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

借助随机扰动的非线性牛顿运动方程，可以逼真地描述各种真实世界的现象，如动物种群的聚集/分散或机械系统中的耗散效应。这类系统的定性行为通常由非线性和位置相关的耗散摩擦力决定。该项目主要集中于分析两种典型模型：Cucker-Smear类型的群集模型和非线性位置依赖摩擦下的运动力学模型。这些模型被馈送了弱Lévy扰动，这是最一般的一类白噪声，包括布朗运动和稳定的Lévy过程，它们在微观时间尺度上运行。彻底分析在更长的宏观时间尺度上可以察觉到的非线性和随机动态之间的新的相互作用效应是该项目的关键。我们将观察到这样的渐近区，其中极限过程是扩散(扩散近似区)或不连续的Lévie型过程(非线性Lévy滤波区)。作为我们分析的主要数学工具，我们将开发新的数学技术，其基于“长步长半鞅”回归方案，它将允许捕捉完全耦合的多尺度系统的遍历行为。最终，我们将开发易于处理的工具来分析羊群的集体行为对各种控制政策的反应，例如宏观时间尺度上的温和控制项，或者根据当前羊群被摧毁的风险来审查部分扰动。该项目所获得的结果将对多尺度随机系统的一般渐近理论做出重大贡献，并将促进对物理、生物和应用科学的现实随机模型中的非线性效应的理解。

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.