Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus
Abstract
We establish asymptotic upper and lower bounds for the Wasserstein distance of any order $p\ge 1$ between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index $H$ and the dimension $d$ of the state space, with a "phasetransition" in the rates when $d=2+1/H$, akin to the AjtaiKomlósTusnády theorem for the optimal matching of i.i.d. points in twodimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discretetime approximations of the process, as well as a lower bound for the same problem on $\mathbb{R}^d$.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 arXiv:
 arXiv:2205.01025
 Bibcode:
 2022arXiv220501025H
 Keywords:

 Mathematics  Probability;
 Mathematics  Analysis of PDEs
 EPrint:
 Comments very welcome