The first Dirichlet eigenvalue of birth-death process on tree

TL;DR

This paper studies the principal eigenvalue of a birth-death process on a tree, providing variational formulas and criteria for positivity, which relate to the process's convergence rate.

Contribution

It introduces new variational formulas for the eigenvalue and establishes a positivity criterion for birth-death processes on trees with a specific structure.

Findings

Derived three variational formulas for the eigenvalue

Established a criterion for positivity of the first eigenvalue

Connected eigenvalue estimates to convergence rates

Abstract

This paper investigates the birth-death ("B-D" for short) process on tree with continuous time, 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 presented. As an application, we obtain a criterion for positivity of the first eigenvalue for B-D processes on tree with one branch after some layer.