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

• URL: http://arxiv.org/abs/2608.23462v1
• Date: Mon, 24 Aug 2026 16:30:59 GMT
• Title: Strong Averaging Principle and Long-Time Dynamics for Fast-Slow SDEs with Increasing Time-Scale Separation and Degenerate Noise
• Abstract summary: We establish a strong averaging principle for fast-slow differential equations with a time-dependent scale-separation parameter.<n>In contrast to approaches based on noise-induced smoothing or regularity, our approach relies on dissipativity of the frozen fast dynamics.
• Score: 1.0026496861838448
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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