发表机构
Huazhong University of Science and Technology（华中科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明布朗驱动McKean-Vlasov方程的Euler-Maruyama格式转移密度以O(n^{-1/2})速率收敛，通过离散参数化展开与连续展开比较实现。
AI 中文摘要
本文建立了布朗驱动McKean-Vlasov随机微分方程的Euler-Maruyama格式的转移密度的收敛性，其中系数是一致椭圆的、空间光滑的，并且在2-Wasserstein距离下关于分布是Lipschitz的。证明了n步格式的密度p_n在Gaussian加权上确界范数下以O(n^{-1/2})的速率收敛到McKean-Vlasov方程的密度p，即关于步长为一阶半阶，因此也在空间上一致收敛。证明过程为Euler链发展了一个离散的参数化（Levi-Hadamard）展开，并将其与极限方程的连续参数化展开沿着测度流、时间离散化和格点修正的层级进行比较；离散与连续测度流之间的差异在2-Wasserstein距离下为O(n^{-1/2})阶，且不会降低最终速率。
英文摘要
Convergence of the transition density of the Euler-Maruyama scheme for a Brownian-driven McKean-Vlasov stochastic differential equation is established, the coefficients being uniformly elliptic, spatially smooth, and Lipschitz with respect to the law in the 2-Wasserstein distance. The density p_n of the n-step scheme is shown to converge to the density p of the McKean-Vlasov equation in a Gaussian-weighted supremum norm at rate O(n^{-1/2}), that is, of order one half in the step size, and hence also uniformly in space. The proof develops a discrete parametrix (Levi-Hadamard) expansion for the Euler chain and compares it with the continuous parametrix of the limiting equation along a hierarchy of measure-flow, time-discretisation and lattice corrections; the discrepancy between the discrete and the continuous measure flows is of order O(n^{-1/2}) in the 2-Wasserstein distance and does not degrade the final rate.
Comments60 pages
