Poincar\'{e} and Weak Poincar\'{e} Inequalities for the Mixed Poisson Measures

TL;DR

This paper investigates functional inequalities for mixed Poisson measures using the Mecke identity, establishing Poincaré and weak Poincaré inequalities, and disproving a related inequality under certain conditions.

Contribution

It introduces new Poincaré and weak Poincaré inequalities for mixed Poisson measures and clarifies the limitations of these inequalities.

Findings

Established Poincaré inequalities for mixed Poisson measures

Proved weak Poincaré inequalities under specific conditions

Disproved a Poincaré type inequality under certain assumptions

Abstract

By using the Mecke identity, we study a class of birth-death type Dirichlet forms associated with the mixed Poisson measure. Both Poincar\'{e} and weak Poincar\'{e} inequalities are established, while another Poincar\'{e} type inequality is disproved under some reasonable assumptions.