Mittag--Leffler Euler integrator and large deviations for stochastic space-time fractional diffusion equations

TL;DR

This paper develops a super-convergent Mittag--Leffler Euler integrator for stochastic space-time fractional diffusion equations and analyzes its large deviation principles, demonstrating convergence of the rate functions.

Contribution

It introduces a novel decomposition method for handling singularities and establishes the super-convergence order of the integrator, along with large deviation analysis.

Findings

Super-convergence order achieved for the integrator

Decomposition method effectively handles solution singularities

Large deviation rate function of the integrator converges to that of the original equation

Abstract

Stochastic space-time fractional diffusion equations often appear in the modeling of the heat propagation in non-homogeneous medium. In this paper, we firstly investigate the Mittag--Leffler Euler integrator of a class of stochastic space-time fractional diffusion equations, whose super-convergence order is obtained by developing a helpful decomposition way for the time-fractional integral. Here, the developed decomposition way is the key to dealing with the singularity of the solution operator. Moreover, we study the Freidlin--Wentzell type large deviation principles of the underlying equation and its Mittag--Leffler Euler integrator based on the weak convergence approach. In particular, we prove that the large deviation rate function of the Mittag--Leffler Euler integrator $\Gamma$-converges to that of the underlying equation.