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

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.