# Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus

@inproceedings{Huesmann2022WassersteinAF, title={Wasserstein Asymptotics for the Empirical Measure of Fractional Brownian Motion on a Flat Torus}, author={Martin Huesmann and Francesco Mattesini and Dario Trevisan}, year={2022} }

. We establish asymptotic upper and lower bounds for the Wasserstein distance of any order p ≥ 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 “phase-transition” in the rates when d = 2 + 1 /H , akin to the Ajtai-Koml´os-Tusn´ady theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof…

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