Measure estimation on a manifold explored by a diffusion process

• Published in: arXiv.org 2024-10
• Main Authors: Divol, Vincent, Guérin, Hélène, Dinh-Toan Nguyen, Tran, Viet Chi
• Format: Article
• Language: English
• Subjects: Convergence; Diffusion rate; Estimation; Sobolev space
• ISSN: 2331-8422

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.