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

2022/05/02 by Martin Huesmann, Francesco Mattesini, Huesmann, Martin +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2205.01025

openalex publication_date 2022/05/02 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28

Abstract

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-Komlós-Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on ℝ^d.

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