On the global maximum of the solution to a stochastic heat equation with compact-support initial data

TL;DR

This paper investigates the long-term behavior of solutions to a stochastic heat equation with initial data of compact support, revealing that the peaks of the solution become highly concentrated over time, with bounds related to the diffusivity parameter.

Contribution

It provides explicit bounds on the growth rate of the supremum of the solution's second moment, demonstrating peak concentration and intermittency in the stochastic heat equation.

Findings

The growth rate of the supremum's logarithm is bounded away from zero and infinity by multiples of 1/κ.

Peaks of the solution are highly concentrated at large times.

In the parabolic Anderson model, this peaking describes physical intermittency.

Abstract

Consider a stochastic heat equation $\partial_t u = \kappa \partial^2_{xx}u+\sigma(u)\dot{w}$ for a space-time white noise $\dot{w}$ and a constant $\kappa>0$. Under some suitable conditions on the the initial function $u_0$ and $\sigma$, we show that the quantity \limsup_{t\to\infty}t^{-1}\ln\E(\sup_{x\in\R} |u_t(x)|^2) is bounded away from zero and infinity by explicit multiples of $1/\kappa$. Our proof works by demonstrating quantitatively that the peaks of the stochastic process $x\mapsto u_t(x)$ are highly concentrated for infinitely-many large values of $t$. In the special case of the parabolic Anderson model--where $\sigma(u)= \lambda u$ for some $\lambda>0$--this "peaking" is a way to make precise the notion of physical intermittency.