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

TL;DR

This paper investigates the asymptotic behavior of the Wasserstein distance between the empirical measure of fractional Brownian motion on a flat torus and the uniform measure, revealing phase transitions influenced by the Hurst index and dimension.

Contribution

It provides new bounds for Wasserstein distances involving fractional Brownian motion, highlighting a phase transition phenomenon and combining PDE and probabilistic methods.

Findings

Identifies phase transition in Wasserstein rates at dimension d=2+1/H.

Establishes bounds for empirical measures of fractional Brownian motion.

Extends results to discrete-time approximations and lower bounds on Euclidean space.

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 "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 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 $\mathbb{R}^d$.