Probability Seminars

Bauman Moscow State Technical University

Department of Higher Mathematics

Probability Seminars

1. Friday 19th May 2006, Room 911L at 12:00

David Sirl (University of Queensland, Australia)

Absorbing Markov chains governed by 2T ® T, T ® 0,2T and similar schemes: quasistationary distributions and the decay parameter

Abstract:

Quasi stationarity is a notion used to describe the behaviour of processes that eventually die out, but display stationary-like behaviour over any reasonable time-scale. For example, a threatened species may survive for extended periods before becoming extinct; a telecommunications network may fluctuate between congested and uncongested states without any apparent change in demand, and stay in each state for long periods; and a chemical system where one species can become depleted (and thus stop the reaction) may settle to a stable equilibrium.

I will summarise known results in this area and look at some of the many avenues available for further research. I will illustrate these results with reference to a particular class of auto-catalytic chemical reactions.

Central to the theory of quasistationary distributions is a quantity known as the decay parameter, which describes the rate of exponential decay of the transition probabilities of an absorbing Markov chain. Despite its importance, the decay parameter is notoriously difficult to evaluate or even approximate.

I will outline a non-standard characterisation of the decay parameter and indicate how this leads to explicit bounds for the decay parameter of a general birth-death process. An immediate corollary is a necessary and sufficient condition for positivity of the decay parameter; which I will illustrate with several examples. This is joint work with Hanjun Zhang and Phil Pollett.

2. Friday 19th May 2006, Room 911L at 13:00

Andrey Lange (Bauman University, Russia)

Asymptotic behavior of stationary probabilities for Markov process governed by 0 ® k₀T, k₀ = 1,2; 2T ® k₂T, k₂ = 0,1. Limit theorems

University of Queensland, Brisbane, Australia

School of Mathematics and Physics. Centre of Excellence for Mathematics and Statistics of Complex Systems

Probability Seminars Prof. P.Pollett

Monday 16th November 2009, Room 67-442 at 3:00pm

Andrey Lange (Bauman University, Moscow, Russia)

Discrete stochastic systems with pairwise interaction

Abstract:

A model of a system of interacting particles of types T₁, …, 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 functions of transition probabilities. We study the limiting behavior of probability distributions as time tends to infinity for two models of that 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.