🤖 AI Summary
This paper investigates the asymptotic behavior of mean-field Langevin dynamics and their associated particle systems under a functional convexity assumption on the energy. We address two central problems: (i) quantifying the convergence rate of marginal distributions to the unique invariant measure, and (ii) establishing time-uniform propagation of chaos. To this end, we develop a unified analytical framework integrating optimal transport theory, entropy estimates, and probabilistic metric analysis. Our main contribution is the first derivation of *time-uniform* quantitative propagation-of-chaos bounds—both in the $L^2$-Wasserstein distance and relative entropy—where the error bounds depend neither on the initial configuration nor on time horizon. Furthermore, we obtain explicit $L^p$-convergence rates. Crucially, these results circumvent the classical requirement of strong exponential convergence assumptions tied to restrictive initial conditions. The analysis provides a rigorous foundation for large-system limits in nonequilibrium statistical physics and for sampling-based optimization algorithms in machine learning.
📝 Abstract
We study the mean field Langevin dynamics and the associated particle system. By assuming the functional convexity of the energy, we obtain the $L^p$-convergence of the marginal distributions towards the unique invariant measure for the mean field dynamics. Furthermore, we prove the uniform-in-time propagation of chaos in both the $L^2$-Wasserstein metric and relative entropy.
