A Near-Optimal Rate of Periodic Homogenization for Convex Hamilton–Jacobi Equations

We consider a Hamilton–Jacobi equation where the Hamiltonian is periodic in space and coercive and convex in momentum. Combining the representation formula from optimal control theory and a theorem of Alexander, originally proved in the context of first-passage percolation, we find a rate of homogenization which is within a log-factor of optimal and holds in all dimensions.