Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise

Abstract

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by L\'evy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations are established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that H\gamma (\scrO ) is compactly embedded in L2 (\scrO ) with \gamma \in (0, 1). Moreover, the uniqueness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of stochastic PDEs perturbed by small L\'evy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee--Infante equations.