The Annals of Probability

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

Daniel Conus, Mathew Joseph, and Davar Khoshnevisan

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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.

Article information

Source
Ann. Probab., Volume 41, Number 3B (2013), 2225-2260.

Dates
First available in Project Euclid: 15 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1368623524

Digital Object Identifier
doi:10.1214/11-AOP717

Mathematical Reviews number (MathSciNet)
MR3098071

Zentralblatt MATH identifier
1286.60060

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

Keywords
Stochastic heat equation chaos intermittency

Citation

Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar. On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41 (2013), no. 3B, 2225--2260. doi:10.1214/11-AOP717. https://projecteuclid.org/euclid.aop/1368623524


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