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(α)tu(t,x)=Bu+u⋅W˙H, where D(α)t is the Caputo fractional derivative of order α∈(0,1) with respect to the time variable t, B is a second order elliptic operator with respect to the space variable x∈Rd and W˙H a time homogeneous fractional Gaussian noise of Hurst parameter H=(H1,⋯,Hd). We obtain conditions satisfied by α and H, so that the square integrable solution u exists uniquely.