Dimension-Free Harnack Inequality and its Applications

Abstract

This paper presents a self-contained account concerning a dimensionfree Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical Li Yau s Harnack inequality did, but also provides a direct way to describe various dimension-free properties of finite and infinite-dimensional diffusion semigroups. The author starts with a standard weighted Laplace operator on a Riemannian manifold with curvature bounded from below, and then move further to the unbounded below curvature case and its infinite-dimensional settings.