黎曼流形上的混沌传播
Propagation of Chaos on Riemannian Manifolds
浏览论文内容
中文总结 AI 辅助
该论文将反射耦合方法推广到完备黎曼流形，实现均匀时间混沌传播，在端点指数和径向拉普拉斯界下获得速率$N^{-1/2}$的传播结果，并证明McKean--Vlasov半群的指数收缩与平衡收敛。
中文摘要 AI 辅助
我们将反射耦合从欧几里得空间扩展到完备黎曼流形上，用于均匀时间混沌传播。沿最小测地线的内在反射取代了欧几里得差分过程，相关的端点指数项量化了曲率、约束和相互作用之间的竞争。在端点指数界和径向拉普拉斯界下，我们建立了全局适定性、均匀矩估计以及修正Wasserstein距离下速率$N^{-1/2}$的均匀时间混沌传播。我们进一步获得了非线性McKean--Vlasov半群的指数收缩、在具有有限二阶矩的概率测度中不变概率测度的唯一性，以及向平衡的指数收敛。尽管Ricci下界是一个重要的特例，我们的框架也包括具有$\inf_M\operatorname{Ric}=-\infty$的曲率尖峰类。
英文摘要
We extend reflection coupling for uniform-in-time propagation of chaos from Euclidean space to complete Riemannian manifolds. Intrinsic reflection along minimizing geodesics replaces the Euclidean difference process, and the associated endpoint-index term quantifies the competition between curvature, confinement, and interaction. Under an endpoint-index bound and a radial Laplacian bound, we establish global well-posedness, uniform moment estimates, and uniform-in-time propagation of chaos with rate $N^{-1/2}$ in a modified Wasserstein distance. We further obtain exponential contraction of the nonlinear McKean--Vlasov semigroup, uniqueness of its invariant probability measure among probability measures with finite second moment, and exponential convergence to equilibrium. Although a Ricci lower bound is an important special case, our framework also includes a curvature-spike class with $\inf_M\operatorname{Ric}=-\infty$.
发表机构
- Hankyong National University（韩国女子大学）
- Texas State University（德克萨斯州立大学）
- Southern University of Science and Technology（南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。
