Dynamics and Large Deviations for Fractional Stochastic Partial Differential Equations with Lévy Noise
Jiaohui Xu, Tomás Caraballo, José Valero
Abstract
This paper is mainly concerned with a kind of fractional stochastic evolution equa-tions driven by L\\'evy noise in a bounded domain. We first state the well-posedness of the problem viaiterative approximations and energy estimates. Then, the existence and uniqueness of weak pullbackmean random attractors for the equations are established by defining a mean random dynamical sys-tem. Next, we prove the existence of invariant measures when the problem is autonomous by meansof the fact thatH\\gamma (\\scrO ) is compactly embedded inL2(\\scrO ) with\\gamma \\in (0,1). Moreover, the unique-ness of this invariant measure is presented, which ensures the ergodicity of the problem. Finally, alarge deviation principle result for solutions of stochastic PDEs perturbed by small L\\'evy noise andBrownian motion is obtained by a variational formula for positive functionals of a Poisson randommeasure and Brownian motion. Additionally, the results are illustrated by the fractional stochasticChafee--Infante equations.