Reciprocal classes of continuous time Markov chains

Motivated by the Schrödinger problem, we study reciprocal classes. Given a continuous time Markov chain, the associated reciprocal class is the convex hull of its bridges. Its elements are in general non Markovian, but rather Markov fields indexed over time.

We propose two different characterisations of it: the first one base on short time asymptotic and the second one based on Integration by parts(duality formulae). A key notion behind both approaches is that of reciprocal characteristics which, in some sense, replaces that of the generator for the original chain.

If time allows, the relation between global bounds on the characteristics and the fluctuations of a bridge are discussed.