Reconstructing dynamical systems as zero-noise limits

The paper’s Theorem 4 and its proof strategy are essentially correct: (i) every stationary measure of the process (6) lies within a δ-neighborhood of h(X), and h(X) lies within a coarse 2δ-neighborhood of that support; and (ii) when the cover is a partition, the one-step kernel concentrates to the deterministic image as δ→0 via a spread→0 argument using Lemma 4.1 and a quantitative spread bound (Lemma 4.3) . However, the text states that the unique stationary distribution of the S-chain is uniform ((1/m,…,1/m)), which is not generally implied by their construction of the (column-stochastic) matrix P; the unique stationary distribution exists by irreducibility (Lemma 4.2) but need not be uniform . In addition, the quantitative spread bound (Lemma 4.3) relies on a Lipschitz constant ∥f∥Lip without this regularity assumption being clearly stated as part of the theorem’s hypotheses . The candidate model’s solution correctly establishes (i.a) and the zero-noise limit in (ii) via a spread bound and Lemma 4.1, but (i.b) incorrectly identifies the stationary law on S as πj=µ(Vj) for general covers and (implicitly) treats P as row-stochastic without the normalization needed when the cover is not a partition. Thus, both the paper and the model contain gaps: the paper overclaims uniform stationarity and omits an explicit regularity assumption for the bound, while the model misidentifies the S-stationary distribution outside the partition case. The main conclusions, though, remain salvageable with the correct hypotheses and arguments (paper: follower–predecessor argument and Q-pushforward calculus; model: irreducibility for S plus positivity of mass in each cell) .