Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice

TL;DR

This paper investigates the asymptotic spread of particles in a supercritical catalytic branching random walk on multidimensional lattices, establishing the shape and density of the propagation front as time approaches infinity.

Contribution

It extends previous one-dimensional results to multidimensional settings, characterizing the propagation front and its density in supercritical catalytic branching random walks.

Findings

Particles are confined within a convex surface called the propagation front.

Under certain conditions, particles are found on the propagation front asymptotically.

The propagation front is densely populated in the limit.

Abstract

For a supercritical catalytic branching random walk on Z^d (d is positive integer) with an arbitrary finite catalysts set we study the spread of particles population as time grows to infinity. Namely, we divide by t the position coordinates of each particle existing at time t and then let t tend to infinity. It is shown that in the limit there are a.s. no particles outside the closed convex surface in R^d which we call the propagation front and, under condition of infinite number of visits of the catalysts set, a.s. there exist particles on the propagation front. We also demonstrate that the propagation front is asymptotically densely populated and derive its alternative representation. Recent strong limit theorems for total and local particles numbers established by the author play an essential role. The results obtained develop ones by Ph.Carmona and Y.Hu (2014) devoted to the spread of catalytic branching random walk on Z. Keywords and phrases: branching random walk, supercritical regime, spread of population, propagation front, many-to-one lemma.