Parabolic Anderson model on critical Galton-Watson trees in a Pareto environment

Abstract

The parabolic Anderson model is the heat equation with some extra spatial randomness. In this paper we consider the parabolic Anderson model with i.i.d. Pareto potential on a critical Galton-Watson tree conditioned to survive. We prove that the solution at time t is concentrated at a single site with high probability and at two sites almost surely as t → ∞. Moreover, we identify asymptotics for the localisation sites and the total mass, and show that the solution u(t, v) at a vertex v can be well-approximated by a certain functional of v. The main difference with earlier results on Z d is that we have to incorporate the effect of variable vertex degrees within the tree, and make the role of the degrees precise.