Diffusion Approximation for Symmetric Birth-and-Death Processes with Polynomial Rates

Abstract

The symmetric birth and death stochastic process on the non-negative integers x ∈ Z + with polynomial rates x α , α ∈ [1, 2], x 6= 0, is studied. The process moves slowly and spends more time in the neighborhood of the state 0. We prove the convergence of the scaled process to a solution of stochastic differential equation without drift. Sticking phenomenon appears at the limiting process: trajectories, starting from any state, take finite time to reach 0 and remain there indefinitely.