Reciprocal class of random walks on a Abelian group

Processes having the same bridges as a given reference Markov process constitute its reciprocal class. In this paper we study the reciprocal class of a continuous time random walk with values in a countable Abelian group, compute explicitly its reciprocal characteristics and present an integral characterization of it. Our main tool is a new iterated version of the celebrated Mecke’s formula from the point process theory, which allows us to study, as transformation of the path space, the addition of random loops. Thanks to the lattice structure of the set of loops, we even obtain a sharp characterization. At the end, we discuss several examples to illustrate the richness of reciprocal classes. We observe how their structure depends on the algebraic properties of the underlying group.