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

arXiv: https://arxiv.org/abs/2608.23462 · HTML: https://mlsanity.com/paper/2608.23462/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 31 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 4 of 4)

- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Non-convergence of Adam and other adaptive stochastic gradient descent optimization methods for non-vanishing learning rates](https://mlsanity.com/paper/2407.08100/) (2024)
- [ODE approximation for the Adam algorithm: General and overparametrized setting](https://mlsanity.com/paper/2511.04622/) (2025)

## Abstract (arXiv)

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.