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

.This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy 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 \(H^\gamma (\mathcal{O})\) is compactly embedded in \(L^2(\mathcal{O})\) 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 Lévy 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.KeywordsFractional Laplacian operatorLévy noiseBrownian motionweak mean random attractorsinvariant measuresergodicitylarge deviation principleMSC codes35R1135Q3065F0860H1565F10