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

We consider Hamilton-Jacobi equations in one space dimension with Hamiltonians of the form $H(p,x,ω) = G(p) + βV(x,ω)$, where $V(\cdot,ω)$ 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 $G$, $β$ and the law of $V(\cdot,ω)$, and discuss a generalization to even-symmetric triple-well Hamiltonians.

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