Probability Versions of Li-Yau Type Inequalities and Applications

TL;DR

This paper develops probabilistic versions of Li-Yau inequalities for diffusion semigroups on manifolds with boundaries, providing new estimates that extend previous results and offering a simpler proof technique using stochastic analysis.

Contribution

It introduces probabilistic Li-Yau inequalities on manifolds with boundary, incorporating curvature and boundary conditions, and simplifies the proof method compared to traditional maximum principle approaches.

Findings

Derived explicit probabilistic Li-Yau inequalities involving curvature and boundary bounds.

Extended global and local estimates to manifolds with boundary, improving existing results.

Provided a more straightforward stochastic analysis proof technique.

Abstract

By using stochastic analysis, two probability versions of Li-Yau type inequalities are established for diffusion semigroups on a manifold possibly with (non-convex) boundary. The inequalities are explicitly given by the Bakry-Emery curvature-dimension, as well as the lower bound of the second fundamental form if the boundary exists. As applications, a number of global and local estimates are presented, which extend or improve existing ones derived for manifolds without boundary. Compared with the maximum principle technique developed in the literature, the probabilistic argument we used is more straightforward and hence considerably simpler.