Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

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Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector φ. The goal of the paper is to give bounds on the amplitude φ/φ. Two approaches are proposed: one using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen in [7].