H\"older continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise

TL;DR

This paper establishes the existence, continuity, and optimal H"older regularity of solutions to the Parabolic Anderson Model driven by space-time Gaussian noise with general covariance, under minimal conditions.

Contribution

It proves the existence and H"older continuity of solutions with optimal exponents under minimal spectral measure conditions, improving previous results.

Findings

Solution exists and is continuous in $L^p(\,\Omega)$

Sample paths are H"older continuous with optimal exponents

Results hold under minimal spectral measure conditions

Abstract

In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(\Omega)$. Then, we show that the solution has a modification whose sample paths are H\"older continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered in the case of the white noise in time). This improves similar results which were obtained in Hu, Huang, Nualart and Tindel (2015), and Song (2017) under more restrictive conditions, and with sub-optimal exponents for H\"older continuity.