#
__Preprint:__

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
∂_{t}u = £u+σ(u)w
in (1+1) dimensions, where w is space-time white noise,
σ:**R**→**R** 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 u_{0} 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

*Last Update: October 1, 2010*

© 2010 - Daniel Conus and Davar Khoshnevisan