Abstract
We prove that if the Ricci tensor of a geodesically complete Riemannian manifold M, endowed with the Riemannian distance ρ and the Riemannian measure , is bounded from below by a continuous function whose negative part satisfies, for every , the exponential integrability condition
then the lifetime of Brownian motion on M starting in any is a.s. infinite. This assumption on k holds if belongs to the Kato class of M. We also derive a Bismut–Elworthy–Li derivative formula for for every and along the heat flow with generator , yielding its --regularization as a corollary.
Moreover, given the stochastic completeness of M, but without any assumption on k except continuity, we prove the equivalence of lower boundedness of by k to the existence, given any , of a coupling of Brownian motions on M starting in such that a.s.,
holds for every with , involving the “average” of k along geodesics from u to v.
Our results generalize to weighted Riemannian manifolds, where the Ricci curvature is replaced by the corresponding Bakry–Émery Ricci tensor.
Funding Statement
The first author was funded by the European Research Council through the ERC-AdG “RicciBounds”.
Acknowledgments
The authors’ collaboration arose from a discussion at the Japanese-German Open Conference on Stochastic Analysis at the University of Fukuoka, Japan, in September 2019. The authors gratefully acknowledge the warm hospitality of this institution.
The authors thank the anonymous reviewers for their helpful comments and corrections which lead to a significant improvement of the paper’s quality.
Citation
Mathias Braun. Batu Güneysu. "Heat flow regularity, Bismut–Elworthy–Li’s derivative formula, and pathwise couplings on Riemannian manifolds with Kato bounded Ricci curvature." Electron. J. Probab. 26 1 - 25, 2021. https://doi.org/10.1214/21-EJP703
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