Fractional Diffusion in Gaussian Noisy Environment

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 u(t, x) = Bu + u · Ẇ , where D 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 ∈ R and Ẇ 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.