Chen Mufa
The study of the convergent rate (spectral gap) in the L2-sense is motivated from several different fields: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in[7] for the estimate of the convergent rate and as a continuation of[4],[5],[7-9],[23] and[24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper[10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings[4] is also extended in the paper.