Estimate of Exponential Convergence Rate in Total Variation by Spectral Gap

<正> This note is devoted to study the exponential convergence rate in the total variation forreversible Markov processes by comparing it with the spectral gap. It is proved that in a quite generalsetup, with a suitable restriction on the initial distributions, the rate is bounded from below by thespectral gap. Furthemore, in the compact case or for birth-death processes or half-line diffosions,the rate is shown to be equal to the spectral gap.