Uniform Positivity Improving Property, Sobolev Inequalities, and Spectral Gaps

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.