Report on MASCOS Workshop on Markov Chains

The University of Queensland
Monday 4th April 2005

General   Markov chains are the simplest mathematical models for random phenomena that evolve over time. Their structure is sufficiently simple that one can say a great deal about their behaviour, yet, at the same time, the class of Markov chains is rich enough to serve in many applications. Indeed, Markov chains are the first and most important examples of stochastic models which arise in areas as diverse as population biology, chemical kinetics and telecommunications.

This workshop, sponsored by the ARC Centre of Excellence for Mathematics and Statistics of Complex Systems (MASCOS), brought together practitioners and mathematicians to examine recent developments in the theory and applications Markov chains.

Invited speakers

• Anyue Chen (University of Greenwich)
• Tony Pakes (University of Western Australia)
• Phil Pollett (MASCOS, University of Queensland)
• David Sirl (MASCOS, University of Queensland)
• Hanjun Zhang (MASCOS, University of Queensland)

[There were no contributed papers]

Venue   Seminar Rooms 1 and 2, Emmanuel College, St Lucia Campus, University of Queensland.

Organizers   David Sirl and Phil Pollett (MASCOS, University of Queensland)

Programme  

	09:15	Registration
	09:30	David Sirl	Markov chains: an introduction/review
	10:30	Break	[Refeshments provided]
	11:00	Hanjun Zhang	Quasi-stationary distributions and the decay parameter
	12:00	Tony Pakes	Long-range Markovian dependence
	13:00	Lunch Break	[Lunch provided]
	14:00	Anyue Chen	Uniqueness, extinction, and explosion properties of the Weighted Collision Branching Process
	15:00	Break	[Refeshments provided]
	15:30	Phil Pollett	Which Markov chains have a given invariant measure?
	16:30	Close

Abstracts

• Anyue Chen

  Uniqueness, extinction and explosion properties of the Weighted Collision Branching Process

  Abstract: The basic questions of uniqueness, extinction and explosivity of one kind of interacting branching system, the Weighted Collision Branching Process (WCBP), are addressed in this talk. It is proved that in the super-explosive case, the process is honest if and only if the mean death rate is greater than or equal to the mean birth rate, while, in the sub-explosive case, it is always honest. Explicit expressions for the extinction probability, and the mean and the conditional mean extinction times are presented. Explosivity of the WCBP is investigated and an explicit expression for mean explosion time is established. We shall see that the super-explosive and the sub-explosive WCBPs exhibit substantially different behaviour. [This is joint work with Junping Li, Phil Pollett and Hanjun Zhang.]

• Tony Pakes

  Long-range Markovian dependence

  Abstract: The title is a tease in that it alludes to geographical long-range influences on two topics I will address. The first of these relates to the fact, shown by Darlington and Pollett, that if a Markov process has an absorbing state which it hits with probability less than one, then any limiting conditional law is not quasi-stationary. Nevertheless, there is some nice theory for this situation. I will discuss it, together with an illustration. The second topic is about a uniqueness question for nonlinear (i.e. weighted) Markov branching processes which arose first from a remark made electronically to me by Anyue Chen.

• Phil Pollett

  Which Markov chains have a given invariant measure?

  Abstract: I will consider the following problem: given a stable, conservative q-matrix Q of transition rates over a denumerable state-space S, together with a subinvariant measure m for Q, determine all Q-processes for which m is an invariant measure. I will review recent work on this problem, giving particular attention to the case when Q is a single-exit q-matrix. I will also examine the case when S consists of a single absorbing state 0 and an irreducible class C, and consider the problem of constructing Q-processes for which a given measure m is m-invariant on C. [The latter is joint work with Hanjun Zhang.]

  [talk]

• David Sirl

  Markov chains: an introduction/review

  Abstract: I will give a brief account of the theory of Markov chains, from discrete time, finite state-space to continuous time, countable state processes. I will recall various quantities of interest, including hitting probabilities, expected hitting times, and stationary and quasi-stationary distributions, and illustrate these concepts with numerous examples.

  [talk]

• Hanjun Zhang

  Quasi-stationary distributions and the decay parameter

  Abstract: The existence of quasi-stationary distributions (qsds) is addressed in this talk. I will review some work on this problem and present some new results. The decay parameter, which is closely related to qsds, will be discussed. I will give a sufficient condition for the decay parameter to be positive for a general Markov chain.

  [talk]