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

 Daniel Conus, Mathew Joseph, & Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. conus@math.utah.edu, joseph@math.utah.edu & davar@math.utah.edu

Last Update: March 29, 2011
© 2011 - Daniel Conus, Mathew Joseph, and Davar Khoshnevisan