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, ${\mathrm{limsup}}_{|x|\to \infty }{u}_t(x)/(\log |x|{)}^{1/6}>0$ 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.

Published in at http://dx.doi.org/10.1214/11-AOP717 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Statistics of a Passive Scalar Advected by a Large-Scale 2D Velocity Field: Analytic Solution
• On the global maximum of the solution to a stochastic heat equation with compact-support initial data

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