We study non-Markovian stochastic differential equations with additive noise characterized by a Poisson point process with arbitrary pulse shapes and exponentially distributed intensities. Specifically, analytic results concerning transitions between different correlation regimes and the long-time asymptotic probability distribution functions are shown to be controlled by the shape of the pulses and dissipative parameter as time progresses. This program is motivated by the study of stochastic partial differential equations perturbed by space Poisson disorder and becomes the main focus of applications of the present exact functional approach.

中文翻译：

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泊松噪声、泊松飞行和泊松波动的精确结果

我们研究具有加性噪声的非马尔可夫随机微分方程，其特征是具有任意脉冲形状和指数分布强度的泊松点过程。具体而言，关于不同相关机制和长期渐近概率分布函数之间的转换的分析结果表明，随着时间的推移，受到脉冲形状和耗散参数的控制。该计划的动机是研究受空间泊松障碍扰动的随机偏微分方程，并成为当前精确泛函方法应用的主要焦点。