Periodic Homogenization of Hamilton-Jacobi Equations for Infinite Systems of Indistinguishable Particles

TL;DR

This paper establishes homogenization results for first-order Hamilton-Jacobi equations in infinite-dimensional spaces, characterizing the effective Hamiltonian and providing convergence rates for systems of infinitely many particles.

Contribution

It extends homogenization theory to infinite-dimensional Hamilton-Jacobi equations with nonconvex Hamiltonians, including quantitative convergence rates.

Findings

Characterized the effective Hamiltonian via a cell problem.

Proved solutions converge at rate O(ε^{1/3}) to the homogenized limit.

Addressed nonconvex Hamiltonians in infinite-dimensional settings.

Abstract

We study the homogenization of first-order Hamilton-Jacobi equations on an infinite-dimensional Hilbert space, motivated by systems of infinitely many indistinguishable particles on the torus. A central difficulty is that the analysis takes place in an infinite-dimensional setting, where the compactness arguments available in finite dimensions break down. The problem is further complicated by the possible nonconvexity of the Hamiltonian, which prevents the direct use of variational methods. Under suitable assumptions on the Hamiltonian and the initial data, we characterize the effective Hamiltonian through an associated cell problem and prove that the solutions converge to those of the limiting equation at rate $O(\varepsilon^{1/3})$. This yields a qualitative and quantitative homogenization result for a class of possibly nonconvex Hamilton-Jacobi equations in infinite dimensions.