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

We study a family of non-linear stochastic heat equations in (1+1) dimensions, driven by the generator of a Levy process and space-time white noise. We assume that the underlying Levy process has finite exponential moments in a neighborhood of the origin and that the initial condition has exponential decay at infinity. Then we prove that under natural conditions on the non-linearity: (i) The absolute moments of the solution to our stochastic heat equation grow 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 non-linear stochastic heat equation under the present setting. Finally, we show that these results extend to the stochastic wave equation driven by Laplacian.