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We consider Hamilton-Jacobi equations in one space dimension withHamiltonians 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. Thehomogenization 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), wegive a new and fully constructive proof of homogenization which yields aformula for the effective Hamiltonian $\overline H$. We use this formula toprovide a complete list of the heights at which the graph of $\overline H$ hasa 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 toeven-symmetric triple-well Hamiltonians.

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