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

TL;DR

This paper demonstrates the existence and regularity of strict critical subsolutions for stationary ergodic Hamiltonians, showing their density and smoothness under certain conditions using advanced mathematical tools.

Contribution

It establishes the existence of critical subsolutions that are strict outside a generalized Aubry set in the stationary ergodic setting, including smoothness results for Tonelli Hamiltonians.

Findings

Critical subsolutions exist and are dense among all critical subsolutions.

Strict subsolutions are of class C^{1,1} for Tonelli Hamiltonians.

Use of Lax--Oleinik semigroups enables regularity results.

Abstract

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.