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

TL;DR

This paper provides bounds on the amplitude of the first Dirichlet eigenvector in finite and certain infinite Markov processes, aiding the analysis of convergence to quasi-stationarity.

Contribution

It introduces two methods for bounding the eigenvector amplitude, extending spectral estimates to non-reversible and infinite state space Markov processes.

Findings

Derived bounds using path methods for finite Markov generators.

Extended spectral approach to infinite birth-death processes.

Facilitated analysis of convergence to quasi-stationarity.

Abstract

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. Perron-Frobenius theory ensures the existence of a corresponding positive eigenvector $\varphi$. The goal of the paper is to give bounds on the amplitude $\max \varphi/\min\varphi$. 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].