Criteria for super- and weak-Poincaré inequalities of ergodic birth-death processes

Abstract

Criteria for the super-Poincare inequality and the weak-Poincare inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 (p. 15) of Prof. Mu-Fa Chen’s book “Eigenvalues, Inequalities and Ergodic Theory” (Springer, London, 2005). As a byproduct, we conclude that only ergodic birth-death processes on finite state space satisfy the Nash inequality with index 0 < ν ≤ 2.