论文标题
与McKean-Vlasov方程相关的平均场相互作用粒子系统的长期行为
Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations
论文作者
论文摘要
在本文中，我们研究了泊松方程的梯度估计和Wasserstein Metric $ w_ {1，d_ {l^1}} $的指数收敛性，粒子数量均匀，以及与McKean-vlasov equagation相关的平均粒子相互作用的chaos均匀的Chaos均匀传播。通过已知的近似构图反射耦合并借助一些新的成本函数，我们获得了这三个问题的明确估计，避免了已知结果中的技术条件。当限制潜力$ v $有许多井时，我们的结果适用，交互潜力$ w $已限制了第二个混合衍生物$ \ nabla^2_ 2_ {xy} w $，这应该不太大，因此没有相位过渡。作为应用，我们获得了平均场相互作用粒子系统与明确和尖锐常数的浓度不平等，时间均匀。提供了几个示例来说明结果。
In this paper, we investigate gradient estimate of the Poisson equation and the exponential convergence in the Wasserstein metric $W_{1,d_{l^1}}$, uniform in the number of particles, and uniform-in-time propagation of chaos for the mean-field weakly interacting particle system related to McKean-Vlasov equation. By means of the known approximate componentwise reflection coupling and with the help of some new cost function, we obtain explicit estimates for those three problems, avoiding the technical conditions in the known results. Our results apply when the confinement potential $V$ has many wells, the interaction potential $W$ has bounded second mixed derivative $\nabla^2_{xy}W$ which should be not too big so that there is no phase transition. As an application, we obtain the concentration inequality of the mean-field interacting particle system with explicit and sharp constants, uniform in time. Several examples are provided to illustrate the results.
