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 H 0 > 1 / 2 and fractional in space with index 0 < H < 1 / 2 such that H 0 + 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.