Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-29T12:52:08.420Z Has data issue: false hasContentIssue false
  • English
  • Français

A description of the long-term behaviour of absorbing continuous-time Markov chains using a centre manifold

Published online by Cambridge University Press:  01 July 2016

P. K. Pollett  and
A. J. Roberts
P. K. Pollett*
Affiliation:
The University of Queensland
A. J. Roberts*
Affiliation:
The University of Adelaide
*
∗Postal address: Department of Mathematics, The University of Queensland, St. Lucia, QLD 4067, Australia.
∗∗Postal address: Department of Applied Mathematics, The University of Adelaide, G.P.O. Box 498, Adelaide, SA 5001, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

We use the notion of an invariant manifold to describe the long-term behaviour of absorbing continuous-time Markov processes with a denumerable infinity of states. We show that there exists an invariant manifold for the forward differential equations and we are able to describe the evolution of the state probabilities on this manifold. Our approach gives rise to a new method for calculating conditional limiting distributions, one which is also appropriate for dealing with processes whose transition probabilities satisfy a system of non-linear differential equations.

Keywords

INVARIANT MANIFOLDSQUASISTATIONARY DISTRIBUTIONS
Type
Research Article
Information
Advances in Applied Probability , Volume 22 , Issue 1 , March 1990 , pp. 111 - 128
Copyright
Copyright © Applied Probability Trust 1990 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bartlett, M. S. (1957) On theoretical models for competitive and predatory biological systems. Biometrika 44, 27–42.Google Scholar
Bellman, R. (1970) Methods of Non-Linear Analysis, Vol. I. Academic Press, New York.Google Scholar
Carr, J. (1981) Applications of Centre Manifold Theory. Applied Math. Sci. 35, Springer-Verlag, Berlin.Google Scholar
Cavender, J. A. (1978) Quasistationary distributions for birth-and-death processes. Adv. Appl. Prob. 10, 570–586.Google Scholar
Coullet, P. H. and Spiegel, E. A. (1983) Amplitude equations for systems with competing instabilities. SIAM J. Appl. Math. 43, 776–821.CrossRefGoogle Scholar
Darroch, J. N. and Seneta, E. (1967) On quasi-stationary distributions in absorbing continuous-time finite Markov chains. J. Appl. Prob. 4, 192–196.CrossRefGoogle Scholar
Davis, E. B. (1980) One-parameter Semigroups. Academic Press, London.Google Scholar
Dunford, N. and Schwartz, J. (1965) Linear Operators, I. Interscience, New York.Google Scholar
Flaspohler, D. C. (1974) Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains. Ann. Inst. Statist. Math. 26, 351–356.Google Scholar
Gillespie, D. T. (1977) Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361.Google Scholar
Gortz, R. and Walls, D. F. (1976) Steady state solutions of master equations without detailed balance. Z. Physik B25, 423–427.Google Scholar
Istratescu, V. I. (1981) Introduction to Linear Operator Theory. Marcel Dekker, New York.Google Scholar
Keitzer, J. (1977) Master equations, Langevin equations, and the effect of diffusion on concentration fluctuations. J. Chem. Phys. 67, 1473–1476.Google Scholar
Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. (3) 13, 337–358.CrossRefGoogle Scholar
Malek-Mansour, M. and Nicolis, G. (1975) A master equation description of local fluctuations. J. Statist. Phys. 13, 197–217.Google Scholar
Mandl, P. (1960) On the asymptotic behaviour of probabilities within groups of states of a homogeneous Markov process. Cas. Pest. Mat. 85, 448456.Google Scholar
Parsons, R. W. and Pollett, P. K. (1987) Quasi-stationary distributions for auto-catalytic reactions. J. Statist. Phys. 46, 249–254.CrossRefGoogle Scholar
Pollett, P. K. (1986) On the equivalence of µ-invariant measures for the minimal process and its q-matrix. Stoch. Proc. Appl. 22, 203–221.Google Scholar
Pollett, P. K. (1988) Reversibility, invariance and µ-invariance. Adv. Appl. Prob. 20, 600–621.CrossRefGoogle Scholar
Pollett, P. K. (1989a) On the problem of evaulating quasistationary distributions for open reaction schemes. J. Statist. Phys. 53, 1207–1215.CrossRefGoogle Scholar
Pollett, P. K. (1989b) The generalized Kolmogorov criterion. Stoch. Proc. Appl. To appear.Google Scholar
Reuter, G. E. H. (1957) Denumerable Markov processes and the associated contraction semigroups on l . Acta Math. 97, 146.CrossRefGoogle Scholar
Roberts, A. J. (1985) Simple examples of the derivation of amplitude equations for systems of equations possessing bifurcations. J. Aust. Math. Soc. B27, 48–65.Google Scholar
Roberts, A. J. (1988a) The application of centre manifold theory to the evolution of systems which vary slowly in space. J. Aust. Math. Soc. B29, 480–500.Google Scholar
Roberts, A. J. (1988b) A formal centre manifold description of the evolution of slowly-varying waves. Report, University of Adelaide.Google Scholar
Roberts, A. J. (1988c) The description of interacting nonlinear waves through a formal centre manifold procedure. Report, University of Adelaide.Google Scholar
Sijbrand, J. (1985) Properties of centre manifolds. Trans. Amer. Math. Soc. 289, 431–469.CrossRefGoogle Scholar
Turner, J. W. and Malek-Mansour, M. (1978) On the absorbing zero boundary problem in birth and death processes. Physica 93A, 517–525.Google Scholar
Tweedie, R. L. (1974) Some ergodic properties of the Feller minimal process. Quart. J. Math. Oxford (2) 25, 485–495.CrossRefGoogle Scholar
Vere-Jones, D. (1968) Ergodic properties of non-negative matrices II. Pacific J. Math. 26, 601–620.Google Scholar
Vere-Jones, D. (1969) Some limit theorems for evanescent processes. Aust. J. Statist. 11, 67–78.Google Scholar