January 31, 2024 | Pierre Monmarché¹, Zhenjie Ren², and Songbo Wang³
This paper investigates time-uniform log-Sobolev inequalities (LSI) for solutions of semilinear mean-field equations and their applications to propagation of chaos (PoC). The authors establish two methods to derive time-uniform LSI: one using bounded-Lipschitz perturbation with respect to a reference measure, and another using a coupling approach at high temperature. These results are then applied to mean-field equations, yielding sharp marginal PoC estimates in smooth cases and time-uniform global PoC in cases of vortex interactions with quadratic confinement. The paper also addresses the question of uniform-in-time LSI and PoC for singular (log or Riesz) interactions in $ \mathbb{R}^d $.
The first method, based on the bounded-Lipschitz perturbation argument, applies to drifts that can be decomposed into a constant and a perturbation with bounded derivatives. The second method, applicable at high temperature, requires that the drift satisfies certain curvature conditions. The authors show that under these conditions, the family of solutions satisfies a uniform LSI, which in turn implies uniform PoC estimates.
The paper also extends the results of Lacker and Le Flem to more general cases, including non-convex potentials on $ \mathbb{R}^d $. The key idea is to use the uniform LSI to derive uniform PoC estimates, which are then applied to various models, including those with log and Riesz interactions. The authors demonstrate that for certain types of interactions, such as those with strong convexity outside a compact set, the uniform PoC estimates hold even when the temperature is high or the interaction is weak.
The paper concludes with a discussion of the implications of these results for the study of mean-field equations and the propagation of chaos in high-dimensional systems. The results provide a framework for understanding the long-time behavior of such systems and highlight the importance of uniform LSI in establishing uniform PoC estimates.This paper investigates time-uniform log-Sobolev inequalities (LSI) for solutions of semilinear mean-field equations and their applications to propagation of chaos (PoC). The authors establish two methods to derive time-uniform LSI: one using bounded-Lipschitz perturbation with respect to a reference measure, and another using a coupling approach at high temperature. These results are then applied to mean-field equations, yielding sharp marginal PoC estimates in smooth cases and time-uniform global PoC in cases of vortex interactions with quadratic confinement. The paper also addresses the question of uniform-in-time LSI and PoC for singular (log or Riesz) interactions in $ \mathbb{R}^d $.
The first method, based on the bounded-Lipschitz perturbation argument, applies to drifts that can be decomposed into a constant and a perturbation with bounded derivatives. The second method, applicable at high temperature, requires that the drift satisfies certain curvature conditions. The authors show that under these conditions, the family of solutions satisfies a uniform LSI, which in turn implies uniform PoC estimates.
The paper also extends the results of Lacker and Le Flem to more general cases, including non-convex potentials on $ \mathbb{R}^d $. The key idea is to use the uniform LSI to derive uniform PoC estimates, which are then applied to various models, including those with log and Riesz interactions. The authors demonstrate that for certain types of interactions, such as those with strong convexity outside a compact set, the uniform PoC estimates hold even when the temperature is high or the interaction is weak.
The paper concludes with a discussion of the implications of these results for the study of mean-field equations and the propagation of chaos in high-dimensional systems. The results provide a framework for understanding the long-time behavior of such systems and highlight the importance of uniform LSI in establishing uniform PoC estimates.