Parabolic Anderson model with rough dependence in space

TL;DR

This paper investigates a one-dimensional parabolic Anderson model driven by Gaussian noise with rough spatial dependence, deriving chaos expansions, Feynman-Kac formulas, and asymptotic bounds for moments.

Contribution

It introduces a novel analysis of the model with fractional Brownian motion in space, providing explicit formulas and bounds for moments.

Findings

Derived Wiener chaos expansion of the solution

Established Feynman-Kac formula for moments

Obtained sharp asymptotic bounds for moments

Abstract

This paper studies the one-dimensional parabolic Anderson model driven by a Gaussian noise which is white in time and has the covariance of a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{4}, \frac{1}{2})$ in the space variable. We derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the $n$th moment of the solution.