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

2026-08-24

AI Summary

本文解决了快慢随机微分方程在时间尺度分离参数逐渐减小至零情况下的强平均原理问题，采用冻结快速动态的耗散性方法，并允许退化噪声的存在。

Abstract

We establish a strong averaging principle for fast-slow stochastic differential equations with a time-dependent scale-separation parameter $(\varepsilon_t)_{t \geq 0}$ satisfying $\varepsilon_t \to 0$ as $t \to \infty$. 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 $1/2$. Under an additional decay condition on $(\varepsilon_t)_{t \ge 0}$, 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.

Problem

• fast-slow SDEs
• time-scale separation
• degenerate noise
• averaging principle
• dissipativity

Innovation

• strong averaging principle
• fast-slow SDEs
• degenerate noise
• dissipativity of frozen fast dynamics
• asymptotic pseudo-trajectory