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ASYMPTOTIC BEHAVIOUR OF RANDOM MARKOV CHAINS WITH TRIDIAGONAL GENERATORS

Part of: Stochastic analysis Special matrices Equations and systems with randomness Markov processes

Published online by Cambridge University Press:  30 March 2012

PETER E. KLOEDEN  and
VICTOR S. KOZYAKIN
PETER E. KLOEDEN*
Affiliation:
Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main, Germany (email: kloeden@math.uni-frankfurt.de)
VICTOR S. KOZYAKIN
Affiliation:
Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane, 19, 101447 Moscow, Russia (email: kozyakin@iitp.ru)
*
✉For correspondence; e-mail: kloeden@math.uni-frankfurt.de
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Abstract

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Continuous-time discrete-state random Markov chains generated by a random linear differential equation with a random tridiagonal matrix are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses comparison theorems for Carathéodory random differential equations and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself with respect to the Hilbert projective metric. It does not involve probabilistic properties of the sample path and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transition probabilities, in which case the attractor is a periodic path.

Keywords

random differential equationsrandom Markov chainspositive conesuniformly contracting cocyclesrandom attractors

MSC classification

Secondary: 34F05: Equations and systems with randomness 60H25: Random operators and equations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 15B48: Positive matrices and their generalizations; cones of matrices 15B51: Stochastic matrices 15B52: Random matrices
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 87 , Issue 1 , February 2013 , pp. 27 - 36
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

Footnotes

P. E. Kloeden is partially supported by DFG grant KL 1203/7-1, the Spanish Ministerio de Ciencia e Innovación project MTM2011-22411, the Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under the Ayuda 2009/FQM314 and the Proyecto de Excelencia P07-FQM-02468. V. S. Kozyakin is partially supported by the Russian Foundation for Basic Research, project no. 10-01-93112.

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