Wasserstein Convergence for Conditional Empirical Measures of Subordinated Dirichlet Diffusions on Riemannian Manifolds

TL;DR

This paper studies the convergence rate of conditional empirical measures for subordinated Dirichlet diffusions on Riemannian manifolds under Wasserstein distance, extending previous work to a non-local setting with boundary conditions.

Contribution

It establishes the sharp convergence rate and the precise limit for conditional empirical measures in a non-local diffusion setting on manifolds, building upon and deviating from Wang's framework.

Findings

Established the sharp rate of convergence for conditional empirical measures.

Proved the precise limit for a large class of initial distributions.

Extended the analysis to non-local subordinated Dirichlet diffusions with boundary.

Abstract

The asymptotic behaviour of empirical measures has plenty of studies. However, the research on conditional empirical measures is limited. Being the development of Wang \cite{eW1}, under the quadratic Wasserstein distance, we investigate the rate of convergence of conditional empirical measures associated to subordinated Dirichlet diffusion processes on a connected compact Riemannian manifold with absorbing boundary. We give the sharp rate of convergence for any initial distribution and prove the precise limit for a large class of initial distributions. We follow the basic idea of Wang, but allow ourselves substantial deviations in the proof to overcome difficulties in our non-local setting.