This paper is devoted to study of time-fractional elliptic equations driven by a multiplicative noise. By combining the eigenfunction expansion method for symmetry elliptic operators, the variation of constant formula for strong solutions to scalar stochastic fractional differential equations, Ito's formula and establishing a new weighted norm associated with a Lyapunov–Perron operator defined from this representation of solutions, we show the asymptotic behaviour of solutions to these systems in the mean square sense. As a consequence, we also prove existence, uniqueness and the convergence rate of their solutions.

中文翻译：

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乘性白噪声驱动的时分椭圆型方程解的渐近行为

本文致力于研究由乘性噪声驱动的时间分数阶椭圆方程。通过将对称椭圆算子的本征函数展开法，标量随机分数阶微分方程强解的常数公式的变型，伊藤公式结合起来，并建立了与由这种解表示式定义的Lyapunov–Perron算子相关的新加权范数，我们证明了这些系统在均方意义上的解的渐近行为。结果，我们还证明了其解的存在性，唯一性和收敛速度。