In this note we show that for a killed Markov process determined by a certain selfadjoint operator on a bounded domain, there are two quite different quasi-ergodic behaviors, corresponding to the Yaglom limit and a certain “fractional” Yaglom limit respectively. We also show that the high...

This paper investigates the birth–death (“B–D” for short) process on trees, emphasizing on estimating the principal eigenvalue (equivalently, the convergence rate) of the process with Dirichlet boundary at the unique root 0. Three kinds of variational formulas for the eigenvalue are...

In this paper we prove a Faber-Krahn-type inequality for regular trees and give a complete characterization of extremal trees. The main tools are rearrangements and perturbation of regular trees.