Large Deviations of Fractional Stochastic Equations with Non-Lipschitz Drift and Multiplicative Noise on Unbounded Domains

TL;DR

This paper establishes the large deviation principle for fractional stochastic reaction-diffusion equations with non-Lipschitz drift and multiplicative noise on unbounded domains, using weak convergence methods.

Contribution

It extends large deviation analysis to non-local fractional stochastic equations with polynomial drift on unbounded domains, overcoming non-compactness challenges.

Findings

Proved strong convergence of control solutions.

Established convergence in distribution as noise vanishes.

Derived large deviation principles for the equations.

Abstract

This paper is concerned with the large deviation principle of the non-local fractional stochastic reaction-diffusion equation with a polynomial drift of arbitrary degree driven by multiplicative noise defined on unbounded domains. We first prove the strong convergence of the solutions of a control equation with respect to the weak topology of controls, and then show the convergence in distribution of the solutions of the stochastic equation when the noise intensity approaches zero. We finally establish the large deviations of the stochastic equation by the weak convergence method. The main difficulty of the paper is caused by the non-compactness of Sobolev embeddings on unbounded domains, and the idea of uniform tail-ends estimates is employed to circumvent the obstacle in order to obtain the tightness of distribution laws of the stochastic equation and the precompactness of the control equation.