A Probabilistic Approach for Gradient Inequalities on Time-Inhomogeneous Manifolds

TL;DR

This paper introduces a probabilistic method to establish gradient inequalities of Hamilton and Li-Yau types for heat equations on time-inhomogeneous manifolds, simplifying previous proofs and extending the theoretical framework.

Contribution

It provides a novel probabilistic approach to derive gradient inequalities on time-inhomogeneous manifolds, enhancing understanding and simplifying existing proofs.

Findings

Established Hamilton and Li-Yau gradient inequalities probabilistically.

Simplified proofs of previous results by Sun (2011).

Extended gradient inequality theory to time-inhomogeneous settings.

Abstract

Gradient inequalities of the Hamilton type and the Li-Yau type for positive solutions to the heat equation are established from a probabilistic viewpoint, which simplifies the proofs of some results of Sun [{\it Pacific J. Math.}, 253 (2011), pp. 489--510].