AI 中文总结
本文为跳跃驱动的随机微分方程建立线性响应理论，从概率密度、平均逃逸时间和逃逸概率三方面推导一阶响应公式，刻画系统对系数扰动的敏感性。
AI 中文摘要
本文针对一类由跳跃过程驱动的随机微分方程，发展了线性响应理论。我们从三个互补的视角研究了系统对小的时间依赖扰动的响应：概率密度函数、平均逃逸时间和逃逸概率。通过对相应的前向和反向Kolmogorov方程进行扰动分析，我们推导了控制这些统计量的一阶响应方程，并建立了显式的线性响应公式，这些公式刻画了概率分布演化和系统逃逸行为对其系数变化的敏感性。
英文摘要
In this paper, we develop a linear response theory for a class of stochastic differential equations driven by jump processes. We investigate the response of the system to small time-dependent perturbations from three complementary perspectives: probability density functions, mean exit times, and escape probabilities. By performing perturbation analyses of the corresponding forward and backward Kolmogorov equations, we derive the first-order response equations governing these statistical quantities and establish explicit linear response formulas, which characterize the sensitivity of the evolution of the probability distribution and the escape behavior of the system to variations in its coefficients.
CommentsThe numerical experiments have not yet been completed, and some parts of the manuscript contain errors, for example, the definition of the stopping time \(τ_D^{t,x}\) in Theorem 5.3 is incorrect. We will resubmit the manuscript after making the necessary corrections and improvements
