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:

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.

A variational formula for the lower bound of the spectral gap of an elliptic operator is presented in the paper for the first time. The main known results are either recovered or improved. A large number of new examples with sharp estimate are illustrated. Moreover, as an application of the march coupling[4], the Poincare inequality with respect to the absolute distribution of the process is also

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.