Asymptotic behaviour of random Markov chains with tridiagonal generators

Authors

  • Peter Kloeden Goethe University
  • Viktor Kozyakin Russian Academy of Sciences

Keywords:

random differential equations, random Markov chains, positive cones, uniformly contracting cocycles, random attractors

Abstract

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 Carath\'eodory 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 transitions probabilities, in which case the attractor is a periodic path. DOI: 10.1017/S0004972712000160

Author Biographies

Peter Kloeden, Goethe University

Institue of Mathematics, Professor

Viktor Kozyakin, Russian Academy of Sciences

Institute for Information Transmission Problems, research professor

Published

2012-12-17

Issue

Vol. 87 No. 1 (2013)

Section

Articles
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