Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion

2023/12/30 by Marie Bormann, Max von Renesse, Bormann, Marie +3 · 2 citations
Computer Science · Mathematics · #35A23 #58C40 #60J65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2401.00206

openalex publication_date 2023/12/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove geometric upper bounds for the Poincaré and Logarithmic Sobolev constants for Brownian motion on manifolds with sticky reflecting boundary diffusion i.e. extended Wentzell-type boundary condition under general curvature assumptions on the manifold and its boundary. The method is based on an interpolation involving energy interactions between the boundary and the interior of the manifold. As side results we obtain explicit geometric bounds on the first nontrivial Steklov eigenvalue, for the norm of the boundary trace operator on Sobolev functions, and on the boundary trace logarithmic Sobolev constant. The case of Brownian motion with pure sticky reflection is also treated.

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• Super and Weak Poincaré Inequalities for Sticky-Reflected Diffusion Processes
• A gradient flow that is none: Heat flow with Wentzell boundary condition

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