A gradient flow that is none: Heat flow with Wentzell boundary condition

2025/06/27 by Marie Bormann, Léonard Monsaingeon, Bormann, Marie +5
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Mechanics and Entropy

paper · pdf · doi:10.48550/arxiv.2506.22093

openalex publication_date 2025/06/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish a representation of the heat flow with Wentzell boundary conditions on smooth domains as gradient descent dynamics for the entropy in a suitably extended Otto manifold of probability measures with additional boundary parts. Yet it is shown that for weak boundary diffusion, the associated Fokker-Planck dynamics cannot be recovered from any entropy-driven metric JKO-Wasserstein scheme, at least if the underlying point metric satisfies certain natural regularity assumptions. This discrepancy is illustrated in competing large-deviation heuristics in the Sanov and Schilder regimes.

Citations

• A Cheeger-type inequality for the drift Laplacian with Wentzell-type boundary condition
• Large deviations for sticky-reflecting Brownian motion with boundary diffusion
• Variational structures for the Fokker--Planck equation with general Dirichlet boundary conditions
• Sticky-reflecting diffusion as a Wasserstein gradient flow
• Functional Inequalities for Brownian Motion on Riemannian Manifolds with Sticky-Reflecting Boundary Diffusion
• Spectral gap estimates for Brownian motion on domains with\n sticky-reflecting boundary diffusion
• Behavior near walls in the mean-field approach to crowd dynamics
• Coalescing-fragmentating Wasserstein dynamics: particle approach
• Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line
• From the Schrödinger problem to the Monge–Kantorovich problem
• A large deviation approach to optimal transport
• Characterization of absolutely continuous curves in Wasserstein spaces
• Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality

Related

• Euclidean, Metric, and Wasserstein Gradient Flows: an overview
• Gradient flow structure for some nonlocal diffusion equations
• Nonlinear Fokker--Planck--Kolmogorov equations as gradient flows on the space of probability measures
• Trajectorial Otto calculus
• Neumann heat flow and gradient flow for the entropy on non-convex domains