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

Abstract

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 (•, ω) is a stationary & 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 H. We use this formula to provide a complete list of the heights at which the graph of H has a flat piece. We illustrate our results by analyzing basic classes of examples, highlight some corollaries that clarify the dependence of H on G, β and the law of V (•, ω), and discuss a generalization to even-symmetric triple-well Hamiltonians.

Percolation games

“…When H is not assumed to be convex (non-convex case), only very specific frameworks were treated: for level-set convex Hamiltonians [4], when the law of H is invariant by rotation [15], in one-dimension [6,18,12,42], when the law of H satisfies a finite range condition and H is positively homogenous or star-shaped [1,16]. The second author [45] proved recently that homogenization does not hold in the general non-convex case.…”