Logarithmic Harnack inequalities∗

Logarithmic Sobolev inequalities first arose in the analysis of elliptic differential operators in infinite dimensions. Many developments and applications can be found in several survey papers [1, 9, 12]. Recently, Diaconis and Saloff-Coste [8] considered logarithmic Sobolev inequalities for Markov chains. The lower bounds for log-Sobolev constants can be used to improve convergence bounds for random walks on graphs [5, 8]. The problem of bounding log-Sobolev constants tends to be harder than estimating eigenvalues. Logarithmic Harnack inequalities provide a direct approach for estimating the log-Sobolev constant. We will derive lower bounds for log-Sobolev constants for Riemannian manifolds and for large classes of graphs.