Reciprocal classes of random walks on graphs

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. Introduction This article answers the question: “When does a continuous-time random walk on a graph share its bridges with a given Markov walk?”, both in terms of their jump intensities and of Taylor expansions in small time of probabilities of conditioned events. The precise answers are stated at Theorem 2.4 and Corollary 2.6 which are the main results of the article. The set of all path measures which share the bridges of a given Markov measure is called its reciprocal class. In contrast with most of the existing literature about reciprocal classes, i.e. shared bridges, which relies on transition probabilities, in this paper we adopt a measure-theoretical approach: our main objects of interest are path measures, i.e. probability measures on the path space, rather than transition probability kernels. It turns out that this is an efficient way for solving our problem and allows to extend significantly already known results on the subject. Date: Version 6. April 26, 2015. 2010 Mathematics Subject Classification. 60J27,60J75.