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 ∂_tu=κ&part²_xxu+ σ(u)w for a space-time white noise w and a constant κ>0. Under some suitable conditions on the initial function u₀ and σ, we show that the quantity

is bounded away from zero and infinity by explicit multiples of 1/κ. Our proof works by demonstrating quantitatively that the peaks of the stochastic process x → u_t(x) are highly concentrated for infinitely-many large values of t. In the special case of the parabolic Anderson model--where σ(u)=λu forsome λ>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.

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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

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

Mohammud Foondun and Davar Khoshnevisan

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