The first Dirichlet eigenvalue of birth–death process on trees

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 presented. As an application, we obtain a criterion for positivity of the first eigenvalue for B–D processes on trees with one branch after some layer.