Fractional diffusion in Gaussian noisy environment

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic partial equations of the following form: $D_t^{\alpha }u(t,x)=\text{B}u+u\cdot W^H$, where $D_t^{\alpha }$ is the fractional derivative of order $\alpha$ with respect to the time variable $t$, $\text{B}$ is a second order elliptic operator with respect to the space variable $x\in {\mathbb{R}}^d$, and $W^H$ a fractional Gaussian noise of Hurst parameter $H=(H_1,\cdots ,H_d)$. We obtain conditions satisfied by $\alpha$ and $H$ so that the square integrable solution $u$ exists uniquely .