Stochastic fractional diffusion equations with Gaussian noise rough in space

Abstract

In this article, we consider the stochastic fractional diffusion equation

$$\left({\partial }^{\mathit{\beta }}+\frac{\mathrm{\nu }}{2}{\left(-\mathrm{\Delta }\right)}^{\mathit{\alpha }∕2}\right)u(t,x)=\mathit{\lambda }{I}_{0_{+}}^{\mathit{\gamma }}\left[u(t,x)\dot{W}(t,x)\right],t>0,x\in \mathbb{R},$$

where $\mathit{\alpha }>0$, $\mathit{\beta }\in (0,2]$, $\mathit{\gamma }\ge 0$, $\mathit{\lambda }\ne 0$, $\mathrm{\nu }>0$, 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 Itô-Skorohod sense and obtain the lower and upper bounds for the p-th moment. The Hölder regularity of the solution is also studied.