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Short Communications

On the Extinction Probability of a Branching Process with Two Kinds of Interaction of Particles

A. V. Kalinkin

N. E. Bauman Moscow State Technical University

Abstract: A continuous-time Markov branching process with a single type $T$ of particles is considered in which any pair of particles $T+T$ produces offspring independently of all other particles. In addition, any particle of type $T$ also produces children. Representations for the extinction probabilities of this process are obtained under certain assumptions on the offspring distribution. To establish this result we apply the exponential generating function method for the solution of a stationary backward Kolmogorov system of differential equations (see [A. V. Kalinkin, Theory Probab. Appl., 27 (1982), pp. 201–205]).

Keywords: branching process with interaction of particles, extinction probabilities, Laplace equation for the exponential generating function, explicit solutions.

Received: 24.09.1998

DOI: 10.4213/tvp3926

 English version: Theory of Probability and its Applications, 2002, 46:2, 347–352

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