Phil Pollett's Research Pages

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Diffusion approximations and quasi-stationary distributions: a short bibliography

This comprises:

  1. Key papers by Tom Kurtz on diffusion approximations
  2. My own work on diffusion approximations
  3. Some recent work from my group on diffusion approximations
  4. 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