On absorption times and Dirichlet eigenvalues

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on the absorbing point) are nonnegative, we show that T is distributed as a mixture of sums of independent geometric laws whose parameters are successive Dirichlet eigenvalues (starting from the smallest one). The mixture weights depend on the starting law. This result leads to a probabilistic interpretation of the spectrum, in terms of strong random times and local equilibria through a simple intertwining relation. Next this study is extended to the continuous time framework, where geometric laws have to be replaced by exponential distributions having the (opposite) Dirichlet eigenvalues of the generator as parameters. Returning to the discrete time setting we consider the influence of negative eigenvalues which are given another probabilistic meaning. These results generalize results of Karlin and McGregor and Keilson for birth and death chains