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Sticky-reflecting diffusion as a Wasserstein gradient flow

Jean-Baptiste Casteras, Léonard Monsaingeon, Filippo Santambrogio

TL;DR

The paper demonstrates that the Fokker-Planck dynamics of reflected Sticky Brownian Motion can be formulated as a Wasserstein gradient flow of a relative entropy with respect to a mixed interior-boundary reference measure, leading to a coupled bulk-boundary PDE system for densities $(\omega,\gamma)$. By employing a JKO minimizing-movement scheme and an energy-dissipation framework, it proves the existence of dissipative weak solutions, with the entropy driving boundary coupling via a trace compatibility that arises from dissipation rather than a prior constraint. The analysis introduces an $\varepsilon$-regularization to handle Euler–Lagrange conditions at the discrete level and obtains sharp dissipation controls that yield exponential convergence to the equilibrium measure, thanks to a boundary logarithmic-Sobolev inequality. This work extends the gradient-flow paradigm to systems with bulk-boundary mass exchange and provides a rigorous variational foundation for sticky diffusion in a mixed-measure setting, with potential implications for models of surface-bulk interactions in diffusion processes.

Abstract

In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.

Sticky-reflecting diffusion as a Wasserstein gradient flow

TL;DR

The paper demonstrates that the Fokker-Planck dynamics of reflected Sticky Brownian Motion can be formulated as a Wasserstein gradient flow of a relative entropy with respect to a mixed interior-boundary reference measure, leading to a coupled bulk-boundary PDE system for densities . By employing a JKO minimizing-movement scheme and an energy-dissipation framework, it proves the existence of dissipative weak solutions, with the entropy driving boundary coupling via a trace compatibility that arises from dissipation rather than a prior constraint. The analysis introduces an -regularization to handle Euler–Lagrange conditions at the discrete level and obtains sharp dissipation controls that yield exponential convergence to the equilibrium measure, thanks to a boundary logarithmic-Sobolev inequality. This work extends the gradient-flow paradigm to systems with bulk-boundary mass exchange and provides a rigorous variational foundation for sticky diffusion in a mixed-measure setting, with potential implications for models of surface-bulk interactions in diffusion processes.

Abstract

In this paper we identify the Fokker-Planck equation for (reflected) Sticky Brownian Motion as a Wasserstein gradient flow in the space of probability measures. The driving functional is the relative entropy with respect to a non-standard reference measure, the sum of an absolutely continuous interior part plus a singular part supported on the boundary. Taking the small time-step limit in a minimizing movement (JKO scheme) we prove existence of weak solutions for the coupled system of PDEs satisfying in addition an Energy Dissipation Inequality.
Paper Structure (9 sections, 12 theorems, 128 equations, 3 figures)

This paper contains 9 sections, 12 theorems, 128 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Step 1: variation in entropy.
  4. Step 2: transportation cost.
  5. Step 3: conclusion.
  6. Discrete estimates
  7. Convergence to a dissipative weak solution
  8. Appendix
  9. Acknowledgment

Key Result

Lemma 2.2

For any nonnegative, absolutely continuous $f\in L^1({\mathcal{X}})$ with mass $M=\int_{\mathcal{X}} f\,\mathrm d x<\infty$ we write $\mu=f\cdot \mathrm d x$. Then

Figures (3)

  • Figure 1: Counterexample to displacement convexity
  • Figure 2: The $(\varepsilon,\delta)$ boundary layer
  • Figure 3: cylindrical exhaustion $(t_0,t_1)\times \Gamma_\eta$ for $\omega_{t_0},\omega_{t_1},|m_\Omega|$

Theorems & Definitions (25)

  • Definition 2.1: Fisher information AGS
  • Lemma 2.2: Properties of the Fisher information
  • Theorem 1
  • Conjecture 2
  • proof : Proof of Theorem \ref{['theo:slope_controls_Fisher']}
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 15 more