Dimension-free Convergence Rate in Sliced Wasserstein Distance for Empirical Measures of Markov Processes

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.