Preprint: ON THE GLOBAL MAXIMUM OF THE SOLUTION TO THE STOCHASTIC HEAT EQUATION WITH COMPACT-SUPPORT INITIAL DATA

Mohammud Foondun and Davar Khoshnevisan

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

 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
© 2009 - Mohammud Foondun and Davar Khoshnevisan