分段确定性马尔可夫过程方法建模带噪声的干摩擦问题
A piecewise deterministic Markov process approach modeling a dry friction problem with noise
AI总结：
本文提出分段确定性马尔可夫过程方法建模带噪声的干摩擦系统，推导柯尔莫哥洛夫方程，证明遍历性并给出平稳测度表示公式，同时表征静态与动态阶段持续时间的拉普拉斯变换。
摘要：
理解和预测涉及干摩擦的系统的动力学特性是物理学和工程学中的一个主要关注点。干摩擦存在于许多机械过程中，从小提琴产生的声音到粉笔在黑板上发出的刺耳声，再到人类婴儿的爬行动力学以及众多生物体（蛇、细菌、扇贝等）基于摩擦的运动，以及地震等作用下机械结构（建筑物、桥梁、核电站、大型工业基础设施）的位移。令人惊讶的是，即使对于低维系统，在存在随机强迫的情况下干摩擦的建模也尚未被阐明。在本文中，我们提出了一种分段确定性马尔可夫过程方法，对具有干摩擦的系统进行建模，该系统包括静态力和动态力的不同系数。在这个数学框架中，我们推导了相应的柯尔莫哥洛夫方程及相关工具，用于计算与静态（粘滞）和动态阶段分布相关的感兴趣的统计量。我们证明了遍历性，并利用轨迹的独立同分布部分（游程）提供了平稳测度的表示公式。我们还获得了静态和动态阶段持续时间的概率密度函数的拉普拉斯变换的确定性表征。
英文摘要：
Understanding and predicting the dynamical properties of systems involving dry friction is a major concern in physics and engineering. It abounds in many mechanical processes, from the sound produced by a violin to the screeching of chalk on a blackboard to human infant crawling dynamics and friction-based locomotion of a multitude of living organisms (snakes, bacteria, scallops..) to the displacement of mechanical structures (building, bridges, nuclear plants, massive industrial infrastructures) under earthquakes and beyond. Surprisingly, even for low-dimensional systems, the modeling of dry friction in the presence of random forcing has not been elucidated. In this paper, we propose a piecewise deterministic Markov process approach modeling a system with dry friction including different coefficients for the static and dynamic forces. In this mathematical framework, we derive the corresponding Kolmogorov equations and related tools to compute statistical quantities of interest related to the distributions of the static (sticked) and dynamic phases. We show ergodicity and provide a representation formula of the stationary measure using independent identically distributed portions of the trajectory (excursions). We also obtain deterministic characterizations of the Laplace transforms of the probability density functions of the durations of the static and dynamic phases.
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