We investigate a semilinear stochastic time-space fractional subdiffusion equation driven by fractionally integrated multiplicative noise. The equation involves the \(\psi \)-Caputo derivative of order \(\alpha \in (0,1)\) and the spectral fractional Laplacian of order \(\beta \in (\frac{1}{2},1]\). The existence and uniqueness of the mild solution are proved in a suitable Banach space by using the Banach contraction mapping principle. The spatial and temporal regularities of the mild solution are established in terms of the smoothing properties of the solution operators.

中文翻译：

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具有集成乘性噪声的半线性随机次扩散的存在性、唯一性和规律性

我们研究了由分数积分乘性噪声驱动的半线性随机时空分数次扩散方程。该方程涉及阶数为\(\alpha \in (0,1)\)的\(\psi \) -Caputo 导数和阶数为\(\beta \in (\frac{1}{2})的谱分数拉普拉斯算子,1]\) .利用Banach收缩映射原理在适当的Banach空间中证明了温和解的存在性和唯一性，并根据解算子的平滑性质建立了温和解的空间和时间规律性。 。