Parabolic Anderson model with rough noise in space and rough initial conditions

Abstract

In this note, we consider the parabolic Anderson model on R+×R, driven by a Gaussian noise which is fractional in time with index H0>1∕2 and fractional in space with index 0<H<1∕2 such that H0+H>3∕4. Under a general condition on the initial data, we prove the existence and uniqueness of the mild solution and obtain its exponential upper bounds in time for all p-th moments with p≥2.