Asymptotic behaviour of random Markov chains with tridiagonal generators

TL;DR

This paper investigates the long-term behavior of random Markov chains with tridiagonal generators, showing they have a singleton attractor that is a random path, using comparison theorems and contraction properties.

Contribution

It introduces a novel approach to analyze the asymptotic behavior of Markov chains with random tridiagonal generators without relying on probabilistic sample path properties.

Findings

Existence of a singleton random attractor for the Markov chain

The attractor is a random path in the probability simplex

Method applies to nonautonomous deterministic chains as well

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 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.