Spatial averages for the Parabolic Anderson model driven by rough noise

TL;DR

This paper establishes a functional central limit theorem for spatial averages of the parabolic Anderson model driven by rough Gaussian noise with specific Hurst parameters, introducing a novel Feynman-Kac formula for these cases.

Contribution

It provides the first functional CLT for the model driven by rough noise with specified Hurst parameters, along with a new Feynman-Kac representation.

Findings

Proves a functional central limit theorem for spatial averages.

Develops a new Feynman-Kac formula for rough Gaussian noise.

Analyzes the parabolic Anderson model with fractional noise.

Abstract

In this paper, we study spatial averages for the parabolic Anderson model in the Skorohod sense driven by rough Gaussian noise, which is colored in space and time. We include the case of a fractional noise with Hurst parameters $H_0$ in time and $H_1$ in space, satisfying $H_0 \in (1/2,1)$, $H_1\in (0,1/2)$ and $H_0 + H_1 > 3/4$. Our main result is a functional central limit theorem for the spatial averages. As an important ingredient of our analysis, we present a Feynman-Kac formula that is new for these values of the Hurst parameters.