Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise

TL;DR

This paper investigates the large-scale spatial behavior of solutions to a stochastic heat equation driven by Gaussian noise with fractional Brownian motion characteristics, revealing asymptotic properties in a rough noise setting.

Contribution

It establishes the almost sure spatial asymptotics for the solution of the stochastic heat equation with fractional Gaussian noise in one dimension.

Findings

Derived the asymptotic behavior as space becomes large.

Extended understanding of stochastic heat equations with rough Gaussian noise.

Provided rigorous probabilistic analysis for fractional Brownian motion driven noise.

Abstract

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.