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.

翻译：随机时空分数阶扩散方程常出现在非均匀介质中热传播过程的建模中。本文首先研究了一类随机时空分数阶扩散方程的Mittag–Leffler欧拉积分器，通过发展一种适用于时间分数阶积分的分解方法，获得了该积分器的超收敛阶。这里所提出的分解方法是处理解算子奇异性的关键。此外，基于弱收敛方法，我们研究了基础方程及其Mittag–Leffler欧拉积分器的Freidlin–Wentzel型大偏差原理。特别地，我们证明了Mittag–Leffler欧拉积分器的大偏差率函数Γ-收敛于基础方程的大偏差率函数。