On the existence and position of the farthest peaks of a family of stochastic heat and wave equations

Daniel Conus and Davar Khoshnevisan

Abstract. We study the stochastic heat equation ∂tu = £u+σ(u)w in (1+1) dimensions, where w is space-time white noise, σ:RR is Lipschitz continuous, and £ is the generator of a symmetric Lévy process. We assume that the underlying Lévy process has finite exponential moments in a neighborhood of the origin and u0 has exponential decay at ±∞. Then we prove that under natural conditions on σ: (i) The νth absolute moment of the solution to our stochastic heat equation grows exponentially with time; and (ii) The distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. Very little else seems to be known about the location of the high peaks of the solution to the stochastic heat equation. [See, however, Gärtner and Molchanov (1998) and Gärtner et al (2007) for the analysis of the location of the peaks in a different model.] Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.

Keywords. Stochastic PDEs, stochastic heat equation, intermittence

AMS Classification (2000) Primary: 60H15; Secondary: 35R60.

Support. Research supported by grants from the Swiss National Science Foundation Fellowship PBELP2-122879 (D.C.) and the US National Science Foundation grant DMS-0706728 (D.K.)

Pre/E-Prints. This paper is available in

Daniel Conus & Davar Khoshnevisan
Department of Mathematics
University of Utah
155 S, 1400 E JWB 233
Salt Lake City, UT 84112-0090, U.S.A. &

Last Update: October 1, 2010
© 2010 - Daniel Conus and Davar Khoshnevisan