Measure estimation on a manifold explored by a diffusion process

From the observation of a diffusion path (Xt)t∈[0,T] on a compact connected d-dimensional manifold M without boundary, we consider the problem of estimating the stationary measure μ of the process. Wang and Zhu (2023) showed that for the Wasserstein metric W2 and for d≥5, the convergence rate of T-1/(d-2) is attained by the occupation measure of the path (Xt)t∈[0,T] when (Xt)t∈[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 μ with respect to the volume measure of M can be leveraged to obtain faster estimators: when p belongs to a Sobolev space of order ℓ≥2, smoothing the occupation measure by convolution with a kernel yields an estimator whose rate of convergence is of order T-(ℓ+1)/(2ℓ+d-2). We further show that this rate is the minimax rate of estimation for this problem.