Unified Speed Estimation of Various Stabilities

To study some infinite-dimensional subject (the phase transitions in statistical physics, for instance), several mathematical tools are developed. One of them is the speed estimation of various stabilities/instabilities. This paper collects some unexpected, unified, nearly sharp basic estimates of various types of stability/instability for the simplest class of Markov processes, the birth-death processes. Some motivations and a part of extensions are also discussed. The paper is based on a talk presented recently in several international conferences.