et (I) dela version principale en anglais. Nous nous intéressons à l’existence, à l’unicité et à l’équivalence de deuxde solution variationnelle relatives au problème (2). Notre résultat principal est le suivant :Théorème 0.1.Supposons que(C), (K), (Hγ ) et (I) soient satisfaites; alors le problème(2) possède une solutiovariationnelleuI,ϕ de typeI et une solution variationnelleuII ,ϕ de typeII telles queuI,ϕ(·, t) = uII ,ϕ(·, t) p.p. dansL2(D) pour chaquet ∈ [0, T ]. De plus, si la fonctionh est affine,uI,ϕ est l’unique solution de typeI de(2) etuII ,ϕen est l’unique solution de typeII .Nous mettons ainsi en évidence l’existence, l’unicité et l’indiscernabilité des deux notions de solutiontionnelle introduites ci-dessous. Notre démonstration de l’existence d’une solution variationnelleuI,ϕ r pose surl’existence et la convergence d’un schéma de Faedo–Galerkin convenable associé à (2). Nous pouvondémontrer queuI,ϕ est nécessairement une solution variationnelle de type II, ceci grâce à de nouvelles prde continuité de l’intégrale stochastique que nous utilisons conjointement avec certaines propriétés d’mation polynomiale des fonctions test admissibles. Nous obtenons l’unicité deuI,ϕ lorsqueh est une fonctionaffine en utilisant le fait queg est lipschitzienne, puis nous pouvons finalement déduire les autres affirmatiothéorème en observant que toute solution variationnelle de type II de (2) en est une solution variationnelleI. À notre connaissance, le théorème énoncé est nouveau : nous ne connaissons en effet aucun résultatpour les équations différentielles ou les équations intégrales stochastiques fractionnaires, bien que cesaient fait l’objet de plusieurs travaux récents (voir par exemple [6–10,12,14,15] et leurs références), et bplusieurs notions de solution variationnelle existent déjà depuis longtemps pour des problèmes tels que (2le bruit est dérivé d’un processus de Wiener standard (voir par exemple [11,17,18]). Notre travail a été motautres choses par le fait qu’il existe nombre d’applications récentes s’appuyant sur une modélisation mathfaisant intervenir soit des mouvements browniens fractionnaires, soit des équations différentielles ordinairedérivées partielles dirigées par des bruits dérivant de tels mouvements browniens (voir par exemple [4,5,et leurs références). Nous renvoyons le lecteur à la version principale en anglais pour une description plusde nos résultats, dont nous donnons les démonstrations détaillées en [16].1. Introduction, notation and main resultMany applications in engineering, the natural sciences and the world of finance, to name only a few, cmathematical modelling in terms of fractional Brownian motions or in terms of differential equations drivefinite or an infinite-dimensional fractional noise (see, for instance, [4,5,8–10,13] and their references). In thwe present new results regarding a class of non autonomous, semilinear, fractional stochastic partial diequations that have potential applications to the above fields. In what follows we assume that all the fuspaces are real and use the standard notations for the usual spaces of Lebesgue integrable functionsnorms. Ford ∈ N+ let D ⊂ Rd be open, bounded and assume that its boundary∂D be sufficiently smooth. LetCbe a linear, self-adjoint, positive, non degenerate trace-class operator inL2(D). In the following we write(ei)i∈N+for an orthonormal basis ofL2(D) consisting of eigenfunctions of the operatorC and(λi)i∈N+ for the sequence othe corresponding eigenvalues. Let((BHi (t))t∈R+0 )i∈N+ be a sequence of one-dimensional, independent, identidistributed fractional Brownian motions with Hurst parameter H∈ (12,1), defined on the complete probabiliD. Nualart, P.-A. Vuillermot / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 281–286 283alueativenroblemsationalhability.l-t of thespace(Ω,F ,P) and starting at the origin. We define theL2(D)-valued, fractional Wiener process(WH(·, t))t∈R+0byWH(·, t) :=+∞∑i=1λ1/2i ei(·)BHi (t) (1)where the series (1) converges in the strong topology ofL2(D) a.s. by virtue of the basic properties of theBHi (t)’sand the fact thatC is trace-class. LetT ∈ R+ and let us consider the class of real, parabolic, initial-boundary vproblems given bydu(x, t) = (div(k(x, t)∇u(x, t)) + g(u(x, t)))dt + h(u(x, t))WH(x,dt),(x, t) ∈ D × (0, T ],(2)u(x,0) = ϕ(x), x ∈ D,∂u(x, t)∂n(k)= 0, (x, t) ∈ ∂D × (0, T ].In the preceding equations the functionk is matrix-valued and the last relation stands for the conormal derivof u relative tok; moreover, we assume that the following hypotheses hold:(C) We haveei ∈ L∞(D) for everyi and∑+∞i=1 λ1/2i ‖ei‖∞ < +∞.(K) The functionk :D × (0, T ] → Rd2 is Lebesgue-measurable and satisfies the symmetry relationki,j (·) =kj,i(·) for everyi, j ∈ {1, . . . , d}. Moreover, there exist constantsk, k̄ ∈ R+ such that the inequalitiesk|q|2  (k(x, t)q, q)Rd k̄|q|2hold uniformly in (x, t) ∈ D × (0, T ] for all q ∈ Rd , where we have written(·, ·)Rd for the standard Euclideainner product inRd .(Hγ ) The functionsg,h :R → R are Lipschitz continuous; moreover, the derivativeh′ :R → R exists, is Höldercontinuous with exponentγ ∈ (0,1] and bounded; finally we have H∈ ( 1γ+1,1).(I) The initial conditionϕ is anL2(D)-valued random variable.Formally, relations (2) define a class of non autonomous, semilinear, stochastic initial-boundary value pdriven by an infinite-dimensional fractional noise. Our purpose in this Note is to introduce two notions of varisolution for (2) and to present new results regarding their existence, their uniqueness and their indistinguisTo this end we write(·, ·)2 for the standard inner product inL2(D), H 1(D) for the usual Sobolev space of reavalued functions onD and ‖ · ‖1,2 for the corresponding norm. We fix once and for all anα ∈ (1 − H, γγ+1)and introduce the Banach spaceBα,2([0, T ];L2(D)) of all Lebesgue-measurable mappingsu : [0, T ] → L2(D)endowed with the norm‖u‖2α,2 :=(esssupt∈[0,T ]∥∥u(t)∥∥2)2 +T∫0dt( t∫0dτ‖u(t) − u(τ)‖2(t − τ)α+1)2< +∞.Our first notion of variational solution for (2) is one where the admissible test functions are independentime variable.Definition 1.1. We say theL2(D)-valued random field(uI(·, t))t∈[0,T ] defined on(Ω,F ,P) is avariational solu-tion of typeI to problem (2) if the following two conditions hold:(1) We haveuI ∈ L2(0, T ;H 1(D)) ∩Bα,2([0, T ];L2(D)) a.s.284 D. Nualart, P.-A. Vuillermot / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 281–286a.s. Inral in thewhene notionsg-timeerences),, we areal noisefew typesrences).y, oneesl(2) The integral relation(v,uI(·, t))2 = (v,ϕ)2 −d∑i,j=1t∫0dτ(vxi , ki,j (·, τ )uI,xj (·, τ ))2 +t∫0dτ(v,g(uI(·, τ )))2++∞∑i=1λ1/2it∫0(v,h(uI(·, τ ))ei)2BHi (dτ) (3)holds a.s. for everyv ∈ H 1(D) and everyt ∈ [0, T ].From the preceding definition and the above hypotheses, we easily infer that each term in (3) is finiteparticular, we can define each one-dimensional stochastic integral in (3) as a generalized Stieltjes integsense of [14,15,19] or [20].It is well-known that various notions of variational solution already exist for problems of the form (2)the noise is derived from a standard Wiener process (see, for instance, [11] and [17]); over the years theshave played an important rôle in a number of situations, particularly in those dealing with the analysis of lonbehavior phenomena in population dynamics and population genetics (see, for instance, [1–3] and their refaside from being equivalent to other notions of solution as was recently shown in [18]. On the other handnot aware of any works concerning the existence of variational solutions for problems driven by a fractionsuch as (2), though there have been several recent articles devoted to the mathematical analysis of quite aof fractional stochastic differential and integral equations (see, for instance, [6–10,12,14,15] and their refeIn this context there is yet another natural notion of variational solution we can think of for (2), namelwhere the admissible test functions depend on the basic space–time variable(x, t). For everyt ∈ (0, T ] let H 1(D ×(0, t)) be the Sobolev space of all real-valued functionsv ∈ L2(D × (0, t)) that possess distributional derivativvxi ∈ L2(D × (0, t)) for everyi ∈ {1, . . . , d} along with a distributional time-derivativevτ ∈ L2(D × (0, t)), thenorm ofH 1(D × (0, t)) being defined in the usual way.Definition 1.2.We say theL2(D)-valued random field(uII (·, t))t∈[0,T ] defined on(Ω,F ,P) is avariational solu-tion of typeII to problem (2) if the first condition of Definition 1.1 holds, and if the integral relation(v(·, t), uII (·, t))2 =(v(·,0), ϕ)2 +t∫0dτ(vτ (·, τ ), uII (·, τ ))2 −d∑i,j=1t∫0dτ(vxi (·, τ ), ki,j (·, τ )uII ,xj (·, τ ))2+t∫0dτ(v(·, τ ), g(uII (·, τ )))2 ++∞∑i=1λ1/2it∫0(v(·, τ ), h(uII (·, τ ))ei)2BHi (dτ) (4)holds a.s. for everyv ∈ H 1(D × (0, t)) and everyt ∈ [0, T ], wherex → v(x,0) ∈ L2(D) andx → v(x, t) ∈ L2(D)denote the Sobolev traces ofv onD andD × {τ ∈ R: τ = t}, respectively.Again we can prove that every term in (4) is well-defined and finite a.s.Our main result is the following:Theorem 1.3.Assume that Hypotheses(C), (K), (Hγ ) and (I) hold. Then, problem(2) possesses a variationasolutionuI,ϕ of typeI and a variational solutionuII ,ϕ of typeII , such thatuI,ϕ(·, t) = uII ,ϕ(·, t) a.s. inL2(D) foreveryt ∈ [0, T ]. Moreover, ifh is an affine function,uI,ϕ is the only variational solution of typeI to (2) anduII ,ϕ isits only variational solution of typeII .D. Nualart, P.-A. Vuillermot / C. R. Acad. Sci. Paris, Ser. I 340 (2005) 281–286 285eache Ionsists ofethef of itsctatgral inithement ofnowtochasticus whiteus whiteSeminarynam. 2s Inform.2. Remarks and brief sketch of the proof(1) Let (wn)n∈N+ be an orthonormal basis ofL2(D) such that(cnwn)n∈N+ be an orthonormal basis ofH 1(D)for some suitably chosen coefficientscn (such a basis always exists by virtue of standard elliptic theory); forN ∈ N+, let VN be the finite-dimensional subspace ofL2(D) generated by{w1, . . . ,wN } and writeϕN for theorthogonal projection of the initial conditionϕ onto VN . Our existence proof of a variational solution of typrests upon the existence and the convergence of the Faedo–Galerkin scheme associated with (2) that cthe approximation sequence(uN)N∈N+ whereuN ∈ L2(0, T ;VN) ∩ Bα,2([0, T ];VN), whose elements satisfy threlation(wn,uN(·, t))2 = (wn,ϕN)2 −d∑i,j=1t∫0dτ(wn,xi , ki,j (·, τ )uN,xj (·, τ ))2+t∫0dτ(wn,g(uN(·, τ )))2++∞∑i=1λ1/2it∫0(wn,h(uN(·, τ ))ei)2BHi (dτ)a.s. for everyn ∈ {1, . . . ,N} and everyt ∈ [0, T ]. Our proof that such a sequence actually exists is built ongeneral theory of [15] for systems of fractional stochastic ordinary differential equations, while our prooconvergence relies upon the availability of good a priori estimates that are uniform inN and upon a new compaembedding theorem for the Banach spaceL2(0, T ;H 1(D)) ∩Bα,2([0, T ];L2(D)).(2) In order to prove that (2) possesses a variational solution of type II, we argue indirectly by showing thuI,ϕitself is necessarily of type II; we obtain this result by proving new continuity properties of the stochastic inte(4) with respect to the strong topology ofBα,2([0, T ];L2(D)), and by combining these continuity properties wsuitable polynomial approximations of the admissible test functions that appear in (4). The uniqueness statthe theorem whenh is an affine function is then a consequence of the Lipschitz property ofg, while the remainingstatements follow from the fact that every variational solution of type II is trivially of type I. We do not kwhether the uniqueness statement still holds for an arbitraryh satisfying Hypothesis (Hγ ).We refer the reader to [16] for more details and for the complete proofs of the above results.References[1] B. Bergé, I.D. Chueshov, P.-A. Vuillermot, On the behavior of solutions to certain parabolic SPDE’s driven by Wiener processes, SProcess. Appl. 92 (2001) 237–263.[2] I.D. Chueshov, P.-A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneonoise: Stratonovitch’s case, Probab. Theory Related Fields 112 (1998) 149–202.[3] I.D. Chueshov, P.-A. Vuillermot, Long-time behavior of solutions to a class of stochastic parabolic equations with homogeneonoise: Itô’s case, Stochastic Process. Appl. 18 (2000) 581–615.[4] F. Comte, E. Renault, Long memory in continuous time volatility models, Math. Finance 8 (1998) 291–323.[5] N. Cutland, P. Kopp, W. Willinger, Stock price returns and the Joseph effect: a fractional version of the Black–Shole model, in:on Stochastic Analysis, Random Fields and Applications, in: Progr. Probab., vol. 36, Birkhäuser, Basel, 1995, pp. 327–351.[6] T.E. Duncan, B. Maslowski, B. Pasik-Duncan, Fractional Brownian motion and stochastic equations in Hilbert spaces, Stoch. D(2002) 225–250.[7] M. Gubinelli, A. Lejay, S. Tindel, Young Integrals and SPDE’s, IECN-Preprint 33, 2004.[8] Y. Hu, Heat equations with fractional white noise potentials, Appl. Math. Optim. 43 (2001) 221–243.[9] M.L. Kleptsyna, P.E. Kloeden, V.V. Anh, Existence and uniqueness theorems for fBm stochastic differential equations, ProblemTransmission 34 (1999) 332–341.286 D. Nualart, P.-A. Vuillermot / C. R. Acad. Sci. Paris, Ser. 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Variational solutions for a class of fractional stochastic partial differential equations

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Variational solutions for a class of fractional stochastic partial differential equations

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