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

Let $(X_t{)}_{t\ge 0}$ be a diffusion process defined on a compact Riemannian manifold, 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 ${\mathbb{E}}^{\nu }[{\mathrm{W}}_2^2({\mu }_t^{(\alpha )},\mu )]$ for sufficient large $t$, where ${\mathrm{W}}_2$ is the quadratic Wasserstein distance and $\mu$ is the invariant measure of the process. In the particular case $\alpha =1$, 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.