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arXiv 2610.13655math.PR
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线性稳定McKean-Vlasov平衡点附近粒子系统的亚稳约束

Metastable confinement of particle systems near linearly stable McKean-Vlasov equilibria

École normale supérieure, Université PSL（巴黎高等师范学院，巴黎文理研究大学）；Université Paris-Dauphine, PSL（巴黎第九大学，巴黎文理研究大学）

Samuel Chan-ashing, Quentin Cormier

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AI总结：

该研究建立通用亚稳性原理，证明局部指数稳定的McKean-Vlasov平衡点可使粒子系统被指数长时间约束，方法不依赖梯度流结构或可逆性。

摘要：

我们建立了一个通用的亚稳性原理，证明McKean-Vlasov平衡点的局部稳定性意味着相关粒子系统会被指数长时间地约束。考虑在ℝᵈ上的相互作用扩散过程，其具有一般的光滑有界相互作用核和在大距离处具有耗散性的全局Lipschitz漂移项。假设McKean-Vlasov方程至少存在一个局部指数稳定的平衡点，我们证明在时间T之前，经验测度离开所选平衡点的足够小的固定Wasserstein-1邻域的概率被C(1+T)e^(-cN)界定，其中N是粒子数，C、c>0且与N和T无关。因此，对于每个0<γ<c，约束可维持至时间e^(γN)，且退出概率呈指数级小。该证明结合了线性化Fokker-Planck半群的显式Volterra构造、经验涨落的温和表示以及随机卷积的指数极大值估计。我们的方法不要求方程具有梯度流结构或可逆性。

英文摘要：

We establish a general metastability principle showing that the local stability of a McKean-Vlasov equilibrium implies exponentially long confinement of the associated particle system. Consider interacting diffusions on $\mathbb{R}^d$ with a general smooth and bounded interaction kernel and a globally Lipschitz drift that is dissipative at large distances. Assume the McKean-Vlasov equation has at least one equilibrium which is locally exponentially stable. We prove that the probability that the empirical measure leaves a sufficiently small fixed Wasserstein-1 neighborhood of the selected equilibrium before time $T$ is bounded by $C(1+T)e^{-cN}$, where $N$ is the number of particles and $C,c>0$ are independent of $N$ and $T$. Consequently, confinement holds up to times $e^{γN}$, for every $0<γ<c$, with exponentially small exit probability. The proof combines an explicit Volterra construction of the linearized Fokker-Planck semigroup, a mild representation of empirical fluctuations, and exponential maximal estimates for stochastic convolutions. Our approach does not require a gradient flow structure or reversibility of the equations.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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