The eigentime identity for continuous-time ergodic Markov chains

Abstract

The eigentime identity is proved for continuous-time reversible Markov chains with Markov generatorL. When the essential spectrum is empty, let {0 = λ0< λ1≤ λ2≤ ···} be the whole spectrum ofLin L2. Then ∑n≥1λn-1< ∞ implies the existence of the spectral gapαofLin L∞. Explicit formulae are presented in the case of birth–death processes and from these formulae it is proved that ∑n≥1λn-1< ∞ if and only ifα> 0.