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...

Let Xt be a diffusion on D with a generator , where f is a probability density such that f 0 on an open convex set D and vanishes outside. Then [integral operator]D 1/f = [infinity] implies that Xt never leaves D. As an application we extend the asymptotics of Wiener integrals of...