Statistical stability and linear response for random hyperbolic dynamics

The paper proves quenched statistical stability with an explicit O(|ε||log|ε||) modulus (Theorem 3, eq. (16)) and quenched linear response via a backward-telescoping identity and a convergent series for the derivative (Theorem 5, eqs. (34), (36), (38)–(40)). The candidate solution reproduces the same telescoping arguments, norm estimates, and series construction, differing only in minor bookkeeping (e.g., a bootstrap for the remainder term in Part A and an explicit s-norm O(|ε|) bound in Part B). No logical gaps were found; the approaches are essentially the same and lead to the same conclusions, in line with the paper’s statements and proofs. See Theorem 3 and its proof steps (18)–(20) leading to (16) , and Theorem 5 with (34), (36), (37)–(40) .