Stochastic Time-Periodic Tonelli Lagrangian on Compact Manifold

The paper states and proves: (i) measurability and θ-invariance of the Mañé critical value α(ω), with ergodicity implying almost sure constancy (Theorem 1.1), (ii) existence of u^ω via the Lax–Oleinik operator, yielding a viscosity solution of ∂_t u + H = α(ω) together with measurability, 1-periodicity, and θ-covariance (Theorem 1.2), and (iii) existence of global minimizers for each ω (Corollary 7.1). These match the candidate solution’s claims. The paper’s approach uses closed measures and Fubini for Theorem 1.1 and countable approximations of actions and endpoints to prove measurability of T^ω_λ and u^ω, plus standard sub/supersolution arguments for viscosity solutions and a calibration/gluing argument for global minimizers (Theorem 1.1, Theorem 1.2, Lemma 6.3–6.4, and Corollary 7.1 respectively). The candidate solution proves the same results but via a different route: subadditive loop actions and ergodic considerations for α(ω), and a classical weak KAM semigroup/calibration framework for u^ω and global minimizers. One caveat: the paper’s first proof of Theorem 1.1 contains a gap in the step that removes an ω-dependent cutoff χ_ω inside an infimum over a countable dense set of measures; however, the alternative “closed-curves” route and the rest of the results are standard and can be repaired with minor revisions. Overall, both are correct and reach the same conclusions by different proofs (Theorem 1.1 and 1.2 in the paper; definitions and steps are explicit in the introduction and Sections 4, 6, and 7). See Theorem 1.1 and the Lax–Oleinik definition in the introduction, together with the measurability and θ-covariance lemmas (Theorem 1.1, Theorem 1.2, Lemma 6.3–6.4), the viscosity solution argument (Step 1–2), and the global minimizer existence (Corollary 7.1) in the paper’s text .