Stochastic fractional diffusion equations with Gaussian noise rough in space

Abstract

In this article, we consider the following stochastic fractional diffusion equation \begin{equation*} \left(\partial^{\beta}+\dfrac{\nu}{2}\left(-\Delta\right)^{\alpha / 2}\right) u(t, x)= \lambda\: I_{0_+}^{\gamma}\left[u(t, x) \dot{W}(t, x)\right] ,\quad t>0,\: x \in \mathbb{R}, \end{equation*} where $\alpha>0$, $\beta\in(0,2]$, $\gamma \ge 0$, $\lambda\neq0$, $\nu>0$, and $\dot{W}$ is a Gaussian noise which is white or fractional in time and rough in space. We prove the existence and uniqueness of the solution in the It\^o-Skorohod sense and obtain the lower and upper bounds for the $p$-th moment. The H\"older regularity of the solution is also studied.