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

**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

- More recent version (Pdf Format) (Version January 31, 2009)
- Pdf Format (Version January 25, 2009)

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