Analytic-geometric methods for finite Markov chains with applications to quasi-stationarity

For a relatively large class of well-behaved absorbing (or killed) finite Markov chains, we give detailed quantitative estimates regarding the behavior of the chain before it is absorbed (or killed). Typical examples are random walks on box-like finite subsets of the square lattice $\mathbb Z^d$ absorbed (or killed) at the boundary. The analysis is based on Poincar\'e, Nash, and Harnack inequalities, moderate growth, and on the notions of John and inner-uniform domains.