Strong Averaging Principle and Long-Time Dynamics for Fast-Slow SDEs with Increasing Time-Scale Separation and Degenerate Noise

We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter (varepsilont)t ≥ 0 satisfying varepsilont → 0 as t → ∞. In contrast to approaches based on noise-induced smoothing or elliptic regularity, our approach relies on dissipativity of the frozen fast dynamics and therefore permits degenerate diffusion coefficients. We prove a maximal Lp-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order 1/2. Under an additional decay condition on (varepsilont)t ≥ 0, this estimate implies that the slow variable is almost surely an asymptotic pseudo-trajectory of the averaged ODE. As a consequence, we obtain criteria for the identification of possible limit points and for convergence toward asymptotically stable equilibria for the slow variable by analyzing the dynamical behavior of the averaged equation.