In the recent years, a great effort has been made to develop a new ergodic theory for Markov processes. It is mainly concerned with the study on several different inequalities. Some of them are very classical but some of them are rather new. The Liggett-Stroock form of Nash-type inequalities, the related ones and their comparison are discussed. Based on some new isoperimetric or Cheeger's constants, a simple su

By using a coupling technique, this paper presents some lower bounds of the first eigenvalue

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 the L2-convergence for Markov chains and for a diffusion on compact manifold. The estimate of the convergent rate provided by this method can be sharp.

The talk begins with some backgrounds of our study: The spectral gap for four classes of reversible Markov processes and the relation between the spectral gap and the phase transitions. Then, we introduce two aspects of our recent progress: 1) The estimates of the spectral gap (or the first non-trivial eigenvalue) of Laplacian on compact Riemannian manifold. 2) Optimal Markovian couplings. These explain the precise meaning of the vague title. The resulting estimates are quite unexpected, not only recover the known sharp estimates but also produce some new ones without using anything from the previous proofs. The optimal estimates come from the optimal couplings, which are often out of our probabilistic intuition. It seems to the author that the study of couplings is renewed but there is still a lot to be done. We emphasize the ideas, including the applications of the coupling technique, in terms of some simple examples. It is hoped that the materials presented here could be helpful not only for experts but also for newcomers.

One challenging problem in the context of reaction-diffusions is to prove the ergodicity or non-ergodicity for the Schl