Research Article

ASYMPTOTIC BEHAVIOUR OF RANDOM MARKOV CHAINS WITH TRIDIAGONAL GENERATORS

PETER E. KLOEDEN^a1 c1 and VICTOR S. KOZYAKIN^a2

^a1 Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main, Germany (email: kloeden@math.uni-frankfurt.de)

^a2 Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane, 19, 101447 Moscow, Russia (email: kozyakin@iitp.ru)

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 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.

(Received January 25 2012)

2010 Mathematics subject classification

• primary 34F05; secondary 37H10;
• 60H25;
• 60J10;
• 15B48;
• 15B51;
• 15B52

Keywords and phrases

• random differential equations;
• random Markov chains;
• positive cones;
• uniformly contracting cocycles;
• random attractors

Correspondence:

^c1 For correspondence; e-mail: kloeden@math.uni-frankfurt.de

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.