Article Dans Une Revue Probability Theory and Related Fields Année : 2025

Measure estimation on a manifold explored by a diffusion process

Résumé

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.

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Dates et versions

hal-05311560 , version 1 (13-10-2025)

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Vincent Divol, Hélène Guérin, Dinh-Toan Nguyen, Viet Chi Tran. Measure estimation on a manifold explored by a diffusion process. Probability Theory and Related Fields, 2025, ⟨10.1007/s00440-025-01437-x⟩. ⟨hal-05311560⟩
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