Asymptotic Behavior of Wasserstein Distance for Weighted Empirical Measures of Diffusion Processes on Compact Riemannian Manifolds

J.-X. Zhu

2024, v.30, Issue 3, 357-397

ABSTRACT

Let $(X_t)_{t \ge 0}$ be a diffusion process defined on a compact connected Riemannian manifold $M$, and for $\alpha > 0$, let
$$
\mu_t^{(\alpha)} = \frac{\alpha}{t^\alpha} \int_{0}^{t} \delta_{X_s} \, s^{\alpha - 1} \mathrm{d} s
$$
be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^\nu \big[ \mathrm{W}_2^2(\mu_t^{(\alpha)}, \mu) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance, $\nu$ is the law of $X_0$ and $\mu$ is the invariant measure of the process. In the particular case $\alpha = 1$, our result sharpens the limit theorem achieved in \cite{WZ19}. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.


doi:10.61102/1024-2953-mprf.2024.30.3.001

Keywords: Empirical measure, diffusion process, Riemannian manifold, Wasserstein distance

COMMENTS

Please log in or register to leave a comment