This paper starts from a nice application of the coupling method to a traditional topic: the estimation of spectral gap (the first non-trivial eigenvalue). Some new variational formulas for the lower bound of the spectral gap of Laplacian on manifold or elliptic operators in Rd or Markov chains are reported [10]; [15]; [16]. The new formulas are especially powerful for the lower bounds, they have no common points with the classical variational formula (which goes back to Lord S. J. W. Rayleigh (1877) or E. Fischer (1905) and is particularly useful for the upper bounds). No analog of the new formulas ever appeared before. The formulas enable us to recover or improve the main known results. This will be illustrated by a comparison of the new results with the known ones in geometry. Next, we will explain the mathematical tool for proving the results. That is, the trilogy of the recent development of the coupling theory: The Markovian coupling, the optimal Markovian coupling and the construction of distances for coupling. Finally, some related results and some problems for the further study are also mentioned. It is hoped that the paper could be readable not only for probabilists but also for geometers and analysts.

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

This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some su