具有奇异系数的McKean--Vlasov随机系统的长时间行为
Long-time behavior of McKean--Vlasov stochastic systems with singular coefficients
- School of Mathematics and Statistics, Jiangsu Normal University（江苏师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文为具有奇异系数的McKean--Vlasov随机系统建立了定量长时间行为框架，分离不变测度的存在性与唯一性，并通过锚定小性条件传递混合速率，应用于颗粒介质和Curie--Weiss模型以揭示相变与拓扑敏感性。
中文摘要 AI 辅助
我们为具有奇异系数和可能发生相变的McKean--Vlasov随机微分方程的长时间行为建立了一个定量框架。该框架将不变测度的存在性与唯一性分开处理。对于存在性，我们引入了一个带有单级俘获机制的广义Lyapunov条件，该条件仅要求在一个容许的矩水平上具有耗散性，而非全局收缩性。这允许具有超临界增长的分布依赖性，因此更适用于相变模型。对于唯一性和收敛性，我们建立了一个锚定唯一性和定量遍历性原理。一个显式的\(L^1\)-小性条件，通过由锚定冻结半群的导数估计构造的卷积核来表达，可推出唯一性，并将冻结动力学的指数或多项式混合速率传递给非线性McKean--Vlasov系统。此外，该原理对拓扑的选择敏感：不同的距离导致不同的扰动核，从而产生不同的稳定性阈值。局部版本在指定平衡点附近给出局部唯一性和定量吸引性。我们将该框架应用于两个代表性模型。对于非对称颗粒介质动力学，我们推导出两个显式的唯一性阈值，揭示了该理论的拓扑敏感性。对于动态Curie--Weiss模型，Wasserstein-1准则在临界等式处恢复了尖锐的分岔阈值。我们识别了所有不变测度，建立了相变区域中依赖于盆地的指数收敛性，并表明临界点处锚定小性的丧失导致多项式减速。
英文摘要
We develop a quantitative framework for the long-time behavior of McKean--Vlasov stochastic differential equations with singular coefficients and possible phase transitions. The framework separates the existence of invariant measures from their uniqueness. For existence, we introduce a generalized Lyapunov condition with a one-level trapping mechanism, which requires dissipativity at only one admissible moment level rather than global contraction. This permits distribution dependence with supercritical growth and thus is more compatible with phase-transition models. For uniqueness and convergence, we establish an anchored uniqueness and quantitative ergodicity principle. An explicit \(L^1\)-smallness condition, expressed through a convolution kernel built from derivative estimates of the anchored frozen semigroup, yields uniqueness and transfers exponential or polynomial mixing rates of the frozen dynamics to the nonlinear McKean--Vlasov system. Moreover, the principle is sensitive to the choice of topology: different distances lead to different perturbation kernels and hence different stability thresholds. A local version gives local uniqueness and quantitative attraction near a prescribed equilibrium. We apply the framework to two representative models. For non-symmetric granular media dynamics, we derive two explicit uniqueness thresholds that reveal the topology-sensitive nature of the theory. For the dynamical Curie--Weiss model, the Wasserstein-1 criterion recovers the sharp bifurcation threshold up to the critical equality. We identify all invariant measures, establish basin-dependent exponential convergence in the phase-transition regime, and show that the loss of anchored smallness at criticality leads to polynomial slowing down.
