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Article Dans Une Revue Stochastic Processes and their Applications Année : 2023

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

Résumé

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.
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hal-03999507 , version 1 (21-02-2023)

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Eleanor Archer, Anne Pein. Parabolic Anderson model on critical Galton–Watson trees in a Pareto environment. Stochastic Processes and their Applications, 2023, 159, pp.34-100. ⟨10.1016/j.spa.2023.01.010⟩. ⟨hal-03999507⟩

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