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

In this note, we consider the parabolic Anderson model on ${\mathbb{R}}_{+}\times \mathbb{R}$, driven by a Gaussian noise which is fractional in time with index $H_0>1\mathrm{/}2$ and fractional in space with index $0<H<1\mathrm{/}2$ such that $H_0+H>3\mathrm{/}4$. 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 $p\ge 2$.