Discrete Stochastic Systems with Pairwise Interation
by Andrey Lange
Abstract: A model of a system of interacting particles of types T_1, ... , T_n is considered as a continuous-time Markov process on a countable state space. Forward and backward Kolmogorov systems of differential equations are represented in a form of partial differential equations for the generating â€¨functions of transition probabilities. We study the limiting behavior of probability distributions as time tends to infinity for two models of thatâ€¨ type.
First model deals with an open system with pairwise interaction. New particles T immigrate either one or two particles at a time, and the interaction T+T leads to the death of either one or both of the interacting particles. The distribution of the number of particles is studied as the time tends to infinity. The exact solutions of the stationary Kolmogorov equations were found in terms of Bessel and hypergeometric functions. The asymptotics for the expectation and variance as well as the asymptotic normality of the stationary distribution were obtained when the intensity of new particles arrival is high.
The second model describes a system with particles T1 and T2. Particles of the two types appear either as the offspring of a particle of type T1 or as a result of interaction T1+T1. The distribution of the final number of particles T2 is considered when the subpopulation of particles T1 becomes extinct. Under certain restrictions on the distribution of the number of appearing particles, the asymptotics for the expectation and variance as well as the asymptotic normality of the final distribution are obtained when the initial number of particles T1 is large.
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