Existence and regularity of strict critical subsolutions in the stationary ergodic setting

We prove that any continuous and convex stationary ergodic Hamiltonian admits critical subsolutions, which are strict outside the random Aubry set. They make up, in addition, a dense subset of all critical subsolutions with respect to a suitable metric. If the Hamiltonian is additionally assumed of Tonelli type, then there exist strict subsolutions of class $\CC^{1,1}$ in $\R^N$. The proofs are based on the use of Lax--Oleinik semigroups and their regularizing properties in the stationary ergodic environment, as well as on a generalized notion of Aubry set.