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