Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound

TL;DR

This paper establishes a log-Harnack inequality and gradient inequality for diffusion semigroups on Riemannian manifolds, linking these inequalities to pointwise curvature bounds and boundary conditions.

Contribution

It introduces explicit inequalities involving local geometry, proving their equivalence to pointwise curvature lower bounds and boundary convexity or absence.

Findings

Log-Harnack inequality with local geometric quantities

Equivalence between inequalities and curvature bounds

Applications to diffusion processes on manifolds

Abstract

By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curvature lower bound condition together with the convexity or absence of the boundary. Some applications of the log-Harnack inequality are also introduced.