Estimate of exponential convergence rate in total variation by spectral gap

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.