Regularity of the stationary density for systems with fast random switching

The paper’s Theorem 2.8 asserts C^k_0-regularity of the stationary density for fast switching under Assumption S and a 1-Hörmander condition on K_Q, with a locally bounded threshold Λ_k(Q) (Theorem 2.8) . Its proof hinges on (i) a uniform submersion/smoothing construction near K_Q (Proposition 5.1 and Proposition 3.3) , (ii) a “good” stopping time with high success probability for large λ, and (iii) the stopping-time renewal identity (Proposition 3.1) together with a mapping result (Proposition 3.5) that sends C^∞_0 inputs to C^k_0 outputs for fast switching . The candidate solution follows the same architecture: local submersion and smoothing, a good-window stopping time, a stopping-time decomposition of the stationary law, and the fast-switching regime to localize smoothing near K_Q. Minor issues are that the candidate assumes strictly positive off-diagonals (stronger than the irreducibility used in the paper) and informally justifies local boundedness of Λ_k via spectral-gap claims not needed in the paper’s argument. These do not affect the core logic, which is essentially the same as the paper’s.