[Paper Review] Stochastic homogenization of level-set convex Hamilton-Jacobi equations

Scott N. Armstrong, Panagiotis E. Souganidis|arXiv (Cornell University)|Mar 28, 2012

Advanced Mathematical Modeling in Engineering19 references4 citations

TL;DR

This paper presents a novel, constructive proof for the stochastic homogenization of level-set convex Hamilton-Jacobi equations in stationary ergodic environments, avoiding reliance on explicit solutions or subcorrectors. By analyzing the asymptotic behavior of metric problems (1.4) and (1.5) via subadditive ergodic theory, the authors derive the effective Hamiltonian and establish local uniform convergence of solutions, extending prior results to the quasiconvex setting with improved qualitative insight into the effective nonlinearity.

ABSTRACT

We present a simple new proof for the stochastic homogenization of quasiconvex (level-set convex) Hamilton-Jacobi equations set in stationary ergodic environments. Our approach, which is new even in the convex case, yields more information about the qualitative behavior of the effective nonlinearity.

Motivation & Objective

• To provide a new, constructive proof of stochastic homogenization for level-set convex Hamilton-Jacobi equations without relying on explicit solutions or subcorrectors.
• To extend the homogenization result to the quasiconvex (level-set convex) case, generalizing prior results that required convexity.
• To offer deeper qualitative understanding of the effective Hamiltonian through the asymptotic analysis of metric problems.
• To establish convergence of solutions to the homogenized equation via a 'one level-set at a time' approach using maximal solutions to eikonal-type equations.

Proposed method

• Analyze the metric problems (1.4) and (1.5), which are eikonal-type equations with level-set convex Hamiltonians, to study the asymptotic behavior of solutions in random environments.
• Use the subadditive ergodic theorem to prove almost sure convergence of normalized solutions along rays, yielding deterministic limits $\overline{m}_\mu$ and $\overline{n}_\mu$.
• Construct approximate super- and subcorrectors from the maximal solutions $m_\mu$ and $n_\mu$ for $\mu = \overline{H}(p)$, leveraging their stationarity and subadditivity.
• Establish the homogenization of the original equation by showing that the difference $m_\mu(y,x,\omega) - p\cdot y$ acts as an approximate supercorrector and $-n_\mu(x,y,\omega) - p\cdot y$ as an approximate subcorrector.
• Use weak convergence and Mazur’s lemma to pass to the limit in nonlinear Hamiltonians, ensuring the homogenization limit is well-defined even for quasiconvex $H$.
• Leverage the maximality of $m_\mu$ and $n_\mu$ to control the behavior of viscosity solutions and derive the effective Hamiltonian $\overline{H}$.

Experimental results

Research questions

• RQ1 Can stochastic homogenization of level-set convex Hamilton-Jacobi equations be proven without relying on explicit formulae or the existence of subcorrectors?
• RQ2 How does the asymptotic behavior of the metric problems (1.4) and (1.5) relate to the homogenization of the original equation?
• RQ3 What qualitative properties of the effective Hamiltonian $\overline{H}$ can be deduced from the subadditive limits of $m_\mu$ and $n_\mu$?
• RQ4 Is it possible to construct approximate correctors directly from the solutions of the metric problems, and how does this lead to homogenization?
• RQ5 How does the proposed method extend to quasiconvex Hamiltonians where standard convexity-based arguments fail?

Key findings

• The effective Hamiltonian $\overline{H}$ is level-set convex and coercive, and the homogenization result holds almost surely in $\omega$.
• The viscosity solutions $u^\varepsilon$ of the original equation converge locally uniformly to the constant solution of $u + \overline{H}(Du) = 0$ as $\varepsilon \to 0$.
• The asymptotic limits $\overline{m}_\mu(y) = \lim_{t\to\infty} t^{-1} m_\mu(ty, 0, \omega)$ and $\overline{n}_\mu(y) = \lim_{t\to\infty} t^{-1} n_\mu(ty, 0, \omega)$ exist almost surely and are deterministic, subadditive, and positively homogeneous.
• The maximal solutions $m_\mu$ and $n_\mu$ serve as building blocks for constructing approximate correctors, enabling the proof of homogenization without prior knowledge of subcorrector existence.
• The method provides a constructive approach to homogenization that reveals deeper structure in $\overline{H}$, particularly in the strictly convex case where $\overline{H}$ inherits strict convexity.
• The proof avoids the Lions-Souganidis method of constructing mean-zero subcorrectors, instead deriving their existence as a consequence of the homogenization result.