Averaging and Mixing for Stochastic Perturbations of Linear Conservative Systems

The paper proves convergence in law for the interaction-representation process a^ε to the solution of an effective SDE with averaged drift R=⟪P⟫ and constant diffusion B (BB*=A), via tightness, a Khasminskii time-discretization to average the fast drift, identification through a martingale problem, and uniqueness by Yamada–Watanabe. The candidate solution mirrors this structure: interaction representation, tightness from moment/modulus bounds, martingale/drift split, oscillatory integral limits for the noise covariance, drift averaging on time-windows of length δ≈ε1/2, generator identification via Itô for test functions, and uniqueness of the effective SDE. Differences are stylistic (e.g., Fejér approximation for uniform averaging versus the paper’s deterministic averaging lemma), not substantive. Key steps and limiting objects (R, A from (4.3)) match the paper exactly.