Representations for the decay parameter of Markov chains (Q527480)

According to Kingman's classical result under some assumptions for the Markov chains there exists the limit \[ -\lambda _1=\lim _{t\to \infty }t^{-1}\log P_{ij}(t), \] which is independent of \(i\) and \(j\). If the chain is transient or null recurrent, then \(\lambda _1\) characterizes the exponential decay rate of the process. Let \[ \lambda ^{*}=-\lim _{t\to \infty }t^{-1}\log \| P(t)\| _u, \] where \(\| P(t)\| _u\) is the norm of the operator \(P(t)\), then \(\lambda _1\geq \lambda ^*\). In many typical cases, \(\lambda ^*\) really is in the spectrum. The aim of the paper is to provide variational representation for \(\lambda _1\) for more general chains and is motivated by the works of \textit{M. D. Donsker} and \textit{S. R. S. Varadhan} [Proc. Natl. Acad. Sci. USA 72, 780--783 (1975; Zbl 0353.49039); Commun. Pure Appl. Math. 29, 595--621 (1976; Zbl 0356.35065)]. Using the Donsker-Varadhan \(I\)-functional some dual representations are given, they are connected with the Perron-Frobenius eigenvalue. Connections with quasi-stationarity and quasi-ergodicity are discussed.