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

Mauro Mariani, Dario Trevisan

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.

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 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 . 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.
Paper Structure (28 sections, 30 theorems, 484 equations)

This paper contains 28 sections, 30 theorems, 484 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Comments on the proof technique
  4. Further questions and conjectures
  5. Structure of the paper
  6. Notation and Basic facts
  7. General notation
  8. Occupation measures and hitting times
  9. Negative Sobolev norms
  10. Total variation distance
  11. Optimal transport
  12. Concentration inequalities
  13. Brownian motion
  14. Brownian motion on $\mathbb{R}^d$
  15. Brownian motion on $\mathbb{T}^d$
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $d \in \left\{ 3, 4 \right\}$ and $p \in (0, (d-2)/2)$, or $d \ge 5$ and $p>0$. Then, there exists a constant $\mathsf{c}(\mathcal{I}, d,p) \in (0, \infty)$ such that the following holds. Let $\Omega\subseteq \mathbb{R}^d$ be a bounded connected domain with $C^2$ boundary (or $\Omega = Q$ be a

Theorems & Definitions (58)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4: The case of a manifold
  • Remark 2.5: Duality and the concave case
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • ...and 48 more