Extending the Lions–Souganidis weak-convergence proof of stochastic homogenization to fully nonlinear second-order equations with convex nonlinearities, and resolving the degenerate invariant-measure difficulty for its variational formula

In a 2010 note, Lions and Souganidis gave a direct, control-free proof of homogenization in probability for first-order and "viscous" Hamilton–Jacobi equations with convex Hamiltonians in stationary ergodic random media, and proved that the effective Hamiltonian admits a variational formula over invariant measures of the environment process. The note itself flags the limitations that define this problem: the proof "would work" for fully nonlinear uniformly elliptic equations with convex nonlinearity "provided we could obtain an appropriate estimate guaranteeing the (weak) convergence, as ε → 0, of the Hessians", and, for the formula, the invariant measures may not be unique when the diffusion matrix is degenerate. The homogenization statements for the fully nonlinear class are known by other methods (obstacle-problem and metric-problem approaches), so the surviving open core is not the truth of homogenization but the completion of the direct program: an a priori Hessian weak-convergence estimate and the extension of the direct proof to fully nonlinear second-order equations with convex structure, together with a resolution of the invariant-measure non-uniqueness difficulty that blocks the variational formula from serving as an independent proof of homogenization in the degenerate-diffusion case. Both threads remain open as of 2026.

Background

Setting. Homogenization studies the limit, as $\varepsilon \to 0$, of solutions of equations whose coefficients oscillate at the fast scale $x/\varepsilon$; the goal is to identify a homogenized equation with deterministic, non-oscillatory coefficients. In a stationary ergodic random medium the coefficients are random fields over a probability space $(\Omega, \mathcal F, \mathbb P)$ carrying a group of measure-preserving translations $(\tau_y)_{y \in \mathbb R^N}$: stationarity means the law of the coefficients is invariant under spatial shifts, and ergodicity means every shift-invariant event has probability $0$ or $1$. Convergence of solutions then holds in probability or almost surely rather than deterministically.

The equations. A first-order Hamilton–Jacobi equation has the form $H(Du, x, x/\varepsilon, \omega) = 0$, where $H$ is the Hamiltonian and $Du$ the gradient. A "viscous" Hamilton–Jacobi equation adds a second-order term $-\varepsilon\delta\,\mathrm{tr}\,(A(x, x/\varepsilon, \omega)D^2u^\varepsilon)$, where $\delta \ge 0$ is a fixed parameter ($\delta = 0$ gives the first-order equation, $\delta > 0$ the viscous one), $A$ is a nonnegative symmetric matrix and $\mathrm{tr}$ is the trace, so the equation is degenerate elliptic. A fully nonlinear uniformly elliptic equation has the form $F(D^2u, Du, u, x, x/\varepsilon, \omega) = 0$, where $F$ is uniformly elliptic — bounded above and below, in its matrix argument, by fixed multiples of the identity — and depends nonlinearly on the Hessian $D^2u$; "convex nonlinearity" means $F$ is convex jointly in its nonlinear arguments $(D^2u, Du, u)$, the standard Hamilton–Jacobi–Bellman structure in which $F$ arises as a supremum of linear operators. Solutions throughout are viscosity solutions (Crandall–Ishii–Lions), the standard weak notion for fully nonlinear degenerate elliptic equations, in which a function counts as a subsolution if every smooth test function touching it from above satisfies the equation inequality; the notion provides comparison and stability without any assumed second-derivative regularity. A Hamilton–Jacobi–Bellman (HJB) equation is an equation of one of the above forms arising as the dynamic programming equation of a stochastic optimal control problem; convex nonlinearities admit such representations.

Effective Hamiltonian and correctors. The homogenized limit of $u_t + H(Du, x, x/\varepsilon, \omega) = 0$ (or of its stationary analog) is $u_t + \bar H(Du) = 0$ with $\bar H$ deterministic: the effective Hamiltonian, whose identification is the analytic heart of the problem. The standard device is the approximate cell problem: for a slope $p \in \mathbb R^N$, solve the approximate corrector equation

$$\varepsilon v^\varepsilon - \delta\,\mathrm{tr}\,(A(y, \omega)D^2v^\varepsilon) + H(Dv^\varepsilon + p, y, \omega) = 0 \quad \text{in } \mathbb R^N,$$

with discount $\varepsilon > 0$ tending to $0$ and fixed diffusion parameter $\delta \ge 0$ (the source’s equation (2.2); later literature often re-parameterizes the discount as $\delta$); homogenization is essentially equivalent to $\varepsilon v^\varepsilon(0, \cdot)$ converging, as $\varepsilon \to 0$, in probability and uniformly on balls of radius $O(1/\varepsilon)$, to the constant $-\bar H(p)$. A Hamiltonian is coercive if $H(\xi, y, \omega) \ge c_1|\xi|^\alpha - c_2$ for some $\alpha > 1$; this superlinear growth yields $L^\alpha$ bounds on the gradients of subsolutions.

Prior results. Homogenization of the viscous equation

$$-\varepsilon\delta\,\mathrm{tr}\,(A(x, x/\varepsilon, \omega)D^2u^\varepsilon) + H(Du^\varepsilon, u^\varepsilon, x, x/\varepsilon, \omega) = 0 \quad \text{(the source’s equation (1.1), with fixed } \delta \ge 0\text{)}$$

with convex Hamiltonian was first proved independently by Lions and Souganidis (CPDE 2005) via corrector methods and by Kosygina, Rezakhanlou and Varadhan (CPAM 2006; KRV) via the stochastic control representation, the environment process "seen from the particle" (the Markov process on the probability space obtained by shifting the medium with the controlled particle’s motion), and the ergodic theorem. The KRV argument yields a variational formula for $\bar H$: writing $\tilde H^*$ for the Legendre transform of $H$ in the momentum variable (the Lagrangian of the control problem),

$$\bar H(p) = \sup_{(b, \varphi) \in \mathcal E}\ \mathbb E\big[\varphi\,(\langle p, b\rangle - \tilde H^*(b))\big],$$

where $\mathcal E$ is the set of pairs of a drift field $b$ and a probability density $\varphi$ on the probability space that is invariant for the environment process with drift $b$ and diffusion matrix $\tilde A$ (the environment’s diffusion written on the probability space). For $A = I$ this is the KRV formula. The 2010 note revisited here gave a new, direct proof of homogenization in probability for (1.1) under convexity of $H$ in the momentum variable, coercivity, and an assumed uniform modulus of continuity, without any control representation. The mechanism: sup- and inf-convolutions (regularizations of a viscosity solution by suprema/infima over small balls, producing semiconvex or semiconcave envelopes with almost-everywhere classical second derivatives) plus coercivity give $L^\alpha$ bounds on the gradients; convexity of $H$ lets weak $L^\alpha$ limits of the gradients be passed through $H$; a construction of sublinearly growing subsolutions and the ergodic theorem (which, combined with the uniform modulus of continuity of the correctors, upgrades convergence in probability at the origin to convergence uniform on balls of radius $O(1/\varepsilon)$) identify the limit. Theorem B of the note then proves that the formula above holds for possibly degenerate $\tilde A$ — and even for the first-order case $\delta = 0$ — as a consequence of the homogenization theorem, by a min–max duality argument.

The gaps flagged in the source. (i) The note states that "the same proof would work" for the fully nonlinear uniformly elliptic equation

$$F(D^2u^\varepsilon, Du^\varepsilon, u^\varepsilon, x, x/\varepsilon, \omega) = 0 \quad \text{(the source’s equation (1.2))}$$

with convex nonlinearity "provided we could obtain an appropriate estimate guaranteeing the (weak) convergence, as $\varepsilon \to 0$, of the Hessians" — weak convergence meaning convergence in the sense of distributions, after testing against smooth compactly supported functions, the only notion available since no uniform second-derivative bounds are known. (ii) After the proof of the formula: "The difficulty, however, encountered, when $\tilde A$ is degenerate, is that the invariant measures $\tilde\varphi$ may not be unique", which blocks the converse route — deriving homogenization from the formula via the ergodic theorem, as in KRV — for degenerate diffusions.

Adjacent literature. Caffarelli, Souganidis and Wang (CPAM 2005) had proved, before the note, homogenization of fully nonlinear uniformly elliptic equations without gradient dependence by an obstacle-problem method (comparison with maximal subsolutions and exploitation of the spreading of contact sets), and Armstrong and Smart (2014) extended such arguments to gradient-dependent $F$ with almost-sure convergence. So the homogenization statements for the fully nonlinear class are known — but by methods that neither supply the missing Hessian estimate nor follow the weak-convergence route, and that produce no invariant-measure or duality formula for the effective operator; the metric-problem program of Armstrong and Souganidis for first-order and level-set convex equations is, in its authors’ words, "strongly first-order" and does not extend to viscous equations. The problem is therefore about methods and formulas inside a class whose homogenization is otherwise settled.

Problem Statement

Let $(\Omega, \mathcal F, \mathbb P, (\tau_y)_{y \in \mathbb R^N})$ be a stationary ergodic probability space. Following Lions–Souganidis (2010), consider equation (1.1) of the source, for a fixed parameter $\delta \ge 0$ ($\delta > 0$: the viscous equation; $\delta = 0$: the first-order equation),

$$-\varepsilon\delta\,\mathrm{tr}\,(A(x, x/\varepsilon, \omega)D^2u^\varepsilon) + H(Du^\varepsilon, u^\varepsilon, x, x/\varepsilon, \omega) = 0,$$

and the fully nonlinear uniformly elliptic equation (1.2) of the source,

$$F(D^2u^\varepsilon, Du^\varepsilon, u^\varepsilon, x, x/\varepsilon, \omega) = 0,$$

where $F$ is stationary ergodic in the fast variables, uniformly elliptic, and convex jointly in its nonlinear arguments $(D^2u, Du, u)$. The problem, in the source’s own two parts, with the common objective of completing the direct, control-free weak-convergence program for the class of convex-structure second-order equations and removing the degeneracy obstruction to its variational formula:

(1) The Hessian estimate and the extension of the direct proof. Find an appropriate a priori estimate guaranteeing the weak convergence as $\varepsilon \to 0$ of the Hessians of the approximate correctors for $F$ (the analogs, for $F$, of the discounted cell-problem solutions above), at least along subsequences, strong enough that weak limits can be passed through the convex nonlinearity $F$; and use it to extend the Lions–Souganidis direct proof of homogenization in probability to equation (1.2) with convex nonlinearity. The sufficiency criterion for the estimate is this extension itself: the weak limits of the Hessians must be passable through the convex nonlinearity $F$ in a sense compatible with the viscosity-solution framework — a Jensen-type inequality for the weak limit, in whatever weak notion the estimate yields — so that the direct argument goes through and homogenization in probability for (1.2) follows without any control representation.

(2) Degenerate invariant measures for the environment process. For equation (1.1) when the diffusion matrix $\tilde A$ of the environment process (the diffusion $A$ written on the probability space) is degenerate, resolve the difficulty flagged in the source: the invariant measures $\tilde\varphi$ entering the formula may not be unique, and when the conclusion of the homogenization theorem is not known a priori the formula can be turned into a proof of homogenization only via the ergodic theorem, as in KRV, for which the ergodic identification of the lower bound in the formula requires uniqueness of the invariant measure for a given drift. The goal is to make the variational formula an independent proof of homogenization in probability for degenerate $\tilde A$, not relying on a homogenization theorem established first by other methods.

Scope note (source-faithful): homogenization of the fully nonlinear class (1.2) is already known by obstacle-problem methods, so the open target is the direct weak-convergence route, not the truth of homogenization. A third remark of the source — a forthcoming paper announcing four general and independent groups of hypotheses giving uniform moduli of continuity — was absorbed by later regularity literature.

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