Stochastic homogenization conjecture for the Hamiltonian $H_{\ell}$

Let $\ell$ be a positive integer. Assume (A1)–(A4), including stationarity and ergodicity, coercivity, continuity and boundedness, and stable pairing for the Hamiltonians involved. Let $H_{\ell}(p,x,\omega)$ be the Hamiltonian defined by the preceding min–max construction and monotone ordering.

Stochastic homogenization conjecture. The Hamiltonian $H_{\ell}(p,x,\omega)$ is stochastically homogenizable.

The assumptions (A1)–(A3) are standard in stochastic homogenization of Hamilton–Jacobi equations, while (A4) excludes strict saddle points that could prevent homogenization. The source presents this as a conjectural assertion; no resolution is supplied in the provided text.