Asymptotic behavior of Wasserstein distance for weighted empirical measures of diffusion processes on compact Riemannian manifolds

Abstract

Let $(X_t)_{t \geq 0}$ be a diffusion process defined on a compact Riemannian manifold, and for $α> 0$, let $$ μ_t^{(α)} = \fracα{t^α} \int_{0}^{t} δ_{X_s} \, s^{α- 1} \mathrm{d} s $$ be the associated weighted empirical measure. We investigate asymptotic behavior of $\mathbb{E}^ν\big[ \mathrm{W}_2^2(μ_t^{(α)}, μ) \big]$ for sufficient large $t$, where $\mathrm{W}_2$ is the quadratic Wasserstein distance and $μ$ is the invariant measure of the process. In the particular case $α= 1$, our result sharpens the limit theorem achieved in [26]. The proof is based on the PDE and mass transportation approach developed by L. Ambrosio, F. Stra and D. Trevisan.