Li-Yau Type Gradient Estimates and Harnack Inequalities by Stochastic Analysis

TL;DR

This paper employs stochastic analysis techniques to derive local Li-Yau gradient estimates and Harnack inequalities for positive solutions of the heat equation on Riemannian manifolds, emphasizing local geometric dependence.

Contribution

It introduces a stochastic analysis approach to obtain local Li-Yau type estimates, highlighting the natural connection between stochastic tools and geometric PDE analysis.

Findings

Derived local gradient bounds depending only on universal constants and local geometry

Established Li-Yau type inequalities using stochastic methods

Provided a framework for local estimates in geometric analysis

Abstract

In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local estimates in the sense that the gradient bound at given point depends only on universal constants and the geometry of the Riemannian manifold locally about this point.