This paper considers the space-time fractional stochastic partial differential equation (SPDE, for short) with fractionally integrated additive noise, which is general and includes many (fractional) SPDEs with additive noise. Firstly, the existence, uniqueness and temporal regularity of the mild solution are presented. Then the Mittag--Leffler Euler integrator is proposed as a time-stepping method to numerically solve the underlying model. Two key ingredients are developed to overcome the difficulty caused by the interaction between the time-fractional derivative and the fractionally integrated noise. One is a novel decomposition way for the drift part of the mild solution, named here the integral decomposition technique, and the other is to derive some fine estimates associated with the solution operator by making use of the properties of the Mittag--Leffler function. Consequently, the proposed Mittag--Leffler Euler integrator is proved to be convergent with order $\min\{ \frac{1}{2} + \frac{\alpha}{2\beta}(r+\lambda-\kappa), 1 \}$ if $\alpha + \gamma = 1$, otherwise order $\min\{ \frac{\alpha}{2\beta}\min\{\kappa,r+\lambda\} + (\gamma-\frac{1}{2})^{+}, 1-\varepsilon\}$ in the sense of $L^2(\Omega,H)$-norm for the nonlinear case. In particular, the corresponding convergence order can attain $\min\{ \alpha + \frac{\alpha r}{2\beta} + (\gamma-\frac{1}{2})^{+}-\varepsilon, \alpha+\gamma -\varepsilon, 1 \}$ for the linear case.
翻译：本文考虑的是空间- 时间分解部分偏差方程式( SPDE, 短部分), 与微量集成的添加噪声( 普通的) 包括许多( 折叠) SPDE 和添加噪声。 首先, 提出温度溶液的存在、 独特性和时间规律性。 然后提出 Mittag- Leffler Euler 整合器, 作为一种时间步骤方法, 以数字方式解析基本模型。 开发了两个关键元素, 以克服时间偏差衍生物与分解噪声之间相互作用造成的困难。 一个是微度溶液漂移部分的新变异方式, 此处命名为集成脱色技术, 另一个是通过使用 Mittag- Leffler 函数的特性来得出一些与溶液操作员相关的精细估计值。 因此, 提议的 Mittag- Leffler Euler 内溶解器被证明符合 $( ⁇ 2 ⁇ _ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ) 和 数 ⁇ ⁇ ⁇ ⁇ 的 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 的 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 的 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 的 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ 的 ⁇ 的 ⁇ ⁇ ⁇ ⁇ 的 ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇ ⁇
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