Stochastic Rotating Waves

The paper proves an exponential-in-κ^2/ε^2 concentration bound for the transverse fluctuation vt := ut − T_{βt}u* under a variational phase constraint on SE(2), via time-window freezing, semigroup decay on the stable subspace, quadratic Taylor remainders, and exponential martingale bounds; see Theorem 4.3 with the choice Δt = (log(4C))/b and a union bound across windows . The candidate solution reproduces this structure: the same SE(2)-modulation, the same window-length ensuring a 1/4 contraction on the stable subspace, the same control of nonlinear remainders (H2-algebra/Nemytskii), and an exponentially small tail for the stochastic convolution. Differences are present only in technical packaging (e.g., invoking Pinelis-type martingale tails instead of Brzeźniak–Peszat/Chernoff bounds), but the logic and estimates align with the paper’s proof.