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

TL;DR

This paper investigates the asymptotic behavior of optimal transportation costs for Brownian motion occupation measures on flat tori, establishing bounds linked to Brownian interlacements and suggesting broader applicability to diffusion processes.

Contribution

It provides new bounds for transportation costs of Brownian occupation measures and proposes a conjecture on their sharpness, extending potential analysis to other diffusion processes.

Findings

Established a global upper bound for transportation costs.

Linked the bounds to Brownian interlacements on .

Suggested the bounds are sharp and applicable to broader processes.

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.