The eigentime identity for continuous-time ergodic Markov chains

Abstract

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