Abstract
The chapter applies stochastic approaches to the analysis on Riemannian manifolds. It presents a complete formulation of Kendall-Cranston's coupling for diffusion processes on manifolds, and then applies this coupling to obtain explicit lower bound estimates of the first eigenvalue of the generators and gradient estimates of diffusion semigroups. The chapter also studies the estimation of the first two Dirichlet eigenvalues and the Liouville property of Riemannian manifolds. As an application of the gradient estimate, a correspondence between the Poincaré inequality and the isoperimetric inequality as well as a dimension-free Harnack inequality are presented.
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