Leonid V. Bogachev

Asymptotics of branching symmetric random walk on the lattice with a single source

Abstract We study the long-time asymptotics of continuous-time branching random walk on ℤd (d ≥ 1) with a single source (i.e., branching site). The random walk is assumed homogeneous, symmetric, irreducible, and having zero mean and finite variance of jumps. We find the limiting extinction probability and the asymptotics of all integer moments for the total population size and for the number of particles at a fixed site.

Branching Random Walk in a Catalytic Medium. I. Basic Equations

We consider a continuous-time branching random walk on the integer lattice ℤd (d ≥ 1 ) with a finite number of branching sources, or catalysts. The random walk is assumed to be spatially homogeneous and irreducible. The branching mechanism at each catalyst, being independent of the random walk, is governed by a Markov branching process. The quantities of interest are the local numbers of particles (at each site) and the total population size. In the present paper, we derive and analyze the Kolmogorov type backward equations for the corresponding Laplace generating functions and also for the successive integer moments and the process extinction probability. In particular, existence and uniqueness theorems are proved and the problem of explosion is studied in some detail. We then rewrite these equations in the form of integral equations of renewal type, which may serve as a convenient tool for the study of the process long-time behavior. The paper also provides a technical foundation to some results published before without detailed proofs.