Stochastic Homogenization of HJ Equations: a Differential Game Approach
Andrea Davini, Raimundo Saona, Bruno Ziliotto
TL;DR
The paper develops a stochastic homogenization theory for a broad class of nonconvex, noncoercive first-order Hamilton–Jacobi equations in stationary ergodic environments, focusing on Hamiltonians that admit a max–min structure arising from differential-game theory. By representing solutions as two-player value functions and exploiting oriented dynamics plus finite-range dependence, it proves almost-sure homogenization with a deterministic effective Hamiltonian $\overline H$, and provides a quantitative convergence rate for linear initial data via a concentration inequality. The authors also extend the results to Lipschitz Hamiltonians not globally expressible in max–min form, using local max–min representations and a localization argument, and demonstrate stability under dense limits of Hamiltonians. These results broaden homogenization theory beyond coercive/convex settings and connect stochastic homogenization with differential-game methods, offering both qualitative and quantitative insights into the effective behavior of solutions as the microscale parameter $\varepsilon$ vanishes.
Abstract
We prove stochastic homogenization for a class of non-convex and non-coercive first-order Hamilton-Jacobi equations in a finite-range-dependence environment for Hamiltonians that can be expressed by a max-min formula. Exploiting the representation of solutions as value functions of differential games, we develop a game-theoretic approach to homogenization. We furthermore extend this result to a class of Lipschitz Hamiltonians that need not admit a global max-min representation. Our methods allow us to get a quantitative convergence rate for solutions with linear initial data toward the corresponding ones of the effective limit problem.