Stochastic fractional partial differential equations driven by Poisson white noise

We study a stochastic fractional partial differential equations of order α > 1 driven by a compensated Poisson measure. We prove existence and uniqueness of the solution and we study the regularity of its trajectories. Équations aux dérivées partielles fractionnaires stochastiques dirigées par un bruit poissonnien Résumé On étudie une équation aux dérivées partielles stochastiques fractionnaires d’ordre α > 1 dirigée par une mesure de Poisson compensée. On montre l’existence et l’unicité de la solution et on étudie la régularité de ses trajectoires.


Introduction
In recent years fractional calculus knew a great progress considering their applications in different fields of science, including numerical analysis, physics, engineering, biology, economics and finance.Partial differential equations is one of the mathematical domains where the fractional calculus is strongly used (see Podlubny [8]).However, few publications treat stochastic partial differential equations involving fractional derivatives.Most of them investigate evolution type equations, driven by a fractional power of the Laplacian (see [3], [10]).These operators generate symmetric stable semigroups when the order of derivation is less than 2. Our idea is inspired by the paper of Debbi and Dozzi [5] in which the authors studied a nonlinear stochastic fractional partial differential equations of high order containing also derivatives of entire order and perturbed by

S. Hajji
Gaussian white noise.The aim of this paper is to generalize the result of [1] to nonlinear stochastic fractional partial differential equations.We study the existence, uniqueness and regularity of the solution of the following one-dimensional stochastic fractional partial differential equation, formally given by ∂u ∂t for t ∈ (0, +∞), x ∈ R, where (U, B(U), q) is a σ finite measure space, D α δ is the fractional differential operator with respect to the spacial variable, One rigorous formulations of the equation (1.1) can be given by the following integral equation for t ∈ [0, +∞) and x ∈ R, where {G α (t, x), x ∈ R, t ∈ [0, +∞)} stands for the fundamental solution of the operator D α δ .Some equations similar to equation (1.1) are studied also in [6] and [2], but with the Laplacian operator in place of the operator D α δ .We assume that the functions f and g satisfy the following growth and Lipschitz conditions: (H) For all T > 0, there exists a constant The remaining of the paper is structured as follows.In Section 2, we give the definitions of stochastic integral with respect to Ñ (ds, dy, dz) and the derivative operator D α δ , then we recall some inequalities satisfied by the fundamental solution G α and we precise the notion of solutions of the equation (1.1).Section 3 is dedicated to the proof of existence and uniqueness of the solution.In Section 4 we prove the spatial Hölder continuity of the solution.

Preliminaries and definitions
Let (Ω, F, P ) be a probability space equipped with a filtration {F t } satisfying the usual conditions.Let (U, B(U), q) be a σ finite measure space and let N be a Poisson measure on R + × R × U with intensity measure ν(ds, dx, dz) = dsdxq(dz).The compensated Poisson measure is denoted by Ñ = N − ν.Now, we define the stochastic integrals we will use.

Stochastic integral with respect to Ñ (ds, dy, dz)
Let us introduce the following class: it is known (see Ikeda and Watanabe [7]) that for any t > 0, the stochastic integral t + 0 R U h(s, y, z) Ñ (ds, dy, dz), for h ∈ F pred , can be well defined.The stochastic integral has the following isometry property Thus for any t > 0, t + 0 R U h(s, y, z) Ñ (ds, dy, dz) ∈ L 2 (Ω).This integral can be extended to a more general class F of integrands without the predictable property, in the following manner : A function h is said to be of class F if is F t adapted, and there exists a sequence h n ∈ F pred such that for any t > 0

S. Hajji
For h ∈ F , for any t > 0, t + 0 R U h(s, y, z) Ñ (ds, dy, dz) is defined as the L 2 (Ω)-limit of the following Cauchy sequence (see Albeverio et al. [1] for more details).

Definition of the operator D α δ
The fractional differential operator D α δ is an extension of the inverse of the generalized Riesz-Feller Potential when α > 2. It is given for α > 0 by Definition 2.1.The fractional differential D α δ is given by is the largest even integer less or equal to α, and δ = 0 when α ∈ 2N+1.F (respectively F −1 ) is the Fourier (respectively Fourier inverse) transform.The operator D α δ is a closed, densely defined operator on L 2 (R) and it is the infinitesimal generator of a semigroup which is in general not symmetric and not a contraction.It is selfadjoint only when δ = 0 and in this case, it coincides with the fractional power of the Laplacian.When α = 2 it is the Laplacian itself (see [4,5]).
The Green function G α (t, x) associated to the equation (1.1) is the fundamental solution of the Cauchy problem Using Fourier's calculus we obtain Let us recall some well-known properties, (see [5]), of the Green kernel G α which will be used later.
) is adapted for every x.
• {u(t, x)}, as a family of L 2 (Ω, F, P )-valued random variables, is right continuous and has left limits in the variable t ∈ [0, T ], namely,

S. Hajji
In this paper we simply call such a u modified càdlàg in t (after the French acronym).
A solution u on [0, T ] is said to be L p (Ω)bounded if All positive constants appearing in this paper are called C. They may change from one line to the next one.

Existence and uniqueness
The main result of this section is the following Theorem 3.1.Let p ≥ 2 and T > 0 be fixed.Under condition (H) and the assumption that the initial condition u 0 is L p (Ω) bounded, equation (1.1) has a unique mild solution u on [0, T ] which is L p (Ω) bounded.
For the proof we need the following lemma.Fix T > 0 and let Then X and Y belong to E.
Proof.Let us first show that X is well defined.For any fixed t ∈ [0, T ], let, for (s, y, z) By condition(H) we have : Thus h n t,x ∈ F pred .On the other hand, by the condition (H) we have By the definition of v n and the fact that v is modified càdlàg we have Furthermore by Lemma 2.2, property (7), we have Then by the Lebesgue's dominated convergence theorem we obtain that Hence h t,x ∈ F for any fixed t ∈ [0, T ], so the integral y)g(s, v(s−, y), z) Ñ (ds, dy, dz) is well defined.Using the same techniques, as in [1], we show that X is F t adapted and modified càdlàg.
Further more, by Burkholder's inequality and condition (H), we have By Lemma 2.2, property (7), the last integral is finite.Then It is well known that Y is measurable F t adapted and modified càdlàg.By Hölder inequality with respect to the measure |G α (t − s, x − y)|dsdy and the growth assumption on f , we have and finally Y ∈ E.
Proof of Theorem 3.1.
• Proof of the existence: Let the sequence (u n , n ≥ 1) be given by: It is easy to see that u 1 ∈ E, so by Lemma 3.2, we have for all n ≥ 1, u n ∈ E. Let By condition (H) and Hölder inequality with respect to the measure |G α (t − s, x − y)|dsdy, we obtain By condition (H), Burkholder inequality and Hölder inequality with respect to the measure G 2 α (t − s, x − y)dsdy we obtain By the same argument as in the proof of existence, we get : Applying Gronwall Lemma, we obtain that H(T ) = 0.This clearly implies that u(t, x, .)= v(t, x, .)a.s.