A semi-discrete approximation for first-order stationary mean field games

The paper proves (i) existence of m_τ via Kakutani’s fixed point on τ-holonomic minimizers and (ii) convergence (along subsequences) of (u_{τ,m_τ}, m_τ) to a stationary MFG solution, by establishing tightness, uniform Lipschitz bounds, convergence to the Hamilton–Jacobi equation, and passing τ-holonomy to closed measures to obtain the continuity equation. These steps are explicit in Theorem 1 and Propositions 6–11 (tightness, Kakutani map upper semicontinuity, viscosity limit, closed-limit/Mather measure, and divergence-free flow) . The candidate solution establishes the same two claims with a slightly different route: standard viscosity stability (Crandall–Ishii–Lions), a calibration/Fenchel–Young argument to identify v = ∂_p H on the limit measure’s support, and a direct distributional pass-to-the-limit of the τ-holonomy constraint. Aside from minor imprecisions (e.g., streamlining the proof that the limit measure is minimizing for L_{m_0}), the model matches the paper’s result and logic.