Fractional Stochastic Partial Differential Equations Research Articles

Moment stability of fractional stochastic evolution equations with Poisson jumps

Fractional differential equations have wide applications in science and engineering. In this paper, we consider a class of fractional stochastic partial differential equations with Poisson jumps. Sufficient conditions for the existence and asymptotic stability in pth moment of mild solutions are derived by employing the Banach fixed point principle. Further, we extend the result to study the asymptotic stability of fractional systems with Poisson jumps. An example is provided to illustrate the effectiveness of the proposed results.

On a jump-type stochastic fractional partial differential equation with fractional noises

We study a class of stochastic fractional partial differential equations of order α>1 driven by a (pure jump) Lévy space–time white noise and a fractional noise. We prove the existence and uniqueness of the global mild solution by the fixed point principle under some suitable assumptions.

Impulsive Noise Driven One-Dimensional Higher-Order Fractional Partial Differential Equations

We are going to study a kind of stochastic fractional partial differential equation driven by an impulsive noise, which is singular not only in time but also in space. We first study the existence and uniqueness of the solution and then investigate the regularity of the solution in its spatial variable which depends on the order of the fractional operator, and deeply relies on the precise analysis of the kernel generated by our operator. In addition, we also discuss the stochastic flow property of the solutions.

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.

On the solution process for a stochastic fractional partial differential equation driven by space–time white noise

Let { u ( t , x ) : t ≥ 0 , x ∈ R } be the solution process for the following Cauchy problem for the stochastic fractional partial differential equation taking values in R d : ∂ ∂ t u ( t , x ) = D α δ u ( t , x ) + W ̇ ( t , x ) , t > 0 , x ∈ R ; u ( 0 , x ) = u 0 ( x ) , where D α δ ( 1 < α < 3 , | δ | ≤ min { α − [ α ] , 2 + [ α ] 2 − α } ) is the fractional differential operator with respect to the spatial variable x (see below for a definition), W ̇ ( t , x ) is an R d -valued space–time white noise, and u 0 is an initial random datum defined on R . In this paper, we study the sample path properties of the solution process. We first find the dimensions in which the process hits points, and then determine the Hausdorff and packing dimensions of the range, the graph and the level sets of the process. Our results generalize those of Mueller and Tribe (2002) and Wu and Xiao (2006) for random string processes.

Regularity of a fractional partial differential equation driven by space-time white noise

We will deal with one dimensional stochastic fractional order partial differential equation driven by space-time white noise. The existence and uniqueness of the solution and especially some regularities of the solution are investigated. The regularities of the solution in its time and space variables depend on the relation of the fractional order of its operator and coefficients.

On a stochastic fractional partial differential equation driven by a Lévy space-time white noise

We study the existence and uniqueness of the global mild solution for a stochastic fractional partial differential equation driven by a Lévy space-time white noise. Moreover, the flow property for the solution is also studied.

Fractional SPDEs with non-Lipschitz coefficients

We study in this paper a class of fractional stochastic partial differential equations with non-Lipschitz coefficients in the whole real line. Existence, uniqueness and regularity of the solution are proved.

Stochastic fractional partial differential equations driven by Poisson white noise

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.

On the solutions of nonlinear stochastic fractional partial differential equations in one spatial dimension

Existence, uniqueness and regularity of the trajectories of mild solutions of one-dimensional nonlinear stochastic fractional partial differential equations of order α > 1 containing derivatives of entire order and perturbed by space–time white noise are studied. The fractional derivative operator is defined by means of a generalized Riesz–Feller potential.

Variational solutions for a class of fractional stochastic partial differential equations

In this Note we present new results regarding the existence, the uniqueness and the equivalence of two notions of variational solution related to a class of non autonomous, semilinear, stochastic partial differential equations defined on an open bounded domain D ⊂ R d . The equations we consider are driven by an infinite-dimensional noise derived from an L 2 ( D ) -valued fractional Wiener process W H with Hurst parameter H ∈ ( 1 γ + 1 , 1 ) , where γ ∈ ( 0 , 1 ] denotes the Hölder exponent of the derivative of 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).