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:

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthermore, in the compact case or for birth-death processes or half-line diffusions, the rate is shown to be equal to the spectral gap.

This paper studies the equivalence of exponential ergodicity and L2-exponential convergence mainly for continuous-time Markov chains. In the reversible case, we show that the known criteria for exponential ergodicity are also criteria for L2-exponential convergence. Until now, no criterion for L2-exponential convergence has appeared in the literature. Some estimates for the rate of convergence of exponentially ergodic Markov chains are presented. These estimates are practical once the stationary distribution is known. Finally, the reversible part of the main result is extended to the Markov processes with general state space.

This is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the in finiteimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).

This is the second one of a series of three papers. The ideas introduced in the last paper are used to study the estimate of spectral gap and four classes of typical eigenvalue problems on manifolds. The comparison with the known optimal estimates are given, some new progress is reported and some open problems are proposed.