A constrained Wasserstein gradient flow for mean-field Langevin dynamics

In the continuity of [1, 2, 3], we study a class of constrained mean-field Langevin dynamics formulated as a Wasserstein gradient flow with an energy constraint. We show that the constrained evolution therefore defines a well-posed dissipative dynamics in the Wasserstein space ${\mathcal{P}}_2({\mathbb{R}}^d)$. We prove that the flow preserves probability mass and the energy constraint, dissipates the free energy monotonically, and converges exponentially to a unique constrained equilibrium under geodesic convexity assumptions. Using synchronous coupling, under Lipschitz assumptions, we demonstrate the quantitative $L^2$-propagation of chaos with a Fournier-Guillin bound, and under conditions of sufficient dissipativity, we derive an explicit time-uniform bound. We also demonstrate exponential controls in the ${\mathcal{W}}_2$ Wasserstein metric and the contraction of the invariant measure map. Using log-Sobolev inequalities, we show that the mean-field entropies decay exponentially along the constrained flow. The results are applied to several concrete and original examples. Keywords and phrases: Wasserstein gradient flows $\bullet$ mean-field Langevin dynamics $\bullet$ constrained nonlinear Fokker-Planck equations $\bullet$ Lagrange multiplier $\bullet$ JKO scheme $\bullet$ propagation of chaos $\bullet$ synchronous coupling $\bullet$ exponential ergodicity $\bullet$ functional inequality $\bullet$ log-Sobolev inequality $\bullet$ transport inequality $\bullet$ Talagrand T 2 -inequality