2026/07/22 by Feng-Yu Wang
#math.PR
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.