Abstract. Consider a stochastic heat equation \(\partial_t u = \kappa \partial^2_{xx}u+\sigma(u)\dot{W}\) for a space-time white noise \(\dot{W}\) and a constant \(\kappa>0\). Under some suitable conditions on the initial function \(u_0\) and \(\sigma\), we show that \[ \limsup_{t\to\infty}\ t^{-1}\ln {\rm E}\left(\sup_{x\in\mathbf{R}} |u_t(x)|^2\right)= \limsup_{t\to\infty}\ t^{-1}\sup_{x\in\mathbf{R}}\ln{\rm E}\left( |u_t(x)|^2\right), \] and this quantity is bounded away from zero and infinity by explicit multiples of \(1/\kappa\). Our proof works by demonstrating quantitatively that the peaks of the stochastic process \(x \to u_t(x)\) are highly concentrated for infinitely-many large values of \(t\). In the special case of the parabolic Anderson model--where \(\sigma(u)=\lambda u\) for some \(\lambda>0\)--this "peaking" is a way to make precise the notion of physical intermittency.
Keywords. The stochastic heat equation.
AMS Classification (2000). Primary. 35R60, 37H10, 60H15; Secondary. 82B44.
Support. The research of DK was supported in part by a grant from the U.S. National Science Foundation.
Pre/E-Prints. This paper is available in
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Mohammud Foondun Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. mohammud@math.utah.edu |
Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |
Last Update: Januray 25, 2009
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2009 - Mohammud Foondun and Davar Khoshnevisan