On sigma-finite measures related to the Martin boundary of recurrent Markov chains

TL;DR

This paper classifies sigma-finite measures associated with the Martin boundary of recurrent Markov chains, extending previous work on stochastic processes like Brownian motion and diffusions.

Contribution

It provides a classification of these measures based on the minimal Martin boundary for recurrent Markov chains, connecting to prior constructions in stochastic process theory.

Findings

Classification of measures based on the minimal Martin boundary

Application to examples from the authors' monograph

Extension of measure construction to recurrent Markov chains

Abstract

In our monograph with B. Roynette and M. Yor, we construct a sigma-finite measure related to penalisations of different stochastic processes, including the Brownian motion in dimension 1 or 2, and a large class of linear diffusions. In the last chapter of the monograph, we define similar measures from recurrent Markov chains satisfying some technical conditions. In the present paper, we give a classification of these measures, in function of the minimal Martin boundary of the Markov chain considered at the beginning. We apply this classification to the examples considered at the end of our monograph.