Wasserstein convergence rates in the invariance principle for deterministic dynamical systems

TL;DR

This paper investigates the rate at which deterministic hyperbolic systems and related models converge in Wasserstein distance within the invariance principle, including applications to homogenization and stochastic differential equations.

Contribution

It establishes Wasserstein convergence rates for a broad class of hyperbolic systems and applies these results to analyze homogenization in fast-slow deterministic systems.

Findings

Wasserstein convergence rates are derived for hyperbolic systems.

Results include applications to intermittent maps and Lorentz gases.

Convergence rates are applied to homogenization problems.

Abstract

In this paper, we consider the convergence rate with respect to Wasserstein distance in the invariance principle for deterministic nonuniformly hyperbolic systems, where both discrete time systems and flows are included. Our results apply to uniformly hyperbolic systems and large classes of nonuniformly hyperbolic systems including intermittent maps, Viana maps, finite horizon planar periodic Lorentz gases and others. Furthermore, as a nontrivial application to homogenization problem, we investigate the $\mathcal{W}_2$-convergence rate of a fast-slow discrete deterministic system to a stochastic differential equation.