Stochastic homogenization and effective Hamiltonians of HJ equations in one space dimension: The double-well case

TL;DR

This paper provides a new constructive proof for the homogenization of one-dimensional Hamilton-Jacobi equations with double-well Hamiltonians, offering explicit formulas for the effective Hamiltonian and analyzing its flat regions.

Contribution

It introduces a fully constructive method for homogenization in the double-well case, including a formula for the effective Hamiltonian and a complete characterization of its flat pieces.

Findings

Derived explicit formula for the effective Hamiltonian ar H.

Identified all heights where ar H has flat segments.

Analyzed dependence of ar H on G, eta, and the potential law.

Abstract

We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,\omega) = G(p) + \beta V(x,\omega)$, where $V(\cdot,\omega)$ is a stationary and ergodic potential of unit amplitude. The homogenization of such equations is established in a 2016 paper of Armstrong, Tran and Yu for all continuous and coercive $G$. Under the extra condition that $G$ is a double-well function (i.e., it has precisely two local minima), we give a new and fully constructive proof of homogenization which yields a formula for the effective Hamiltonian $\overline H$. We use this formula to provide a complete list of the heights at which the graph of $\overline H$ has a flat piece. We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of $\overline H$ on $G$, $\beta$ and the law of $V(\cdot,\omega)$, and discuss a generalization to even-symmetric triple-well Hamiltonians.