AI 中文总结
针对带时变尺度分离参数且满足 $\u03b5_t \to 0$ 的快慢随机微分方程，本文利用冻结快动力学的耗散性建立强平均原理，证明慢变量与平均常微分方程的最大 $L^p$ 估计，给出其收敛到渐近稳定平衡点的判据。
AI 中文摘要
我们建立了带时变尺度分离参数 $(\u03b5_t)_{t \geq 0}$ 的快慢随机微分方程的强平均原理，其中当 $t \to \infty$ 时 $\u03b5_t \to 0$。与基于噪声诱导平滑或椭圆正则性的方法不同，我们的方法依赖于冻结快动力学的耗散性，因此允许退化扩散系数。我们证明了慢变量与平均常微分方程在后期的最大 $L^p$ 估计，其经典强收敛阶为 $1/2$。在 $(\u03b5_t)_{t \geq 0}$ 满足额外衰减条件下，该估计意味着慢变量几乎必然是平均常微分方程的渐近伪轨迹。因此，我们通过分析平均方程的动力学行为，获得了慢变量的可能极限点识别及收敛到渐近稳定平衡点的判据。
英文摘要
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.
