A characterization of exponential functionals in finite Markov chains

This work considers Markov chains with finite state space. It is supposed that the process has a single recurrent class, but the set of transient states is not necessarily empty. In this context, a Varadhan’s function, giving the exponential grow rate of an aggregated cost function, is studied. The main result establishes that this functional is the optimal value of a minimization problem on the Euclidean space whose dimension equals the number of states. Copyright Springer-Verlag 2004