Quenched asymptotics for a 1-d stochastic heat equation driven by a rough spatial noise

TL;DR

This paper studies the one-dimensional stochastic heat equation with rough spatial noise, establishing existence, uniqueness, and asymptotic behavior of solutions through spectral analysis of a related random operator.

Contribution

It provides the first analysis of quenched asymptotics for the 1D stochastic heat equation driven by fractional noise with Hurst parameter less than 1/2.

Findings

Existence and uniqueness of solutions for H<1/2

Asymptotic characterization of the principal eigenvalue

Application of Feynman-Kac representation to analyze solution behavior

Abstract

In this note we consider the parabolic Anderson model in one dimension with time-independent fractional noise $\dot{W}$ in space. We consider the case $H<\frac{1}{2}$ and get existence and uniqueness of solution. In order to find the quenched asymptotics for the solution we consider its Feynman-Kac representation and explore the asymptotics of the principal eigenvalue for a random operator of the form $\frac{1}{2} \Delta + \dot{W}$.