Phil Pollett's Research Pages

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Diffusion approximations and quasi-stationary distributions: a short bibliography

This comprises:

- Key papers by Tom Kurtz on diffusion approximations
- My own work on diffusion approximations
- Some recent work from my group on diffusion approximations
- My own work on quasi-stationary distributions

1. Key papers by Tom Kurtz on diffusion approximations

Kurtz, T.G. (1970)
Solutions of ordinary differential equations as limits of pure jump
Markov processes.
*Journal of Applied Probability* 7, 49-58.

Kurtz, T.G. (1971)
Limit theorems for sequences of jump Markov processes approximating
ordinary differential processes.
*Journal of Applied Probability* 8, 344-356.

Kurtz, T.G. (1973)
The relationship between stochastic and deterministic models in
chemical reactions.
J. Chem. Phys.
*Journal of Chemical Physics* 57, 2976-2978.

Kurtz, T.G. (1976)
Limit theorems and diffusion approximations for density dependent
Markov chains.
*Math. Prog. Study* 5, 67-78.

Kurtz, T.G. (1978)
Strong approximation theorems for density dependent Markov chains.
*Stochastic Process and their Applications* 6, 223-240.

2. My own work on diffusion approximations

Pollett, P.K. (1990) On a model for interference between searching
insect parasites. *Journal of the Australian Mathematical Society,
Series B* 31, 133-150.

Pollett, P.K. (1991) Modelling random fluctuations in a bistable
telecommunications network. In (Ed. P. Hutton) *Proceedings of the
11th National Conference of the Australian Society for Operations
Research*, pp. 11-22.

Pollett, P.K. (1992) Diffusion approximations for a circuit switching
network with random alternative routing. *Australian
Telecommunication Research* 25, 45-52.

Pollett, P.K. (1992) Modelling random fluctuations in a bistable
telecommunications network. In (Ed. W. Henderson) *Proceedings of the
7th Australian Teletraffic Research Seminar*, Teletraffic Research
Centre, University of Adelaide, Adelaide, pp. 335-345.

Pollett, P.K. and Vassallo, A. (1992) Diffusion approximations for
some simple chemical reaction schemes. *Advances in Applied
Probability* 24, 875-893.

Pollett, P.K. (2001) Diffusion approximations
for ecological models. In (Ed. Fred Ghassemi) *Proceedings of the
International Congress on Modelling and Simulation*, Vol.2,
Modelling and Simulation Society of Australia and New Zealand, Canberra,
Australia, pp. 843-848.

3. Some recent work from my group on diffusion approximations

Ross, J.V. (2005) Stochastic models for mainland-island metapopulations
in static and dynamic landscapes.
*Bulletin of Mathematical Biology* 68, 417-449.

Ross, J.V. (2006) A stochastic metapopulation model accounting for
habitat dynamics. *Journal of Mathematical Biology* 52, 788-806.

Ross, J.V., Taimre, T. and Pollett, P.K. (2006) On parameter estimation in
population models. *Theoretical Population Biology* (to appear).

4. My own work on quasi-stationary distributions

Pollett, P.K. (1986) On the equivalence of m-invariant measures for the minimal process and its
q-matrix. *Stochastic Processes and their Applications* 22,
203-221.

Parsons, R.W. and Pollett, P.K. (1987) Quasistationary distributions
for some autocatalytic reactions. *Journal of Statistical
Physics* 46, 249-254.

Pollett, P.K. (1987) On the long-term behaviour of a population that
is subject to large-scale mortality or emigration. In (Ed. S. Kumar)
*Proceedings of the 8th National Conference of the Australian Society
for Operations Research*, pp. 196-207.

Pollett, P.K. (1988) Reversibility, invariance and m-invariance. *Advances in Applied
Probability* 20, 600-621.

Pollett, P.K. (1988) On the problem of evaluating quasistationary
distributions for open reaction schemes. *Journal of Statistical
Physics* 53, 1207-1215.

Pakes, A.G. and Pollett, P.K. (1989) The supercritical birth, death
and catastrophe process: limit theorems on the set of extinction.
*Stochastic Processes and their Applications* 32, 161-170.

Pollett, P.K. (1989) The generalized Kolmogorov criterion.
*Stochastic Processes and their Applications* 33, 29-44.

Pollett, P.K. and Roberts, A.J. (1990) A description of the long-term
behaviour of absorbing continuous-time Markov chains using a centre
manifold. *Advances in Applied Probability* 22, 111-128.

Pollett, P.K. and Vere-Jones, D. (1992) A note on evanescent
processes. *The Australian Journal of Statistics* 34,
531-536.

Nair, M.G. and Pollett, P.K. (1993) On the relationship between m-invariant measures and quasistationary
distributions for continuous-time Markov chains. *Advances in Applied
Probability* 25, 82-102.

Pollett, P.K. (1993) Modelling the long-term behaviour of evanescent
ecological systems. In (Ed. M. McAleer) *Proceedings of the
International Congress on Modelling and Simulation*, Modelling and
Simulation Society of Australia, Perth, Vol. 1, pp. 157-162.

Pollett, P.K. (1993) Recent advances in the theory and application of
quasistationary distributions. In (Eds S. Osaki and D.N.P. Murthy)
*Proceedings of the 1st Australia-Japan Workshop on Stochastic Models
in Engineering, Technology and Management*, World Scientific,
Singapore, pp. 477-486.

Pollett, P.K. (1993) Analytical and computational methods for
modelling the long-term behaviour of evanescent random processes. In
(Eds D.J. Sutton, C.E.M. Pearce and E.A. Cousins) *Decision Sciences:
Tools for Today, Proceedings of the 12th National Conference of the
Australian Society for Operations Research*, Australian Society for
Operations Research, Adelaide, pp. 514-535.

Pollett, P.K. and Stewart, D.E. (1994) An efficient procedure for
computing quasistationary distributions of Markov chains with sparse
transition structure. *Advances in Applied Probability* 26,
68-79.

Pollett, P.K. (1995) The determination of quasistationary
distributions directly from the transition rates of an absorbing Markov
chain. *Mathematical and Computer Modelling ***22**,
279-287.

Bean, N.G., Pollett, P.K. and Taylor, P.G. (1996) The quasistationary
distributions of homogeneous quasi-birth-and-death processes. In (Eds
Richard J. Wilson, D.N. Pra Murthy and Shunji Osaki) *Proceedings of
the 2nd Australia-Japan Workshop on Stochastic Models in Engineering,
Technology and Management*, Technology Management Centre, The
University of Queensland, pp. 44-55.

Elmes, S., Pollett, P.K. and Walker, D. (1996) On the relationship
between m-invariant measures and
quasistationary distributions for absorbing continuous-time Markov
chains when absorption is not certain. In (Eds Richard J. Wilson, D.N.
Pra Murthy and Shunji Osaki) *Proceedings of the 2nd Australia-Japan
Workshop on Stochastic Models in Engineering, Technology and
Management*, Technology Management Centre, The University of
Queensland, pp. 131-140.

Hart, A.G. and Pollett, P.K. (1996) Direct analytical methods for
determining quasistationary distributions for continuous-time Markov
chains. In (Eds C.C. Heyde, Yu V. Prohorov, R. Pyke and S.T. Rachev)
*Athens Conference on Applied Probability and Time Series Analysis,
Volume I: Applied Probability, In Honour of J.M. Gani*, Lecture
Notes in Statistics 114, Springer-Verlag, New York, pp. 116-126.

Hart, A.G. and Pollett, P.K. (1996) New methods for determining
quasistationary distributions for continuous-time Markov chains. In (Eds
Richard J. Wilson, D.N. Pra Murthy and Shunji Osaki) *Proceedings of
the 2nd Australia-Japan Workshop on Stochastic Models in Engineering,
Technology and Management*, Technology Management Centre, The
University of Queensland, pp. 177-186.

Pollett, P.K. (1996) Modelling the long-term behaviour of evanescent
ecological systems. *Ecological Modelling* 86, 135-139.

Bean N.G., Bright, L., Latouche, G., Pearce, C.E.M., Pollett, P.K.
and Taylor, P.G. (1997) The quasistationary behaviour of
quasi-birth-and-death processes. *The Annals of Applied
Probability* 7, 134-155.

Kijima, M., Nair, M.G., Pollett, P.K. and van Doorn, E. (1997)
Limiting conditional distributions for birth-death processes.
*Advances in Applied Probability* 29, 185-204.

Pollett, P.K. (1997) Limiting conditional distributions
for stochastic metapopulation models. In (Eds A.D. McDonald and M.
McAleer) *Proceedings of the International Congress on Modelling and
Simulation*, Vol. 2 (ISBN 0-86422-826-0), Modelling and Simulation
Society of Australia, Hobart, Australia, pp. 807-812.

Roberts, G.O., Jacka, S.D. and Pollett, P.K. (1997) Non-explosivity
of limits of conditioned birth and death processes.
*Journal of Applied Probability* 34, 35-45.

Bean, N.G., Pollett, P.K. and Taylor, P.G. (1998) The
quasistationary distributions of level-independent quasi-birth-and-death
processes. *Stochastic Models* 14 (Special Issue in Honour
of Marcel Neuts), 389-406.

Coolen-Schrijner, P. and Pollett, P.K. (1999)
Quasi-stationarity of discrete-time Markov chains with drift to
infinity. *Methodology and Computing in Applied
Probability* 1, 81-96.

Pollett, P.K. (1999) Quasistationary distributions
for continuous time Markov chains when absorption is not certain.
*Journal of Applied Probability* 36, 268-272.

Pollett, P.K. (1999) Modelling quasi-stationary
behaviour in metapopulations. *Mathematics and Computers in
Simulation* 48, 393-405.

Pollett, P.K. (1999) Quasistationarity in
populations that are subject to large-scale mortality or emigration. In
(Eds Les Oxley, Frank Scrimgeour and Anthony Jakeman) *Proceedings of
the International Congress on Modelling and Simulation*, Vol. 3
(ISBN 0-86422-950-X), Modelling and Simulation Society of Australia and
New Zealand, Hamilton, New Zealand, pp. 667-672.

Bean, N.G., Pollett, P.K. and
Taylor, P.G. (2000) The quasistationary distributions of level-dependent
quasi-birth-and-death processes. *Stochastic Models* 16,
511-541.

Coolen-Schrijner, P., Hart, A.G. and Pollett, P.K.
(2000) Quasistationarity of continuous-time Markov chains with positive
drift. *Journal of the Australian Mathematical Society, Series
B* 41, 423-441.

Darlington, S.J. and Pollett, P.K. (2000) Quasistationarity
in continuous time Markov chains where absorption is not certain.
*Journal of Applied Probability* 37, 598-600.

Elmes, S., Pollett, P.K. and Walker, D. (2000) Further results on the
relationship between m-invariant measures and
quasistationary distributions for absorbing continuous-time Markov
chains. *Mathematical and Computer Modelling* 31, 107-113.

Hart, A.G. and Pollett, P.K. (2000) New methods for determining
quasistationary distributions for continuous-time Markov chains.
*Mathematical and Computer Modelling* 31, 143-150.

Clancy, D., O'Neill, P.D. and Pollett,
P.K. (2001) Approximations for the long-term behaviour of an
open-population epidemic model. *Methodology and Computing in Applied
Probability* 3, 75-95.

Pollett, P.K. (2001) Quasistationarity in
populations that are subject to large-scale mortality or emigration.
*Environment International* 27, 231-236.

Pollett, P.K. (2002) Identifying
*Q*-processes with a given finite m-invariant measure. In (Eds Zhenting Hou, Jerzy A.
Filar and Anyue Chen) *Markov Processes and Controlled Markov
Chains*, Kluwer, pp. 41-55.

Clancy, D. and Pollett, P.K. (2003) A note on
quasi-stationary distributions of birth-death processes and the SIS
logistic epidemic. *Journal of Applied Probability* 40,
821-825.

Pollett, P.K. and Zhang, H. (2004) Existence and
uniqueness of *Q*-processes with a given finite m-invariant measure. In (Eds. Philip K. Pollett and
Peter G. Taylor) *Festschrift in Honour of Daryl Daley*,
*Australian & New Zealand Journal of Statistics* 46,
111-118.

Gray, B., Pollett, P.K. and Zhang, H. (2005) On the
existence of uni-instantaneous *Q*-processes with a given finite
m-invariant measure.
*Journal of Applied Probability* 42, 713-725.

If you have any comments on these pages,

feel free to e-mail
me: * pkp@maths.uq.edu.au *