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

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

Some estimates of logarithmic Sobolev constant for general symmetric forms are obtained in terms of new Cheeger's constants. The estimates can be sharp in some sense.

This is the first one of a series of three papers. They are partially surveys on three aspects: 1) explaining the main ideas of our recent application of the coupling method to the estimation of spectral gap, 2) introducing some more recent progress on the study on some related topics, 3) collecting some open problems for the further study. The technical details are often avoided in order to keep the paper to be readable at the graduate level.

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.