2405.01343
CONDITIONED STOCHASTIC STABILITY OF EQUILIBRIUM STATES ON UNIFORMLY EXPANDING REPELLERS
Bernat Bassols-Cornudella, Matheus M. Castro, Jeroen S.W. Lamb
correctmedium confidence
- Category
- Not specified
- Journal tier
- Strong Field
- Processed
- Sep 28, 2025, 12:56 AM
- arXiv Links
- Abstract ↗PDF ↗
Audit review
The paper proves the local (Theorem 2.10) and global-with-hole (Theorem 2.11) results for uniformly expanding repellers via a strong-Feller/compactness route: (i) quasi-stationary densities m_ε are obtained as fixed points of a normalized transfer operator on C^β, (ii) the annealed Koopman operator P_ε is strong Feller and P_ε^2 is compact, enabling a peripheral-spectral analysis (Appendix A) and a spectral gap, (iii) g_ε is the unique positive eigenfunction, and (iv) the quasi-ergodic measure is ν_ε ∝ g_ε m_ε, with ν_ε → ν_0 and log λ_ε → P(T, φ − log|det dT|, Σ) as ε → 0 (Σ = R_i or Λ) . The candidate solution gives a correct alternative proof strategy based on Lasota–Yorke (Doeblin–Fortet) inequalities on Hölder spaces, quasi-compactness with spectral gap, Krein–Rutman positivity, and Keller–Liverani/Baladi–Young spectral stability, then identifies ν_ε ∝ g_ε μ_ε and shows convergence to the equilibrium state. This matches the paper’s statements (adjoint relation L_ε^* = P_ε; uniqueness/positivity of eigen-elements; quasi-ergodic = product of left/right eigenvectors; pressure limit) though via different tools . Differences: the paper avoids an explicit Lasota–Yorke inequality and instead uses strong Feller + compactness; its stability argument (L_ε m_ε → L m_0) is direct rather than invoking Keller–Liverani (Lemma 3.7, Proposition 3.8) . Under mixing, both derive global quasi-ergodicity, but the paper does so via a graph decomposition and positivity on the recurrent component, not by an explicit Doeblin minorization argument (Propositions 4.1–4.3, 4.15; Theorem 4.16) . Overall, the paper’s arguments are sound and complete for its hypotheses, and the model’s approach is also mathematically valid as an alternative route.
Referee report (LaTeX)
\textbf{Recommendation:} minor revisions \textbf{Journal Tier:} strong field \textbf{Justification:} The paper provides a rigorous and complete framework for conditioned stochastic stability of equilibrium states on uniformly expanding repellers, unifying random perturbations, thermodynamic formalism, and quasi-stationarity. The approach via strong Feller regularization, compactness, and peripheral spectral analysis is technically clean and complements classical Lasota–Yorke treatments. Results are significant and broadly relevant. Minor revisions to improve exposition and to cross-reference alternative operator approaches would enhance accessibility.