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

Abstract

We consider a nonlinear stochastic heat equation $\partial_{t}u=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, 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.