This paper studies a random differential equation with random switch perturbations. We explore how the maximum displacement from the equilibrium state depends on the statistical properties of time series of the random switches. We show a power law dependence between the upper bound of displacement and the frequency of random perturbation switches, and the slope of power law dependence is dependent on the specific distribution of the intervals between switching times. This result suggests a quantitative connection between frequency modulation and amplitude modulation under random perturbations.

中文翻译：

---

随机微分方程中的幂律依赖

本文研究了具有随机开关扰动的随机微分方程。我们探讨了平衡状态的最大位移如何取决于随机开关时间序列的统计特性。我们展示了位移上限与随机扰动开关频率之间的幂律依赖性，并且幂律依赖性的斜率取决于切换时间间隔的具体分布。该结果表明在随机扰动下频率调制和幅度调制之间存在定量联系。