Asymptotic Behavior of Local Particles Numbers in Branching Random Walk

Abstract

Critical catalytic branching random walk on d-dimensional integer lattice is investigated for all d. The branching may occur at the origin only and the start point is arbitrary. The asymptotic behavior, as time grows to infinity, is determined for the mean local particles numbers. The same problem is solved for the probability of particles presence at a fixed lattice point. Moreover, the Yaglom type limit theorem is established for the local number of particles. Our analysis involves construction of an auxiliary Bellman-Harris branching process with six types of particles. The proofs employ the asymptotic properties of the (improper) c.d.f. of hitting times with taboo. The latter notion was recently introduced by the author for a non-branching random walk on an integer lattice.

References in corpus (2)

• Branching random walk in Z^4 with branching at the origin only
• The Hitting Times with Taboo for a Random Walk on an Integer Lattice

Cited by in corpus (5)

• Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice
• Maximum of Catalytic Branching Random Walk with Regularly Varying Tails
• Classification of catalytic branching processes and structure of the criticality set
• Finiteness of hitting times under taboo
• Subcritical catalytic branching random walk with finite or infinite variance of offspring number