论文标题
慢速麦基 - 弗拉索夫方程的大量功能定律和中央限制定理
Functional law of large numbers and central limit theorem for slow-fast McKean-Vlasov equations
论文作者
论文摘要
在本文中，我们研究了完全耦合的慢速麦凯恩 - 维拉索夫随机系统的渐近行为。使用瓦斯汀空间上的非线性泊松方程，我们首先在大型类型的功能定律的平均原理中建立了强收敛。特别是，缓慢过程的扩散系数可以取决于快速运动的分布。然后，我们考虑原始系统围绕其平均值的随机波动，并证明归一化差异将弱收敛到线性McKean-Vlasov Ornstein-Uhlenbeck类型过程，可以将其视为功能性中心极限定理。明确表征涉及期望的额外漂移和扩散系数。此外，还获得了收敛的最佳速率。
In this paper, we study the asymptotic behavior of a fully-coupled slow-fast McKean-Vlasov stochastic system. Using the non-linear Poisson equation on Wasserstein space, we first establish the strong convergence in the averaging principle of the functional law of large numbers type. In particular, the diffusion coefficient of the slow process can depend on the distribution of the fast motion. Then we consider the stochastic fluctuations of the original system around its average, and prove that the normalized difference will converge weakly to a linear McKean-Vlasov Ornstein-Uhlenbeck type process, which can be viewed as a functional central limit theorem. Extra drift and diffusion coefficients involving the expectation are characterized explicitly. Furthermore, the optimal rates of the convergence are also obtained.
