多尺度随机环境中常微分方程的平均原理
Averaging Principle for Ordinary Differential Equations in a Multiscale Random Environment
AI总结：
该研究针对快慢随机系统，在适当条件下证明向量场依赖快慢过程的微分方程解依测度收敛到仅依赖慢过程的平均系统，将经典平均理论推广到受多分量快随机环境影响的系统。
AI中文摘要：
我们研究快慢随机系统，其中快过程在若干遍历分量内演化，驱动在这些分量间切换的慢跳跃过程的转移率。考虑一个向量场依赖于两个过程的微分方程，我们证明，在适当的遍历性和正则性条件下，其解在$D([0,T],\boldsymbol{R}^d)$中依测度收敛到仅依赖慢过程的平均系统。平均漂移通过对每个分量内快过程的不变测度上的原始向量场取平均得到。我们的结果将经典平均理论推广到向量场受具有多分量动力学的快随机环境显式影响的系统。
英文摘要：
We study slow-fast stochastic systems in which a fast process, evolving within several ergodic components, drives the transition rates of a slow jump process switching between these components. Considering a differential equation whose vector field depends on both processes, we show that, under suitable ergodicity and regularity conditions, its solutions converge in law in $D([0,T],\mathbb R^d)$ to an averaged system depending only on the slow process. The averaged drift is obtained by averaging the original vector field over the invariant measure of the fast process within each component. Our results extend classical averaging theory to systems where the vector field is explicitly influenced by a fast random environment with multi-component dynamics.
