Entropy of absorbed Markov chains

TL;DR

This paper investigates the entropy properties of absorbed Markov chains, introducing a canonical stationary chain via quasi-stationary distributions and analyzing the entropy relationships among different representations.

Contribution

It provides a novel stationary representation for absorbed Markov chains using quasi-stationary distributions and 2-stringing, linking entropy measures to the natural probability factors.

Findings

Established a canonical stationary chain for absorbed Markov chains.

Linked entropy measures to natural probability factors.

Provided a method to verify the construction's suitability through entropy balance.

Abstract

We consider a strictly substochastic matrix or an stochastic matrix with absorbing states. By using quasi-stationary distributions one shows there is a canonical associated stationary Markov chain. Based upon $2-$stringing representation of the resurrected chain we supply a stationary representation of the killed and the absorbed chains. The entropies of these representations has a clear meaning when one identifies the probability measure of natural factors. The balance between the entropies of these representations and the canonical chain, serves to check the suitability of the whole construction.