Moreau-Yosida Penalization of McKean-Vlasov Dynamics: Full-Flux Limits and Boundary Reaction Measures

Source abstract

We study McKean-Vlasov Fokker-Planck dynamics on a compact convex set $K\subset\mathbb{R}^d$ through the whole-space Moreau-Yosida penalization $U_λ=\operatorname{dist}(\cdot,K)^2/(2λ)$, 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 $\operatorname{Tr}(ρ)n\,\mathcal{H}^{d-1}|_{\partial K}\,\mathrm{d}t$. 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 $\sqrtλ$ 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.