Second Order Bismut formulae and applications to Neumann semigroups on manifolds

2022/10/18 by Lijuan Cheng, Cheng, Li-Juan, Anton Thalmaier +3 · 1 citation
Mathematics · #58J35 #58J65 #60J60 #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Approximation and Integration #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2210.09607

openalex publication_date 2022/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a complete connected Riemannian manifold with boundary ∂ M, and let P_t be the Neumann semigroup generated by ( 1)/( 2) L where L=Δ+Z for a C¹-vector field Z on M. We establish Bismut type formulae for LP_t f and \rm Hess_Ptf and present estimates of these quantities under suitable curvature conditions. In case when P_t is symmetric in L²(μ) for some probability measure μ, a new type of log-Sobolev inequality is established which links the relative entropy H, the Stein discrepancy S, and relative Fisher information I, generalizing the authors' recent work in the case without boundary.

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