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

Abstract: 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.