Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice

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.

2 figures

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Cited by in corpus (7)

• Maximum of Catalytic Branching Random Walk with Regularly Varying Tails
• Branching Random Walks with Two Types of Particles on Multidimensional Lattices
• Fluctuations of propagation front in catalytic branching walk
• Catalytic Branching Random Walk with Semi-exponential Increments
• Maximal displacement and population growth for branching Brownian motions
• Random walk through a fertile site
• Fluctuations of the rightmost particle in the catalytic branching Brownian motion