Moderate Deviations for Stochastic Fractional Heat Equation Driven by Fractional Noise

Xichao Sun,Ming Li,Wei Zhao

doi:10.1155/2018/7402764

Abstract

We consider a class of stochastic fractional heat equations driven by fractional noises. A central limit theorem is given, and a moderate deviation principle is established.

Highlights

Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models
Cardon-Weber [2] proved a LDP for a Burgers-type SPDE driven by white noise
Jiang et al [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise

Summary

Introduction

Since the work of Freidlin and Wentzell [1], the large deviation principle (LDP) has been extensively developed for small noise systems and other types of models (such as interacting particle systems) (see [2,3,4,5,6,7]). Marquez-Carreras and Sarra [3] proved a LDP for a stochastic heat equation with spatially correlated noise, and Mellali and Mellouk [4] extended Marquez-Carreras and Sarra’s [3] to a fractional operator. Jiang et al [5] proved a LDP for a fourth-order stochastic heat equation driven by fractional noise. There are many works about MDP about stochastic (partial) differential equations; some surveys and literatures could be found in Budhiraja et al [11], Wang and Zhang [12], Li et al [13], Yang and Jiang [14], and the references therein. We investigated the moderate deviations about the stochastic fractional heat equation with fractional noise as follows:. To fill the gap between scale a ε = 1 and scale a ε = 1/ ε, we mainly devote to the moderate deviation when the scale satisfies the following:. In Appendix, some results about the Green kernel are given

Fractional Noise

Main Results and Their Proof

Convergence of the Solution