Reciprocal classes of random walks on graphs

The reciprocal class of a Markov path measure is the set of all the 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 intensity of jump. 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.