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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
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09:15 |
Registration |
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09:30 |
David Sirl |
Markov chains: an introduction/review |
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10:30 |
Break |
[Refeshments provided] |
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11:00 |
Hanjun Zhang |
Quasi-stationary distributions and the decay parameter |
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12:00 |
Tony Pakes |
Long-range Markovian dependence |
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13:00 |
Lunch Break |
[Lunch provided] |
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14:00 |
Anyue Chen |
Uniqueness, extinction, and explosion properties of the Weighted
Collision
Branching Process
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15:00 |
Break |
[Refeshments provided] |
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15:30 |
Phil Pollett |
Which Markov chains have a given invariant measure?
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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]
Participants
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Name |
Email |
Affiliation |
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Moshin Ali |
moshin at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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John Paul Barjaktarevic |
jpb at physics.uq.edu.au |
Department of Physics, University of Queensland |
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Kevin Burrage |
kb at maths.uq.edu.au |
Advanced Computational Modelling Centre, University of
Queensland |
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Ben Cairns |
bjc at maths.uq.edu.au |
MASCOS, University of Queensland |
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Anyue Chen |
A.Chen at gre.ac.uk |
Mathematical Sciences Department, University of
Greenwich |
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Zhao Yang Dong |
zdong at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Selina Fothergill |
selinafo at psy.uq.edu.au |
Key Centre for Applied Cognitive Psychology and Human
Factors, University of Queensland |
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Marcus Gallagher |
marcusg at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Ben Gladwin |
gladwin at maths.uq.edu.au |
MASCOS, University of Queensland |
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Andrei Hryshko |
dushenka at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Naveen Kumar |
naveen at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Dharma Lesmono |
dlesmono at maths.uq.edu.au |
MASCOS, University of Queensland |
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Ariel Liebman |
aliebman at itee.uq.edu.au |
ARC Centre for Complex Systems, University of Queensland |
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Valerie Lim |
lim at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Miao (Jennie) Lu |
lumiao at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Stefan Maetschke |
stefan at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Marissa McBride |
s4033977 at student.uq.edu.au |
Department of Mathematics, University of Queensland |
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Rizah Memisevic |
r.memisevic at uq.edu.au |
Key Centre for Applied Cognitive Psychology and Human
Factors, University of Queensland |
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Sho Nariai |
sho at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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Michael Nielsen |
nielsen at physics.uq.edu.au |
Department of Physics, University of Queensland |
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Tony Pakes |
pakes at maths.uwa.edu.au |
School of Mathematics and Statistics, University of
Western Australia |
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Phil Pollett |
pkp at maths.uq.edu.au |
MASCOS, University of Queensland |
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Tony Roberts |
apr at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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Joshua Ross |
jvr at maths.uq.edu.au |
MASCOS, University of Queensland |
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Asrul Sani |
asani at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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Mark Seeto |
mbs at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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David Sirl |
dsirl at maths.uq.edu.au |
MASCOS, University of Queensland |
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Antony Stace |
aws at maths.uq.edu.au |
MASCOS, University of Queensland |
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Thomas Taimre |
ttaimre at maths.uq.edu.au |
MASCOS, University of Queensland |
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Bill Whiten |
W.Whiten at uq.edu.au |
Julius Kruttschnitt Mineral Research Centre, University
of Queensland |
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Andy Wilkins |
awilkins at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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Burton Wu |
burton.wu at team.telstra.com |
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George (Zhao) Xu |
xuzhao at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Wen-Li Yang |
wenli at maths.uq.edu.au |
Department of Mathematics, University of Queensland |
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Hanjun Zhang |
hjz at maths.uq.edu.au |
MASCOS, University of Queensland |
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Lu Zhe |
luzhe at itee.uq.edu.au |
School of Information Technology and Electrical
Engineering, University of Queensland |
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Justin Zhu |
j.zhu at imb.uq.edu.au |
Institute for Molecular Bioscience, University of
Queensland |
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