Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds

TL;DR

This paper develops a martingale-based method to derive localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds, extending previous results and removing compactness restrictions.

Contribution

It introduces a new martingale approach to Hessian formulas that applies to non-compact manifolds and provides explicit local estimates for heat semigroups and harmonic functions.

Findings

Extended Hessian formulas to non-compact manifolds.

Provided explicit local Hessian estimates.

Removed compactness restrictions in previous formulas.

Abstract

In this article, we develop a martingale approach to localized Bismut-type Hessian formulas for heat semigroups on Riemannian manifolds. Our approach extends the Hessian formulas established by Stroock (1996) and removes in particular the compact manifold restriction. To demonstrate the potential of these formulas, we give as application explicit quantitative local estimates for the Hessian of the heat semigroup, as well as for harmonic functions on regular domains in Riemannian manifolds.