Correctors, gradient-dependent diffusion, and quantitative rates for the stochastic homogenization of viscous Hamilton–Jacobi equations in stationary ergodic media

Lions and Souganidis (2005) proved almost-sure homogenization of "viscous" Hamilton–Jacobi equations in stationary ergodic media — equations of the form $u^{\varepsilon}_t - \varepsilon\,\operatorname{tr}(A\,D^2u^{\varepsilon}) + H(Du^{\varepsilon}, \varepsilon^{-1}x, \omega) = 0$ whose coefficients oscillate on scale $\varepsilon$ while the viscosity amplitude vanishes at the same critical rate. Following the subsequent literature, one of the four issues raised there — whether the structural growth condition used to obtain uniform gradient bounds is necessary — is settled (it is not, for convex Hamiltonians: the proof of Armstrong and Tran requires only an averaged, local coercivity and growth condition), and the one-dimensional nonconvex theory is essentially complete. Three precise open cores survive and constitute the target: (i) a general criterion for the existence of global correctors, that is, sublinearly growing solutions of the macroscopic (cell) problem, currently available only at extreme points of convexified sublevel sets of the effective Hamiltonian and in special environment laws; (ii) homogenization with gradient-dependent diffusion $A(p,\cdot,\omega)$ — the class of geometric front-motion equations such as forced mean curvature motion — under general stationary ergodic laws, beyond the finite-range-of-dependence and radially symmetric cases handled to date; and (iii) quantitative convergence rates and constructive evaluation of the effective Hamiltonian under mixing assumptions weaker than finite range of dependence.

Background

A Hamilton–Jacobi equation is a first-order nonlinear partial differential equation of the general form $u_t + H(Du, x) = 0$, where $H$ is the Hamiltonian and $Du$ is the spatial gradient. Because such equations are degenerate and their solutions are typically nonsmooth, they are understood in the viscosity sense — the standard weak notion under which comparison principles and stability under limits hold. A viscous Hamilton–Jacobi equation adds a second-order term $-\operatorname{tr}(A\,D^2u)$ with a symmetric nonnegative definite, possibly degenerate (vanishing in some directions) matrix $A$.

Homogenization studies the large-scale behavior of equations whose coefficients oscillate rapidly on a microscopic scale $\varepsilon$: one asks whether the solutions $u^{\varepsilon}$ converge, as $\varepsilon \to 0$, to the solution of a deterministic effective equation, and what that equation is. In the periodic setting, which goes back to the 1980s, the effective equation is identified through a cell problem: for each fixed momentum $p$ one seeks a periodic corrector $\chi$ solving $H(p + D\chi, y) = \overline{H}(p)$ on the torus, and the unique constant $\overline{H}(p)$ for which such a solution exists defines the effective Hamiltonian $\overline{H}$; the oscillations of $D\chi$ then absorb the microstructure of $Du^{\varepsilon}$. The random counterpart takes the coefficients to be stationary ergodic random fields: functions $A(y,\omega)$ and $H(p,y,\omega)$ of a spatial variable $y$ and a random outcome $\omega$, drawn from a probability measure on environments that is invariant under spatial translations (stationarity) and for which every translation-invariant event has probability zero or one (ergodicity). Convergence is then expected almost surely — with probability one over environments — and $\overline{H}$ is deterministic. For first-order equations with convex Hamiltonians this was settled around 1999–2000.

Lions and Souganidis (2005) proved the first such result for second-order equations with the critical scaling, the paper considered here:

$$u^{\varepsilon}_t - \varepsilon\,\operatorname{tr}\!\big(A(x, \varepsilon^{-1}x, \omega)\,D^2u^{\varepsilon}\big) + H\big(Du^{\varepsilon}, u^{\varepsilon}, x, \varepsilon^{-1}x, \omega\big) = 0$$

— in which the viscosity amplitude $\varepsilon$ vanishes exactly at the rate at which the coefficients oscillate, hence the quotation marks around viscous. The scaling is critical: the vanishing diffusion provides no useful regularizing effect in the limit and destroys the finite propagation speed on which first-order arguments relied. The hypotheses are: $A$ degenerate elliptic and bounded, with a Lipschitz square root $\Sigma$ (that is, $A = \Sigma\Sigma^T$); $H$ convex and coercive in $p$ (coercive meaning $H(p,\cdot) \to +\infty$ as $|p| \to \infty$), with Lipschitz regularity; and a structural growth condition — an explicit inequality coupling the superlinear growth of $H$ in $p$ with the oscillation of $H$ and $\Sigma$ in the fast spatial variable — which is what produces gradient bounds uniform in $\varepsilon$ and $\omega$; the superquadratic case is treated first and the general coercive case reached by a penalization argument, which adds a superquadratic term to $H$ and lets its weight tend to zero. The conclusions are almost-sure, locally uniform homogenization to a deterministic effective equation; an inf–sup variational formula for $\overline{H}$ over stationary potentials; and applications to quenched (almost-sure, for a fixed typical environment, rather than averaged over environments) large-deviation principles for controlled diffusions in random media, WKB-type (Wentzel–Kramers–Brillouin: the geometric-optics regime in which the logarithm of the solution of the linear equation satisfies a Hamilton–Jacobi equation) asymptotics for linear parabolic equations with random potentials, and front speeds for KPP (Kolmogorov–Petrovskii–Piskunov, that is, monostable reaction–diffusion) equations in random media.

Correctors, however, are unavailable in the random setting. The macroscopic problem $-\operatorname{tr}(A\,D^2\chi) + H(p + D\chi, y, \omega) = \overline{H}(p)$ in $\mathbb{R}^d$ generally has no viscosity solution of strictly sublinear growth at infinity: Lions and Souganidis (2003) showed this for the inviscid corrector equation, and Section 8 of the 2005 paper constructs stationary ergodic potentials of the type arising in the spectral theory of random Schrödinger operators (so-called Russian-model potentials) for which the one-dimensional equation $-\Delta\chi + |D\chi|^m = V(y,\omega) + \overline{H}(0)$, $1 \le m \le 2$, admits no sublinear solution at all, with separated-variable extensions to higher dimensions. The proof of homogenization therefore runs through localized approximate correctors — discounted problems on large boxes — and variational and subadditive ergodic arguments rather than a cell problem. The paper delimits four unresolved directions: a general condition guaranteeing correctors; whether the structural condition is necessary; equations whose diffusion matrix depends on the gradient, that is, geometric front-motion equations; and quantitative or constructive information about $\overline{H}$.

Several terms used in the problem statement: a stationary ergodic law has finite range of dependence if the coefficients evaluated at points farther than a fixed distance apart are independent — a strong mixing assumption, much stronger than stationarity plus ergodicity; a Hamiltonian is positively homogeneous (geometric) if $H(\lambda p, y) = \lambda H(p, y)$ for $\lambda \ge 0$ (and $H(0) = 0$), the structure of level-set geometric equations such as forced mean curvature motion, in which a front propagates with normal velocity equal to a driving force plus its curvature, encoded by a diffusion matrix depending on the direction $Du/|Du|$; a Hamiltonian is quasiconvex if its sublevel sets in $p$ are convex; and quantitative homogenization asks for explicit convergence rates and error bounds rather than bare convergence. Two decades of subsequent work resolved the necessity question and filled parts of the others under independence or symmetry assumptions. The problem statement below is the surviving open core.

Problem Statement

Complete the stochastic-homogenization program for critically scaled viscous Hamilton–Jacobi equations in general stationary ergodic media opened by Lions and Souganidis (2005): identify when correctors exist, whether homogenization holds for gradient-dependent (geometric) diffusion, and how fast the convergence is. The common setting is the equation

$$u^{\varepsilon}_t - \varepsilon\,\operatorname{tr}\!\big(A\,D^2u^{\varepsilon}\big) + H\big(Du^{\varepsilon}, \varepsilon^{-1}x, \omega\big) = 0$$

with $A$ degenerate elliptic — possibly $A = A(Du^{\varepsilon}, \varepsilon^{-1}x, \omega)$, genuinely depending on the gradient, in component (B) — and $H$ convex and coercive in $p$ (positively homogeneous in component (B)), the coefficients being stationary ergodic random fields under a law that is general: not restricted to finite range of dependence, radial symmetry, or one dimension. Convergence is sought almost surely and locally uniformly, to the solution of a deterministic effective equation with effective Hamiltonian $\overline{H}$. The target has three linked components (A)–(C). They are independent: a resolution of any one of (A), (B), (C1), or (C2) below does not depend on the others.

(A) Correctors. Give a criterion — necessary and/or sufficient conditions on the environment law and on the direction $p$ — under which the macroscopic problem $-\operatorname{tr}(A\,D^2\chi) + H(p + D\chi, y, \omega) = \overline{H}(p)$ admits a corrector: a viscosity solution with strictly sublinear growth at infinity. What is known: correctors exist at extreme points of the convex hull of the sublevel set $\{q : \overline{H}(q) \le \overline{H}(p)\}$ under homogenization in probability plus a uniform-Lipschitz assumption (Cardaliaguet and Souganidis 2017); they exist for all directions under radially symmetric environment laws; and they can fail to exist in general — Lions and Souganidis (2003) proved nonexistence for the inviscid corrector equation, and Section 8 of the 2005 paper constructs stationary ergodic potentials of the type arising in the spectral theory of random Schrödinger operators (Russian-model potentials) for which the one-dimensional equation $-\Delta\chi + |D\chi|^m = V(y,\omega) + \overline{H}(0)$, $1 \le m \le 2$, admits no solution of sublinear growth, with separated-variable extensions to higher dimensions. The open core is the interior, non-extreme directions and general stationary ergodic laws. Component (A) is resolved either by such a criterion — necessary, sufficient, or both — covering these directions and laws, or by a proof that no criterion in a non-degenerate, precisely specified class of conditions on the environment law and on the direction $p$ — not so narrow that the impossibility is vacuous — can separate the environments and directions admitting correctors from the known counterexamples.

(B) Gradient-dependent diffusion. Prove or disprove almost-sure homogenization when the diffusion matrix genuinely depends on the momentum, $A = A(Du^{\varepsilon}, \varepsilon^{-1}x, \omega)$, the central case being positively homogeneous Hamiltonians, that is, geometric level-set equations including forced mean curvature motion in stationary ergodic media. What is known covers only finite range of dependence (Armstrong and Cardaliaguet 2018; and, allowing locally pinned interfaces under isotropic finite-range laws, Fischer and Ingmanns 2026) and radially symmetric laws (Cardaliaguet and Souganidis 2017); the case of general stationary ergodic laws — and gradient dependence beyond positive homogeneity — is open.

(C) Quantitative homogenization and effective Hamiltonians. (C1) Establish convergence rates and error bounds for $u^{\varepsilon} - u$, and for approximations of $\overline{H}$, under mixing assumptions strictly weaker than finite range of dependence — or prove by counterexample that no uniform rate exists in such generality. (C2) Provide constructive approximation procedures with rigorous error estimates for evaluating $\overline{H}$ in concrete random models beyond the periodization and perturbation regimes already available.

The open core is delimited by the known counterexamples — nonexistence of sublinear correctors for $-\Delta\chi + |D\chi|^m = V + \overline{H}(0)$, $1 \le m \le 2$, in one dimension; non-homogenization for saddle-type nonconvex Hamiltonians in dimension $d \ge 2$; flat-spot obstructions to quantitative rates at non-exposed points of sublevel sets — and by the known positive classes (extreme points of convexified sublevel sets; radially symmetric laws; finite range of dependence, including the isotropic pinned-interface case; one dimension). Two parts of the source question are excluded as resolved: the necessity of the 2005 structural condition (settled negatively for convex Hamiltonians by Armstrong and Tran 2014) and the one-dimensional theory (essentially complete: quasiconvex cases by Yilmaz 2021 and Davini 2025, general nonconvex stationary ergodic cases by Kosygina and Yilmaz 2025).

References

1. Pierre-Louis Lions and Panagiotis E. Souganidis (2005). Homogenization of "Viscous" Hamilton–Jacobi Equations in Stationary Ergodic Media. Communications in Partial Differential Equations 30(3):335–375. DOI: 10.1081/PDE-200050077, https://doi.org/10.1081/PDE-200050077.
2. Pierre-Louis Lions and Panagiotis E. Souganidis (2003). Correctors for the homogenization of Hamilton–Jacobi equations in the stationary ergodic setting. Communications on Pure and Applied Mathematics 56(10):1501–1524. DOI: 10.1002/cpa.10101, https://doi.org/10.1002/cpa.10101.
3. Elena Kosygina, Fraydoun Rezakhanlou, and S. R. S. Varadhan (2006). Stochastic homogenization of Hamilton-Jacobi-Bellman equations. Communications on Pure and Applied Mathematics 59(10):1489–1521. DOI: 10.1002/cpa.20137, https://doi.org/10.1002/cpa.20137.
4. Elena Kosygina and S. R. S. Varadhan (2008). Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. Communications on Pure and Applied Mathematics 61(6):816–847. DOI: 10.1002/cpa.20220, https://doi.org/10.1002/cpa.20220.
5. Pierre-Louis Lions and Panagiotis E. Souganidis (2010). Stochastic homogenization of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi equations with convex nonlinearities — Revisited. Communications in Mathematical Sciences 8(2):627–637. DOI: 10.4310/cms.2010.v8.n2.a14, https://doi.org/10.4310/cms.2010.v8.n2.a14.
6. Scott N. Armstrong, Pierre Cardaliaguet, and Panagiotis E. Souganidis (2014). Error estimates and convergence rates for the stochastic homogenization of Hamilton-Jacobi equations. Journal of the American Mathematical Society 27(2):479–540. DOI: 10.1090/S0894-0347-2014-00783-9, https://doi.org/10.1090/S0894-0347-2014-00783-9.
7. Scott N. Armstrong and Hung V. Tran (2014). Stochastic homogenization of viscous Hamilton–Jacobi equations and applications. Analysis & PDE 7(8):1969–2007. DOI: 10.2140/apde.2014.7.1969, https://arxiv.org/abs/1310.1749.
8. Scott N. Armstrong and Pierre Cardaliaguet (2015). Quantitative stochastic homogenization of viscous Hamilton-Jacobi equations. Communications in Partial Differential Equations 40(3):540–600. DOI: 10.1080/03605302.2014.971372, https://arxiv.org/abs/1312.7593.
9. Pierre Cardaliaguet and Panagiotis E. Souganidis (2015). Periodic approximations of the ergodic constants in the stochastic homogenization of nonlinear second-order (degenerate) equations. Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire 32(3):571–591. DOI: 10.1016/j.anihpc.2014.01.007, https://doi.org/10.1016/j.anihpc.2014.01.007.
10. Bruno Ziliotto (2017). Stochastic homogenization of nonconvex Hamilton-Jacobi equations: a counterexample. Communications on Pure and Applied Mathematics 70(9):1798–1809. DOI: 10.1002/cpa.21674, https://doi.org/10.1002/cpa.21674.
11. William M. Feldman and Panagiotis E. Souganidis (2017). Homogenization and non-homogenization of certain non-convex Hamilton–Jacobi equations. Journal de Mathématiques Pures et Appliquées 108(5):751–782. DOI: 10.1016/j.matpur.2017.05.016, https://doi.org/10.1016/j.matpur.2017.05.016.
12. Wenjia Jing, Panagiotis E. Souganidis, and Hung V. Tran (2017). Stochastic homogenization of viscous superquadratic Hamilton–Jacobi equations in dynamic random environment. Research in the Mathematical Sciences 4(1), article 6. DOI: 10.1186/s40687-016-0090-9, https://doi.org/10.1186/s40687-016-0090-9.
13. Pierre Cardaliaguet and Panagiotis E. Souganidis (2017). On the existence of correctors for the stochastic homogenization of viscous Hamilton–Jacobi equations. Comptes Rendus Mathématique 355(7):786–794. DOI: 10.1016/j.crma.2017.06.001, https://doi.org/10.1016/j.crma.2017.06.001.
14. Scott N. Armstrong and Pierre Cardaliaguet (2018). Stochastic homogenization of quasilinear Hamilton–Jacobi equations and geometric motions. Journal of the European Mathematical Society 20(4):797–864. DOI: 10.4171/jems/777, https://arxiv.org/abs/1504.02045.
15. Pierre Cardaliaguet, Claude Le Bris, and Panagiotis E. Souganidis (2018). Perturbation problems in homogenization of Hamilton–Jacobi equations. Journal de Mathématiques Pures et Appliquées 117:221–262. DOI: 10.1016/j.matpur.2018.03.005, https://arxiv.org/abs/1701.05440.
16. Elena Kosygina, Atilla Yilmaz, and Ofer Zeitouni (2020). Homogenization of a class of one-dimensional nonconvex viscous Hamilton–Jacobi equations with random potential. Communications in Partial Differential Equations 45(1):32–56. DOI: 10.1080/03605302.2019.1657448, https://doi.org/10.1080/03605302.2019.1657448.
17. Atilla Yilmaz (2021). Stochastic homogenization of a class of quasiconvex viscous Hamilton-Jacobi equations in one space dimension. Journal of Differential Equations 300:660–691. DOI: 10.1016/j.jde.2021.08.004, https://doi.org/10.1016/j.jde.2021.08.004.
18. William M. Feldman, Jean-Baptiste Fermanian, and Bruno Ziliotto (2021). An example of failure of stochastic homogenization for viscous Hamilton-Jacobi equations without convexity. Journal of Differential Equations 280:464–476. DOI: 10.1016/j.jde.2021.01.009, https://doi.org/10.1016/j.jde.2021.01.009.
19. A. Hajej (2016). Error estimates for approximation schemes of effective Hamiltonians arising in stochastic homogenization of Hamilton-Jacobi equations. Numerical Algorithms 73(3):839–868. DOI: 10.1007/s11075-016-0120-0, https://doi.org/10.1007/s11075-016-0120-0.
20. Andrea Davini and Elena Kosygina (2022). Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension. Journal of Differential Equations 333:231–267. DOI: 10.1016/j.jde.2022.06.011, https://doi.org/10.1016/j.jde.2022.06.011.
21. Andrea Davini, Elena Kosygina, and Atilla Yilmaz (2024). Stochastic homogenization of nonconvex viscous Hamilton-Jacobi equations in one space dimension. Communications in Partial Differential Equations 49(7-8):698–734. DOI: 10.1080/03605302.2024.2390836, https://doi.org/10.1080/03605302.2024.2390836.
22. Andrea Davini (2025). Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d. Calculus of Variations and Partial Differential Equations 64(2), article 38. DOI: 10.1007/s00526-024-02870-x, https://doi.org/10.1007/s00526-024-02870-x.
23. Elena Kosygina and Atilla Yilmaz (2025). Homogenization of nonconvex viscous Hamilton–Jacobi equations in stationary ergodic media in one dimension. Nonlinearity 38(7):075017. DOI: 10.1088/1361-6544/adde27, https://arxiv.org/abs/2403.15963.
24. Julian Fischer and Jonas Ingmanns (2026). Quantitative homogenization of forced geometric motions through random fields of obstacles. Preprint, arXiv:2603.12179. https://arxiv.org/abs/2603.12179.
25. Andrea Davini, Raimundo Saona, and Bruno Ziliotto (2026). Stochastic homogenization of HJ equations: A differential game approach. Annales de l'Institut Henri Poincaré C, Analyse Non Linéaire 43(4):883–925. DOI: 10.4171/aihpc/174, https://doi.org/10.4171/aihpc/174.