Optimal markovian couplings and applications

This paper is devoted to studying a new topic: optimal Markovian couplings, mainly for time-continuous Markov processes. The study emphasizes the analysis of the coupling operators rather than the processes. Some constructions of optimal Markovian couplings for Markov chains and diffusions are presented, which are often unexpected. Then, the results are applied to study theL ²-convergence for Markov chains and for a diffusion on compact manifold. The estimate of the convergent rate provided by this method can be sharp. Supported in part by NSFC, the State Education Commission of China, the NSERC operating grant of D.A. Dawson and Centro Vito Volterra.