Equivalence of exponential ergodicity and L2-exponential convergence for Markov chains

This paper studies the equivalence of exponential ergodicity and L2-exponential convergence mainly for continuous-time Markov chains. In the reversible case, we show that the known criteria for exponential ergodicity are also criteria for L2-exponential convergence. Until now, no criterion for L2-exponential convergence has appeared in the literature. Some estimates for the rate of convergence of exponentially ergodic Markov chains are presented. These estimates are practical once the stationary distribution is known. Finally, the reversible part of the main result is extended to the Markov processes with general state space.