Space-time fractional Anderson model driven by Gaussian noise rough in space

Abstract

In this paper, we study a class of space-time fractional Anderson model driven by multiplicative Gaussian noise which is white/colored in time and has the covariance of a fractional Brownian motion with Hurst index [Formula: see text] in space. We prove the existence of the solution in the Skorohod sense and obtain the upper and lower bounds for the [Formula: see text]th moments for all [Formula: see text]. Then we can prove that solution of this equation in the Skorohod sense is weakly intermittent. We also deduce the Hölder continuity of the solution with respect to the time and space variables.