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Cheeger inequalities for absorbing Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  19 September 2016

Gary Froyland  and
Robyn M. Stuart
Gary Froyland*
Affiliation:
University of New South Wales
Robyn M. Stuart*
Affiliation:
University of New South Wales
*
* Postal address: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia. Email address: [email protected]
** Current address: Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark.
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Abstract

We construct Cheeger-type bounds for the second eigenvalue of a substochastic transition probability matrix in terms of the Markov chain's conductance and metastability (and vice versa) with respect to its quasistationary distribution, extending classical results for stochastic transition matrices.

Keywords

Absorbing Markov chaintransient Markov chainsubstochastic transition matrixquasistationary distributionCheeger constantconductancemetastability

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 631 - 647
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Arjas, E.,Nummelin, E. and Tweedie, R. L. (1980).Semi-Markov processes on a general state space: α-theory and quasistationarity.J. Austral. Math. Soc. A 30,187–200.Google Scholar
[2] Asselah, A.,Ferrari, P. A. and Groisman, P. (2011).Quasistationary distributions and Fleming‒Viot processes in finite spaces.J. Appl. Prob. 48,322–332.Google Scholar
[3] Bartlett, M. S. (1960).Stochastic Population Models in Ecology and Epidemiology.Methuen,London.Google Scholar
[4] Cheeger, J. (1970).A lower bound for the smallest eigenvalue of the Laplacian. In Problems in Analysis (Papers dedicated to Salomon Bochner),Princeton University Press, pp.195–199.Google Scholar
[5] Cheong, C. K. (1970).Quasi-stationary distributions in semi-Markov processes.J. Appl. Prob. 7,388–399, 788.Google Scholar
[6] Chung, F. (2005).Laplacians and the Cheeger inequality for directed graphs.Ann. Combinatorics 9,1–19.Google Scholar
[7] Collet, P.,Martínez, S. and San Martín, J. (2013).Quasi-Stationary Distributions: Markov Chains, Diffusions and Dynamical Systems.Springer,Heidelberg.Google Scholar
[8] Darroch, J. N. and Seneta, E. (1965).On quasi-stationary distributions in absorbing discrete-time finite Markov chains.J. Appl. Prob. 2,88–100.CrossRefGoogle Scholar
[9] Darroch, J. N. and Seneta, E. (1967).On quasi-stationary distributions in absorbing continuous-time finite Markov chains.J. Appl. Prob. 4,192–196.Google Scholar
[10] Dellnitz, M. and Junge, O. (1999).On the approximation of complicated dynamical behavior.SIAM J. Numerical Anal. 36,491–515.Google Scholar
[11] Deuflhard, P.,Huisinga, W.,Fischer, A. and Schütte, C. (2000).Identification of almost invariant aggregates in reversible nearly uncoupled Markov chains.Linear Algebra Appl. 315,39–59.Google Scholar
[12] Ewens, W. J. (1963).The diffusion equation and a pseudo-distribution in genetics.J. R. Statist. Soc. B 25,405–412.Google Scholar
[13] Ewens, W. J. (1964).The pseudo-transient distribution and its uses in genetics.J. Appl. Prob. 1,141–156.Google Scholar
[14] Ferrari, P. A. and Marić, N. (2007).Quasi-stationary distributions and Fleming‒Viot processes in countable spaces.Electron. J. Prob. 12,684–702.Google Scholar
[15] Ferrari, P. A.,Kesten, H.,Martinez, S. and Picco, P. (1995).Existence of quasi-stationary distributions. A renewal dynamical approach.Ann. Prob. 23,501–521.Google Scholar
[16] Flaspohler, D. C. (1974).Quasi-stationary distributions for absorbing continuous-time denumerable Markov chains.Ann. Inst. Statist. Math. 26,351–356.Google Scholar
[17] Flaspohler, D. C. and Holmes, P. T. (1972).Additional quasi-stationary distributions for semi-Markov processes.J. Appl. Prob. 9,671–676.Google Scholar
[18] Fleming, W. H. and Viot, M. (1979).Some measure-valued Markov processes in population genetics theory.Indiana Univ. Math. J. 28,817–843.Google Scholar
[19] Froyland, G. (2005).Statistically optimal almost-invariant sets.Phys. D 200,205–219.Google Scholar
[20] Froyland, G. and Padberg, K. (2009).Almost-invariant sets and invariant manifolds–connecting probabilistic and geometric descriptions of coherent structures in flows.Phys. D 238,1507–1523.Google Scholar
[21] Froyland, G.,Pollett, P. K. and Stuart, R. M. (2014).A closing scheme for finding almost-invariant sets in open dynamical systems.J. Comput. Dynamics 1,135–162.Google Scholar
[22] Huisinga, W. and Schmidt, B. (2006).Metastability and dominant eigenvalues of transfer operators. In New Algorithms for Macromolecular Simulation (Lecture Notes Comput. Sci. Eng. 49),Springer,Berlin, pp.167–182.Google Scholar
[23] Kelly, F. P. (1979).Reversibility and Stochastic Networks.John Wiley,Chichester.Google Scholar
[24] Kelly, F. P. (1983).Invariant measures and the q-matrix. In Probability, Statistics and Analysis (London Math. Soc. Lecture Notes Ser. 79),Cambridge University Press, pp.143–160.Google Scholar
[25] Kingman, J. F. C. (1969).Markov population processes.J. Appl. Prob. 6,1–18.Google Scholar
[26] Lawler, G. F. and Sokal, A. D. (1988).Bounds on the L 2 spectrum for Markov chains and Markov processes: a generalization of Cheeger's inequality.Trans. Amer. Math. Soc. 309,557–580.Google Scholar
[27] Levin, D. A.,Peres, Y. and Wilmer, E. L. (2009).Markov Chains and Mixing Times.American Mathematical Society,Providence, RI.Google Scholar
[28] Meyer, C. (2000).Matrix Analysis and Applied Linear Algebra.SIAM,Philadelphia, PA.Google Scholar
[29] Nair, M. G. and Pollett, P. K. (1993).On the relationship between μ-invariant measures and quasi-stationary distributions for continuous-time Markov chains.Adv. Appl. Prob. 25,82–102, 717–719.Google Scholar
[30] Pakes, A. G. (1993).Absorbing Markov and branching processes with instantaneous resurrection.Stoch. Process. Appl. 48,85–106.CrossRefGoogle Scholar
[31] Pollett, P. K. (1988).Reversibility, invariance and μ-invariance.Adv. Appl. Prob. 20,600–621.Google Scholar
[32] Seneta, E. and Vere-Jones, D. (1966).On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states.J. Appl. Prob. 3,403–434.Google Scholar
[33] Sinclair, A. and Jerrum, M. (1989).Approximate counting, uniform generation and rapidly mixing Markov chains.Inf. Comput. 82,93–133.Google Scholar
[34] Steinsaltz, D. and Evans, S. N. (2004).Markov mortality models: implications of quasistationarity and varying initial distributions.Theoret. Pop. Biol. 65,319–337.Google Scholar
[35] Tweedie, R. L. (1974).Some ergodic properties of the Feller minimal process.Quart. J. Math. Oxford Ser. (2) 25,485–495.Google Scholar
[36] Van Doorn, E. A. and Pollett, P. K. (2008).Survival in a quasi-death process.Linear Algebra Appl. 429,776–791.Google Scholar
[37] Van Doorn, E. A. and Pollett, P. K. (2009).Quasi-stationary distributions for reducible absorbing Markov chains in discrete time.Markov Process. Relat. Fields 15,191–204.Google Scholar
[38] Yaglom, A. M. (1947).Certain limit theorems of the theory of branching random processes.Doklady Akad. Nauk SSSR (N.S.) 56,795–798.Google Scholar