We study the stochastic heat equation ∂tu = 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 Lvy process. We assume that the underlying Lvy 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. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.
This is joint work with Daniel Conus (University of Utah)