Abstract: The optimal matching problem is at the interface of analysis and probability theory. Among its different formulations we consider the problem of studying the asymptotic of the transportation cost between the occupation measure of a fractional Brownian motion taking values on a d-dimensional torus and the Lebesgue measure restricted to it. A similar problem has recently been studied for Markovian Gaussian 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 results with the ones already known.