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

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