Asymptotics for stochastic reaction-diffusion equation driven by subordinate Brownian motions

TL;DR

This paper investigates the long-term behavior of a stochastic reaction-diffusion system driven by subordinate Brownian motions, establishing ergodicity, uniqueness of invariant measures, and a large deviation principle despite heavy-tailed noise.

Contribution

It proves ergodicity, strong Feller property, irreducibility, and a large deviation principle for the system driven by subordinate Brownian motions, extending understanding beyond stable noise cases.

Findings

Existence of a unique invariant measure.

Establishment of a large deviation principle.

Demonstration that strong dissipation can overcome heavy-tailed noise effects.

Abstract

We study the ergodicity of stochastic reaction-diffusion equation driven by subordinate Brownian motions. After establishing the strong Feller property and irreducibility of the system, we prove the tightness of the solution's law. These properties imply that this stochastic system admits a unique invariant measure according to Doob's and Krylov-Bogolyubov's theories. Furthermore, we establish a large deviation principle for the occupation measure of this system by a hyper-exponential recurrence criterion. It is well known that S(P)DEs driven by $\alpha$-stable type noises do not satisfy Freidlin-Wentzell type large deviation, our result gives an example that strong dissipation overcomes heavy tailed noises to produce a Donsker-Varadhan type large deviation as time tends to infinity.