Representations for the decay parameter of Markov chains

Abstract

In this paper, we give variational representations for decay parameters of Markov chains. In continuous-time cases, the representation involves Donsker–Varadhan’s famous $I$-functional, from which some dual representations are given, which are expected to he useful in estimating the lower and upper bounds of the decay parameter. As a consequence, dual representations for decay parameters of discrete time Markov chains are derived. For continuous-time chains with finite states, we also give another form of dual formulas, which can be regarded as a version of the one for the Perron–Frobenius eigenvalue, with nonnegative matrices replaced by $Q$-matrices of the chains. Connections with quasi-stationarity and quasi-ergodicity of absorbing Markov chains are discussed. An interpretation for the corresponding variational solutions is given.