Polynomial Convergence Rate for Quasi-Periodic Homogenization of Hamilton-Jacobi Equations and Application to Ergodic Estimates

TL;DR

This paper establishes a polynomial convergence rate for the homogenization of Hamilton-Jacobi equations with quasi-periodic potentials, linking it to the regularity of the effective Hamiltonian and providing new ergodic estimates.

Contribution

It introduces a novel quantitative ergodic estimate for quasi-periodic functions and connects convergence rates to Hamiltonian regularity in homogenization.

Findings

Polynomial convergence rate for quasi-periodic homogenization

New ergodic estimate for bounded quasi-periodic functions

Application to Birkhoff averages of unbounded functions

Abstract

In this paper, we demonstrate a polynomial convergence rate for homogenization of Hamilton-Jacobi equations with quasi-periodic potentials. We establish a connection between the convergence rate of homogenization and the regularity of the effective Hamiltonian, by using a new quantitative ergodic estimate for bounded quasi-periodic functions with Diophantine frequencies. As an application, we also study the convergent rate for Birkhoff average of unbounded quasi-periodic functions.