Reciprocal classes of random walks on graphs

TL;DR

This paper characterizes the reciprocal class of continuous-time Markov random walks on graphs using reciprocal characteristics and Taylor expansions, extending existing results with a measure-theoretical approach and illustrative examples.

Contribution

It provides new characterizations of reciprocal classes for Markov random walks on graphs based on jump intensities and small-time expansions.

Findings

Characterization via reciprocal characteristics depending on jump intensity

Extension of known results using measure-theoretical methods

Illustrative examples demonstrating the theoretical results

Abstract

The reciprocal class of a Markov path measure is the set of all mixtures of its bridges. We give characterizations of the reciprocal class of a continuous-time Markov random walk on a graph. Our main result is in terms of some reciprocal characteristics whose expression only depends on the jump intensity. We also characterize the reciprocal class by means of Taylor expansions in small time of some conditional probabilities. Our measure-theoretical approach allows to extend significantly already known results on the subject. The abstract results are illustrated by several examples.