AI 中文总结
研究马尔可夫过程经验测度在切片瓦瑟斯坦距离下的无维收敛速率，采用特定概率测度诱导的 SW 距离，推导遍历马尔可夫过程在\(\BB\)上经验测度的无维收敛速率，避免“维度诅咒”，结果适用于广泛无限维模型。
AI 中文摘要
为了推导巴拿赫空间上马尔可夫过程经验测度的无维收敛速率，我们采用由对偶空间单位球上具有全支撑的概率测度诱导的切片瓦瑟斯坦距离（SW 距离）。该距离在拓扑上强于有限维分布收敛，在巴拿赫空间为有限维时与瓦瑟斯坦距离拓扑等价。在此距离下，我们推导了遍历马尔可夫过程在\(\BB\)上经验测度的无维收敛速率，具体例子表明其可能是精确的。该研究提供了一种利用马尔可夫过程样本轨迹模拟无限维分布的有效方法，避免了经典瓦瑟斯坦距离出现的“维度诅咒”。主要结果适用于广泛的无限维模型，文末通过部分耗散的随机偏微分方程进行了说明。
英文摘要
To derive dimension-free convergence rates of empirical measures for Markov processes on a Banach space, we adopt the sliced Wasserstein distance (SW distance) induced by a probability measure with full support on the unit ball of the dual space. This distance is topologically stronger than the convergence in finite-dimensional distributions, and is topologically equivalent to the Wasserstein distance when the Banach space is finite-dimensional. Under this distance, we derive dimension-free convergence rates for the empirical measures of ergodic Markov processes on $\BB$, which can be sharp as illustrated by concrete examples. The study provides an efficient way to simulate infinite-dimensional distributions using sample trajectories of Markov processes, so that the $``$curse of dimensionality" appearing to the classical Wasserstein distance is avoided. The main results apply to a broad class of infinite-dimensional models, and are illustrated by partially dissipative SPDEs in the end of the paper.
Comments40 pages
