Translator Disclaimer
May 2013 On the chaotic character of the stochastic heat equation, before the onset of intermitttency
Daniel Conus, Mathew Joseph, Davar Khoshnevisan
Ann. Probab. 41(3B): 2225-2260 (May 2013). DOI: 10.1214/11-AOP717

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.

Citation

Download Citation

Daniel Conus. Mathew Joseph. Davar Khoshnevisan. "On the chaotic character of the stochastic heat equation, before the onset of intermitttency." Ann. Probab. 41 (3B) 2225 - 2260, May 2013. https://doi.org/10.1214/11-AOP717

Information

Published: May 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1286.60060
MathSciNet: MR3098071
Digital Object Identifier: 10.1214/11-AOP717

Subjects:
Primary: 60H15
Secondary: 35R60

Rights: Copyright © 2013 Institute of Mathematical Statistics

JOURNAL ARTICLE
36 PAGES


SHARE
Vol.41 • No. 3B • May 2013
Back to Top