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

Wasserstein Asymptotics for Brownian Motion on the Flat Torus and Brownian Interlacements