LOG-HÖLDER REGULARITY OF STATIONARY MEASURES

The paper establishes log-Hölder regularity for stationary measures under finite logarithmic-moment assumptions by an energy/L2-method with precise kernel estimates and a probabilistic normalization scheme (Theorems 2.3 and 2.8) . In contrast, the model’s argument hinges on a tail bound P(S_n > s) ≤ C n^α s^{-α} and chooses n ≍ √s to obtain s^{-α/2}. This is unjustified for α ∈ (0,1), where E[S_n^α] ≲ n E[X^α] (not n^α), and even for α ≥ 1 the optimization step is not derived; the subsequent bootstrapping step is only sketched. The paper avoids such pitfalls by proving robust energy inequalities (e.g., Proposition 3.8 and Lemma 3.7) and a contraction scheme that yields the claimed bounds without heavy-tail large-deviation estimates .