Bismut-Stroock Hessian formulas and local Hessian estimates for heat semigroups and harmonic functions on Riemannian manifolds-Reference-Cited by-同舟云学术

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

Published:2022-03-23 Issue: Volume: Page:
ISSN:2194-0401
Container-title:Stochastics and Partial Differential Equations: Analysis and Computations
language:en
Short-container-title:Stoch PDE: Anal Comp

Author:

Chen Qing-Qian,Cheng Li-Juan^ORCID,Thalmaier Anton^ORCID

Abstract

AbstractIn 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 (An estimate on the Hessian of the heat kernel: 355–371, 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.

Funder

Natural Science Foundation of Zhejiang Province

Publisher

Springer Science and Business Media LLC

Subject

Applied Mathematics,Modeling and Simulation,Statistics and Probability

Link

https://link.springer.com/content/pdf/10.1007/s40072-022-00241-1.pdf

Reference21 articles.

1. Arnaudon, M., Driver, B.K., Thalmaier, A.: Gradient estimates for positive harmonic functions by stochastic analysis. Stochastic Process. Appl. 117(2), 202–220 (2007)

2. Arnaudon, M., Plank, H., Thalmaier, A.: A Bismut type formula for the Hessian of heat semigroups. C. R. Math. Acad. Sci. Paris 336(8), 661–666 (2003)

3. Bismut, J.-M.: Large deviations and the Malliavin calculus, Progress in Mathematics, vol. 45. Birkhäuser Boston Inc, Boston, MA (1984)

4. Cao, J., Cheng, L.-J., Thalmaier, A.: Hessian heat kernel estimates and Calderón-Zygmund inequalities on complete Riemannian manifolds, arXiv:2108.13058 (2021)

5. Chen, X., Li, X.-M., Wu, B.: Logarithmic heat kernels: estimates without curvature restrictions, arXiv:2106.02746v1 (2021)