Moreau-Yosida Penalization of McKean-Vlasov Dynamics: Full-Flux Limits and Boundary Reaction Measures
Abstract
We study McKean-Vlasov Fokker-Planck dynamics on a compact convex set K⊂ℝd through the whole-space Moreau-Yosida penalization Uλ=dist(·,K)2/(2λ), keeping the external and interaction potentials fixed. At finite times, the penalized densities converge to the unique variational solution of the reflected no-flux equation, strongly in space-time L2 on K and in whole-space L1, and narrowly, uniformly in time. We prove that the singular penalty-force measures converge weakly-* to the purely normal boundary reaction Tr(ρ)n ℋd−1|∂ K dt. This measure cancels exactly the surface singularity in the distributional gradient of the zero-extended density, explaining the emergent no-flux condition at the flux level. At equilibrium, we establish sharp √λ rates in L1 and Wasserstein distances for kernel-free Gibbs states, and in L1 and quadratic Wasserstein distance for interacting fixed points under Dobrushin smallness or positive semidefiniteness, including maximum mean discrepancy interactions. The proof combines a tilted Gaussian supersolution, nonautonomous weighted energy estimates, and normal factorization with a Green-formula identification. We also identify the limit with a nonlinear reflecting diffusion, obtain path-space convergence and λ-uniform propagation of chaos through monotone synchronous coupling, and derive all-time Gibbs domination in the reversible case. Numerical experiments illustrate the boundary reaction and stationary scalings on a constrained maximum mean discrepancy flow.
Related
- From Transition-State Geometry to Gap Times: What Lagrangian Betweenness Measures in Chemical Reaction Dynamics
- Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous Discretizations
- Travelling waves in nonlocal reaction-dispersion equations with diffuse Lévy measures
- Boundary-layer asymptotics for Gaussian-smoothed singular measures
- Large Deviations of Invariant Measures of Fractional Reaction-Diffusion Equations Driven by Multiplicative Noise on R^n
BibTeX
@misc{seo2026moreauyosida,
title = {{Moreau-Yosida Penalization of McKean-Vlasov Dynamics: Full-Flux Limits and Boundary Reaction Measures}},
author = {Geuntaek Seo and Beomjun Choi and Yeoneung Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25025},
eprint = {2610.25025},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25025},
}
RIS
TY - UNPB
TI - Moreau-Yosida Penalization of McKean-Vlasov Dynamics: Full-Flux Limits and Boundary Reaction Measures
AU - Geuntaek Seo
AU - Beomjun Choi
AU - Yeoneung Kim
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25025
UR - https://arxiv.org/abs/2610.25025
AB - We study McKean-Vlasov Fokker-Planck dynamics on a compact convex set K⊂ℝ^d through the whole-space Moreau-Yosida penalization U_λ=dist(·,K)^2/(2λ), keeping the external and interaction potentials fixed. At finite times, the penalized densities converge to the unique variational solution of the reflected no-flux equation, strongly in space-time L^2 on K and in whole-space L^1, and narrowly, uniformly in time. We prove that the singular penalty-force measures converge weakly-* to the purely normal boundary reaction Tr(ρ)n ℋ^d−1|_∂ K dt. This measure cancels exactly the surface singularity in the distributional gradient of the zero-extended density, explaining the emergent no-flux condition at the flux level. At equilibrium, we establish sharp √λ rates in L^1 and Wasserstein distances for kernel-free Gibbs states, and in L^1 and quadratic Wasserstein distance for interacting fixed points under Dobrushin smallness or positive semidefiniteness, including maximum mean discrepancy interactions. The proof combines a tilted Gaussian supersolution, nonautonomous weighted energy estimates, and normal factorization with a Green-formula identification. We also identify the limit with a nonlinear reflecting diffusion, obtain path-space convergence and λ-uniform propagation of chaos through monotone synchronous coupling, and derive all-time Gibbs domination in the reversible case. Numerical experiments illustrate the boundary reaction and stationary scalings on a constrained maximum mean discrepancy flow.
ER -