Uniqueness, Recurrence and Integral-type Functionals for Birth-death Processes on Trees

A birth-death process (abbr. BDP) on trees is a time-continuous and homogeneous random walk in which the transition rate from any state to its father is called death rate and the ones to its offspring are called birth rates. In this paper, we obtain explicit uniqueness and recurrence criteria for BDPs on trees. Meanwhile, we also get an explicit and recursive representation for moments of integral-type functionals for this process. We then study the uniqueness and recurrence for some specific examples of BDPs on trees and apply integral-type functionals' results to more general integral-type functionals. Moreover, polynomial ergodicity is derived and a sufficient condition for a central limit theorem is also obtained.