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*
2*T* ® *T*,
*T* ® 0,2*T*
*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 ® *k*_{0}*T*,
*k*_{0} = 1,2;
2*T* ® *k*_{2}*T*,
*k*_{2} = 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 *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
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 *T*_{1} and *T*_{2}. Particles of
the two types appear either as the offspring of a particle of type *T*_{1} or
as a result of interaction *T*_{1} + *T*_{1}. The distribution of the final number of
particles *T*_{2} is considered when the subpopulation of particles *T*_{1} 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 *T*_{1} is large.

University of Melbourne, Melbourne, Australia

Department of Mathematics and Statistics

**Probability Seminars Prof. K.A.Borovkov**

Thursday 26th November 2009, Room 225 at 2:15pm

**Andrey Lange** (Bauman University, Moscow, Russia)

*Discrete stochastic systems with pairwise interaction*

Monash University, Melbourne, Australia

School of Mathematical Sciences. Centre for Modelling of Stochastic Systems

**Probability Seminars Prof. F.C.Klebaner**

Tuesday 1th December 2009, Room M345 at 3:00pm

**Andrey Lange** (Bauman University, Moscow, Russia)

*Discrete stochastic systems with pairwise interaction*

Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden

School of Mathematical Sciences. Division of Mathematical Statistics

**Probability Seminars Prof. S.A.Zuyev**

Thursday March 10th 2011, Room H3021 at 3:00pm

**Andrey Lange** (Bauman University, Moscow, Russia)

*Discrete stochastic systems with pairwise interaction*

Peter Jagers, Andrey Lange and Serik Sagitov

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