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__Preprint:__

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

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Daniel Conus, Mathew Joseph, and Davar Khoshnevisan

**Abstract.**
We consider a nonlinear stochastic heat equation
\(\partial_t u = \frac12 \partial_{xx} u + \sigma(u)\partial_{xt}W\),
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
technical 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. The
mentioned 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.
**Keywords.**
The stochastic heat equation, chaos, intermittency.

**
AMS Classification (2000)** Primary: 60H15; Secondary: 35R60.

**Support**. Research supported in part by the
Swiss National Science Foundation Fellowship PBELP2-122879 (D.C.),
the NSF grant DMS-0747758 (M.J.),
and the NSF grants DMS-0706728 and DMS-1006903 (D.K.).

**Pre/E-Prints.** This paper is available in

*Last Update: March 29, 2011*

© 2011 - Daniel Conus, Mathew Joseph, and Davar Khoshnevisan