发表机构
Univ Angers; CNRS; LAREMA; SFR MATHSTIC（昂热大学; 法国国家科学研究中心; 昂热数学与应用数学实验室; 数学与科学计算联合研究结构）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究约束平均场Langevin动力学的Wasserstein梯度流，证明其适定性、耗散性、指数收敛及传播混沌，并给出具体应用。
AI 中文摘要
延续文献[1, 2, 3]的工作，我们研究一类受能量约束的约束平均场Langevin动力学，将其表述为带能量约束的Wasserstein梯度流。我们证明，该约束演化在Wasserstein空间P_2(R^d)中定义了一个适定的耗散动力学。我们证明该流保持概率质量和能量约束，单调耗散自由能，并在测地凸性假设下指数收敛到唯一的约束平衡。利用同步耦合，在Lipschitz假设下，我们证明了具有Fournier-Guillin界的定量L^2-传播混沌，并且在充分耗散条件下，我们推导出显式的时间一致界。我们还证明了在W_2 Wasserstein度量下的指数控制以及不变测度映射的收缩性。利用log-Sobolev不等式，我们证明了平均场熵沿约束流指数衰减。这些结果应用于几个具体且新颖的例子。关键词和短语：Wasserstein梯度流、平均场Langevin动力学、约束非线性Fokker-Planck方程、拉格朗日乘子、JKO格式、传播混沌、同步耦合、指数遍历性、泛函不等式、log-Sobolev不等式、传输不等式、Talagrand T_2不等式。
英文摘要
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
