On the chaotic character of the stochastic heat equation, before the onset of intermitttency

TL;DR

This paper demonstrates that solutions to a nonlinear stochastic heat equation exhibit chaotic behavior at fixed times, with their global properties heavily influenced by initial conditions, even before the development of intermittency.

Contribution

It reveals the sensitive dependence of the stochastic heat equation's solutions on initial data, highlighting chaos prior to intermittency onset.

Findings

Finite supremum for compactly supported initial data

Logarithmic growth of solutions at infinity for bounded initial data

Solution behavior is critically dependent on initial conditions

Abstract

We consider a nonlinear stochastic heat equation $\partial_tu=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space-time white noise and $\sigma:\mathbf {R}\to \mathbf {R}$ is Lipschitz continuous. We establish that, at every fixed time $t>0$, the global behavior of the solution depends in a critical manner on the structure of the initial function $u_0$: under suitable conditions on $u_0$ and $\sigma$, $\sup_{x\in \mathbf {R}}u_t(x)$ is a.s. finite when $u_0$ has compact support, whereas with probability one, $\limsup_{|x|\to\infty}u_t(x)/({\log}|x|)^{1/6}>0$ when $u_0$ is bounded uniformly away from zero. This sensitivity to the initial data of the stochastic heat equation is a way to state that the solution to the stochastic heat equation is chaotic at fixed times, well before the onset of intermittency.