Averaging principle for slow-fast systems of rough differential equations via controlled paths

The paper proves an Lp-averaging principle for slow–fast RDEs under (A),(H1)–(H7) via Khasminskii time-discretization, an ergodic bound for the frozen fast SDE, and a controlled-path stability estimate, culminating in Theorem 2.1 . The candidate’s solution follows the same strategy: (i) a priori estimates, (ii) frozen dynamics and invariant measure, (iii) Khasminskii partition and auxiliary fast process, (iv) decomposition of the averaging error, and (v) rough-path stability. Two technical inaccuracies in the candidate’s write-up do not affect the conclusion: (a) they assert global Lipschitz continuity of f̄, while the paper only needs (and proves) boundedness plus local Lipschitz with polynomial weight (Proposition C.5) ; and (b) they bound a Hölder norm by a supremum. Replacing that with the 2β-Hölder bounds used in Lemmas 5.3–5.4 resolves this (decomposition (5.6) and estimates) . Overall, both arguments are substantively the same and correct.