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

Abstract

We study the stochastic heat equation ${\partial_t u = \mathcal{L}u+\sigma(u)\dot W}$ in (1 + 1) dimensions, where ${\dot W}$ is space-time white noise, σ : R → R is Lipschitz continuous, and ${\mathcal{L}}$ is the generator of a symmetric Lévy process that has finite exponential moments, and u ₀ has exponential decay at ±∞. 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 under the present setting (see, however, Gärtner et al. in Probab Theory Relat Fields 111:17–55, 1998; Gärtner et al. in Ann Probab 35:439–499, 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.