On the intermittency front of stochastic heat equation driven by colored noises

TL;DR

This paper investigates the propagation of high peaks in the solution to a stochastic heat equation driven by Gaussian noise with general homogeneous covariance, providing bounds on the intermittency front and analyzing specific cases like Riesz kernels.

Contribution

It introduces new estimates for the propagation speed of intermittency fronts in stochastic heat equations with general Gaussian noise, including precise bounds for Riesz kernel cases.

Findings

Derived upper and lower bounds for propagation speed

Provided more precise bounds for Riesz kernel covariance

Analyzed the solution's intermittency front in detail

Abstract

We study the propagation of high peaks (intermittency front) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R}^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.