Optimal Convergence Rate for Periodic Homogenization of Rearrangement-Invariant Convex Hamilton--Jacobi Equations in Infinite Dimensions

Abstract

We prove the optimal convergence rate $O(\varepsilon)$ for periodic homogenization of convex Hamilton-Jacobi equations arising from infinite systems of indistinguishable particles on the torus, under the assumption that the initial data depend only on the mean configuration. This extends the finite-dimensional result [1], which is based on the large-time behavior of the Lagrangian action metric and a curve-surgery argument. Here, these tools cannot be applied directly because minimizing curves live in an infinite-dimensional Hilbert space, where local compactness and finite-dimensional topology are unavailable. We overcome this difficulty by cutting the finite-dimensional mean-time projection of a minimizing curve and gluing the lifted pieces in the Hilbert space using the compact quotient induced by periodicity and rearrangement invariance. We conclude with an example showing that this rate is sharp.