We introduce a certain kind of strong ergodicity condition to study the existence of spectral gap for Markov generator. We can estimate the spectral gap using the ergodicity condition and a Sobolev type inequality. We apply our results to the Dirichlet form on Wiener spaces, Riemannian manifolds, and loop spaces. 1998 Academic Press
1. Introduction
Let p(t, x, dy)= p(t, x, y) m(dy) be a transition probability of a symmetric Markov semigroup P t which is defined on a probability space (X, B, m). Let us assume that 1 # D(E) and E(1, 1)=0, where E denotes the Dirichlet form. Then if there exists a time t>0 such that inf x, y p(t, x, y)>0, E has a spectral gap and the gap is estimated by sup t>0 1ร2t inf x, y p(t, x, y) from below. Hence in the case of compact Riemannian manifolds, Li-Yau's parabolic Harnack inequality immediately gives a lower bound of the spectral gap (=the second eigenvalue) using the lower bound of the Ricci curvature and the diameter. However usually for all t, inf x, y p(t, x, y)=0 and there may not be a density function. In this paper, we will study a relation between the spectral gap and a certain lower bound of transition probability by using a Sobolev type inequality. Roughly speaking, if we are given a Sobolev type inequality, the spectral gap measures the probability that the transition density takes the small value. Namely under the bound of the Sobolev constant, the condition that the spectral gap becomes small implies the probability that p(t, x, y) takes the small value becomes large. See Corollary 2.13. Our estimate can be applied to the case where the transition probability has no density and the spectral bottom except 0 is not an eigenvalue. See Theorem 2.11. A typical