  We study the large time behavior of the optimal transportation cost towards
the uniform distribution, for the occupation measure of a stationary Brownian
motion on the flat torus in $d$ dimensions, where the cost of transporting a
unit of mass is given by a power of the flat distance. We establish a global
upper bound, in terms of the limit for the analogue problem concerning the
occupation measure of the Brownian interlacement on $\R^d$. We conjecture that
our bound is sharp and that our techniques may allow for similar studies on a
larger variety of problems, e.g. general diffusion processes on weighted
Riemannian manifolds.
