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

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.

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.

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

A general formula for the lower bound of the first eigenvalue on compact Riemannian mani-folds is presented in this paper for the first time. The formula improves the main known sharp estimates including Lichnerowicz's estimate and Zhong-Yang's estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on a probabilistic approach (i.e., the coupling method) introduced by the authors previously.

The study of the convergent rate (spectral gap) in the L2-sense is motivated from several differenti elds: probability, statistics, mathematical physics, computer science and so on and it is now an active research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4], [5], [7] [9], [23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings [4] is also extended in the paper.

This note studies the first non-trivial eigenvalue of second order self-adjoint elliptic operators in Rd. Some lower bounds of the eigenvalue are obtained by using a probabilistic approach and some geometric consideration. In one-dimensional case, an analytic proof is also presented. The resulting bounds can be sharp.

This paper is mainly devoted to estimate the logarithmic Sobolev (abbrev. L.S.) constant for diffusion operators on manifold or in Rd. In most cases, we study the lower bounds but a gener- alization to [9; Theorem 1] for the upper bound is also presented (Theorem 1.5). Based on a simple observation (due to [5]) of the comparison between the L.S. constants for different potentials, the pow-erful Bakry-Emery criterion for the L. S. inequality is improved considerably in the paper, especially for the manifolds with non-positive sectional curvatures (Theorem 1.3 (1)). In terms of our notation:

Abstract. The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case and certain extension is made in the paper.