Asymptotics of transportation cost for the occupation measure of fractional Brownian motion

We establish sharp upper and lower bounds for the Kantorovich optimal transport distance between the uniform measure and the occupation measure of a path of a fractional Brownian motion with Hurst index H taking values in a d-dimensional torus. Similar problems have been recently studied for diffusion processes taking values on a compact connected Riemmanian manifold. We give new insights in the case of fractional Brownian motion taking care of the absence of the Markovian structure by means of recently introduced PDE techniques and compare our result with the ones already known. In particular we show that a phase transition between rates occurs if d=1/H +2, in analogy with the random Euclidean bipartite matching problem, i.e. when the occupation measure is replaced by i.i.d. uniform points (formally given by infinite H).

Joint work with M. Huesmann (WWU Münster) and D. Trevisan (Università degli studi di Pisa)