Asymptotic analysis of multiscale Lévy-driven stochastic Cucker-Smale and non-linear friction models

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Various real-world phenomena such as flocking/dispersion of animal populations or dissipation effects in mechanical systems can be realistically described with the help of randomly perturbed non-linear Newtonian equations of motion. The qualitative behaviour of such systems is often determined by the non-linear and position dependent dissipative friction force. The project mainly focuses on the analysis of two paradigmatic models: a Cucker-Smale type model of flocking and a mechanical model of motion under non-linear position dependent friction. These models are fed with weak Lévy perturbations, the most general class of white noises which includes the Brownian motion and stable Lévy processes, which operate on the microscopic time scale. The thorough analysis of novel interplay effects between the non-linear and stochastic dynamics which become perceivable on the longer macroscopic time scales constitutes the crux of the project. We will observe such asymptotic regimes where the limiting process is either a diffusion (diffusion approximation regime) or a discontinuous Lévy-type process (non-linear Lévy filter regime). As a main mathematical tool for our analysis, we will develop new mathematical techniques based on the ``long-step semi-martingale'' regression scheme which will allow to catch the ergodic behaviour of fully coupled multi-scale systems. Eventually, we will develop easy-to-treat tools to analyse the response of the collective behaviour of a flock to various control policies such as a mild control term at the macroscopic time scale or a censoring a part of the perturbations depending on the current risk of the flock to be ruined. The results obtained in the project will contribute substantially to the general asymptotic theory of multi-scale stochastic systems, and will advance the understanding of the non-linear effects in realistic stochastic models of physics, biology,and applied sciences.