Parabolic Anderson model with colored noise on torus

TL;DR

This paper develops a Gaussian noise framework on the torus to analyze the parabolic Anderson model, providing bounds and regularity results, and exploring the influence of geometry on stochastic PDEs.

Contribution

It introduces an intrinsic colored noise on the torus and studies the PAM with measure-valued initial conditions, advancing understanding of stochastic PDEs on compact manifolds.

Findings

Established sharp bounds for moments of the solution

Proved Hölder continuity in space and time

Developed a foundational framework for stochastic PDEs on the torus

Abstract

We construct an intrinsic family of Gaussian noises on $d$-dimensional flat torus $\mathbb{T}^d$. It is the analogue of the colored noise on $\mathbb{R}^d$, and allows us to study stochastic PDEs on torus in the It\^{o} sense in high dimensions. With this noise, we consider the parabolic Anderson model (PAM) with measure-valued initial conditions and establish some basic properties of the solution, including a sharp upper and lower bound for the moments and H\"{o}lder continuity in space and time. The study of the toy model of $\mathbb{T}^d$ in the present paper is a first step towards our effort in understanding how geometry and topology play an role in the behavior of stochastic PDEs on general (compact) manifolds.