Functional inequalities for transient birth–death processes and their applications

We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth–death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth–death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.