  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 $\mathbb{R}^d$.
