Fractional Diffusion in Gaussian Noisy Environment

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: (D_t^{(alpha)} u(t, x)=textit{B}u+ucdot dot W^H), where (D_t^{(alpha)}) is the Caputo fractional derivative of order (alphain (0,1)) with respect to the time variable (t), (textit{B}) is a second order elliptic operator with respect to the space variable (xinmathbb{R}^d) and (dot W^H) a time homogeneous 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