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This paper investigates the birth-death ("B-D" for short) process on treewith continuous time, emphasizing on estimating the principal eigenvalue(equivalently, the convergence rate) of the process with Dirichlet boundary atthe unique root 0. Three kinds of variational formulas for the eigenvalue arepresented. As an application, we obtain a criterion for positivity of the firsteigenvalue for B-D processes on tree with one branch after some layer.

The first Dirichlet eigenvalue of birth-death process on tree