BMO Martingales and Positive Solutions of Heat Equations

TL;DR

This paper introduces a novel approach linking PDEs and quadratic BSDEs to derive gradient estimates for positive solutions of heat equations on various spaces, extending Li-Yau's estimate.

Contribution

It develops a new method connecting PDEs and quadratic BSDEs to obtain gradient bounds for heat equations on Euclidean and Riemannian spaces.

Findings

Derived gradient estimates for positive solutions of heat equations.

Provided a generalized Li-Yau estimate.

Linked PDE analysis with quadratic BSDE techniques.

Abstract

In this paper, we develop a new approach to establish gradient estimates for positive solutions to the heat equation of elliptic or subelliptic operators on Euclidean spaces or on Riemannian manifolds. More precisely, we give some estimates of the gradient of logarithm of a positive solution via the uniform bound of the logarithm of the solution. Moreover, we give a generalized version of Li-Yau's estimate. Our proof is based on the link between PDE and quadratic BSDE. Our method might be useful to study some (nonlinear) PDEs.