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 $({\epsilon }_t{)}_{t\ge 0}$ satisfying ${\epsilon }_t\to 0$ as $t\to \mathrm{\infty }$. 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 $L^p$-estimate between the slow variable and the averaged ODE at late times, with the classical strong convergence rate of order $1\mathrm{/}2$. Under an additional decay condition on $({\epsilon }_t{)}_{t\ge 0}$, 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.