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 $d\geq 5$, 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 $\ell\geq 2$, 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.

翻译：通过观测无边界紧连通d维流形$\mathcal{M}$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与Zhu（2023）的研究表明，当$(X_t)_{t\in [0,T]}$为朗之万扩散时，对于Wasserstein度量$\mathcal{W}_2$及$d\geq 5$的情形，路径$(X_t)_{t\in [0,T]}$的占据测度能达到$T^{-1/(d-2)}$的收敛速率。我们在多个方向上拓展了他们的结果。首先，我们证明该收敛速率适用于一大类扩散路径，其生成算子满足一致椭圆性条件。其次，平稳测度$\mu$相对于流形$\mathcal{M}$体积测度的密度函数$p$的正则性可被用于构造更快速的估计量：当$p$属于阶数$\ell\geq 2$的Sobolev空间时，通过核卷积对占据测度进行平滑处理所得的估计量，其收敛速率可达$T^{-(\ell+1)/(2\ell+d-2)}$阶。我们进一步证明该速率即为该估计问题的最小最大最优速率。