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.Comment: This second version of the paper is shorter than the first one. It only considers the easiest situation where the graph is non-directed. This restricted setting avoids many technicalities and provides an easier presentation, keeping the essential features of the proof. The reader interested in directed random walks (for instance, on directed trees) should read the first versio

Bridges of Markov counting processes. Reciprocal classes and duality formulas

Processes having the same bridges are said to belong to the same reciprocal class. In this article we analyze reciprocal classes of Markov counting processes by identifying their reciprocal invariants and we characterize them as the set of counting processes satisfying some duality formula