Ten Explicit Criteria of One-Dimensional Processes

Abstract

The traditional ergodicity consists a crucial part in the theory of stochastic processes, plays a key role in practical applications. The ergodicity has much refined recently, due to the study on some inequalities, which are especially powerful in the infinite dimensional situation. The explicit criteria for various types of ergodicity for birth-death processes and one-dimensional diffusions are collected in Tables 8.1 and 8.2, respectively. In particular, an interesting story about how to obtain one of the criteria for birth-death processes is explained in details. Besides, a diagram for various types of ergodicity for general reversible Markov processes is presented.