Moscow Mathematical Journal

Volume 19, Issue 1, January–March 2019 pp. 7–36.

Lattice Birth-and-Death Processes

Authors: Viktor Bezborodov (1), Yuri Kondratiev (1), and Oleksandr Kutoviy (1)
Author institution:(1) Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615 Bielefeld, Germany

Summary:

Lattice birth-and-death Markov dynamics of particle systems with spins from ℤ₊ are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker–Pacala–Dieckmann–Law model and an aggregation model.

2010 Math. Subj. Class. 60K35, 82C22.

Keywords: Birth-death process, interacting particle systems, stochastic equation with Poisson noise, martingale problem, invariant measure, Bolker–Pacala model.

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