Quantitative homogenization of convex Hamilton-Jacobi equations in the Wasserstein space

Abstract

We study a homogenization problem for first-order Hamilton-Jacobi equations in the Wasserstein space with a convex Hamiltonian. We show that the solution $U^\varepsilon$, which is the value function of a mean field control problem, converges uniformly as $\varepsilon \to 0$ to the solution of a limiting Hamilton-Jacobi equation whose Hamiltonian is obtained through a suitable cell problem. Furthermore, we establish quantitative rates of convergence. Under general assumptions with multiscale dependence, we prove that the rate of convergence is $O(\sqrt{\varepsilon})$. When the Hamiltonian depends only on the fast variable and the momentum, we establish the sharp convergence rate $O(\varepsilon)$. To the best of our knowledge, this is the first quantitative convergence result extending the optimal rate for first-order Hamilton-Jacobi equations in finite dimensions to the Wasserstein space. Finally, we show that our analysis extends to dynamic optimal transport problems, where the terminal condition imposes a constraint on the final distribution.