Explicit criteria for several types of ergodicity

TL;DR

This paper reviews recent explicit criteria for various ergodicity types in one-dimensional diffusions and birth-death processes, highlighting solutions to longstanding open problems and connecting the topic to multiple mathematical fields.

Contribution

It provides a detailed survey of newly discovered explicit ergodicity criteria and discusses their mathematical significance and open problems.

Findings

Explicit criteria for several ergodicity types have been established.

The criteria include exponential ergodicity for birth-death processes.

The problem's solution connects multiple branches of mathematics.

Abstract

The explicit criteria for several types of ergodicity of one-dimensional diffusions or birth-death processes have been found out recently in a surprisingly short period. One of the criteria is for exponential ergodicity of birth-death processes. This problem has been opened for a long time in the study of Markov chains. The survey article explains in details the idea which leads to solve the problem just mentioned. It is interesting that the problem is connected with several branches of mathematics. Some open problems for the further study are also proposed.