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

TL;DR

This paper studies the parabolic Anderson model driven by rough Gaussian noise in both time and space, establishing existence, uniqueness, and moment bounds for solutions under specific fractional noise conditions.

Contribution

It proves existence and uniqueness of solutions for the model with rough fractional noise and initial data, extending previous results to more irregular noise settings.

Findings

Existence and uniqueness of mild solutions under fractional noise conditions.

Exponential upper bounds for moments of solutions.

Extension of the model analysis to rough initial conditions.

Abstract

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 $H_0>1/2$ 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 $p\ge 2$.