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

各种现实世界的现象，如动物种群的群集/分散或机械系统中的耗散效应，都可以在随机扰动的非线性牛顿运动方程的帮助下真实地描述。此类系统的定性行为通常由非线性且与位置相关的耗散摩擦力决定。该项目主要集中在两个范例模型的分析：Cucker-Smale型植绒模型和非线性位置相关摩擦下的运动力学模型。这些模型被喂以弱Lévy扰动，最一般的一类白色噪声，其中包括布朗运动和稳定的Lévy过程，在微观时间尺度上操作。深入分析非线性和随机动力学之间的新的相互作用效应，在更长的宏观时间尺度上变得可感知，构成了该项目的关键。我们将观察到这样的渐近状态，其中极限过程是扩散（扩散近似状态）或不连续的Lévy型过程（非线性Lévy滤波器状态）。作为我们分析的主要数学工具，我们将开发新的数学技术的基础上的“长步半鞅”回归计划，这将允许捕捉遍历行为的完全耦合的多尺度系统。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根据羊群当前的风险来审查部分扰动。在该项目中获得的结果将大大有助于多尺度随机系统的一般渐近理论，并将推进物理学，生物学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.