A piecewise deterministic Markov process approach modeling a dry friction problem with noise

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 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. In particular, the analysis of the power spectral density of the velocity reveals a critical value of the noise correlation time below which the correlations of the dynamic behaviors coincide with those of the white noise limit. The existence of such a critical value was already mentioned in the physical literature [Geffert and Just, Phys. Rev. E, 95 (2017), 062111]. © 2023 Society for Industrial and Applied Mathematics.