From the observation of a diffusion path \((X_t)_{t\in [0,T]}\) on a compact connected d-dimensional manifold \(\mathcal {M}\) without boundary, we consider the problem of estimating the stationary measure \(\mu \) of the process. Wang and Zhu (2023) showed that for the Wasserstein metric \(\mathcal {W}_2\) and for \(d\ge 5\) , the convergence rate of \(T^{-1/(d-2)}\) is attained by the occupation measure of the path \((X_t)_{t\in [0,T]}\) when \((X_t)_{t\in [0,T]}\) is a Langevin diffusion. We extend their result in several directions. First, we show that the rate of convergence holds for a large class of diffusion paths, whose generators are uniformly elliptic. Second, the regularity of the density p of the stationary measure \(\mu \) with respect to the volume measure of \(\mathcal {M}\) can be leveraged to obtain faster estimators: when p belongs to a Sobolev space of order \(\ell \ge 2\) , smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order \(T^{-(\ell +1)/(2\ell +d-2)}\) . We further show that this rate is the minimax rate of estimation for this problem.
