From the observation of a diffusion path $(X_t)_{t\in [0,T]}$ on a compact connected $d$-dimensional manifold $M$ 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 $W_2$ 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 $M$ can be leveraged to obtain faster estimators: when $p$ belongs to a Sobolev space of order $\ell>0$, 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.

翻译：通过观测在无边界紧致连通 $d$ 维流形 $M$ 上的扩散路径 $(X_t)_{t\in [0,T]}$，我们考虑估计该过程平稳测度 $\mu$ 的问题。Wang 和 Zhu (2023) 的研究表明，对于 Wasserstein 度量 $W_2$ 且当 $d\ge 5$ 时，路径 $(X_t)_{t\in [0,T]}$ 的占位测度可以达到 $T^{-1/(d-2)}$ 的收敛速率，其中 $(X_t)_{t\in [0,T]}$ 为朗之万扩散。我们在多个方向上扩展了他们的结果。首先，我们证明该收敛速率适用于一大类扩散路径，其生成算子是一致椭圆的。其次，可以利用平稳测度 $\mu$ 相对于流形 $M$ 体积测度的密度函数 $p$ 的正则性来获得更快的估计器：当 $p$ 属于阶数为 $\ell>0$ 的 Sobolev 空间时，通过用核函数对占位测度进行卷积平滑，可以得到收敛速率为 $T^{-(\ell+1)/(2\ell+d-2)}$ 的估计器。我们进一步证明该速率是此估计问题的最小最大速率。