N-point motions in Random Dynamical Systems

随机动力系统结合了经典的、确定性的数学模型，旨在捕捉系统的控制本质，即其主要驱动力，以及称为噪声的外部随机波动，为我们提供了一个更现实的框架来描述广泛的过程[1]。为了研究这些系统，从随机分析、遍历理论和最近的分叉理论发展了稳健的分析工具，不仅允许对模型进行统计解释，还允许对其行为进行动态路径解释。在这种背景下，我们专注于研究随机动力系统中的几个(N)粒子，我们称之为n点运动，以超越通常对模型的单点、统计和概率描述。从两点运动出发，在随机微分方程的情况下，特别是对于微分同胚的随机流，H.Kurita在1990年[2]证明了过程的规律完全由两点运动来刻画，换言之，对系统内任意两个粒子的动力学的了解产生了对流的完全描述。事实上，对随机动力系统中两点运动的研究对于描述同步也是至关重要的，并且与该领域的基本概念如Lyapunov指数或系统的熵等密切相关。已经观察到两点运动不变度量上的分叉和相变之间的密切联系，这提供了对潜在系统及其动力学的更丰富的理解。然而，能够描述这一主题的完整理论还有待开发。本项目的目的是识别和分析这种新的机制，因为我们获得了隐藏在两点运动动力学背后的信息，并继续建立对系统n点运动的完整描述，揭示这种复杂模型的属性。本项目属于EPSRC统计和应用概率以及非线性系统研究领域。

Random dynamical systems combine classical, deterministic, mathematical models designed to capture the governing essence of a system, i.e. its main driving forces, together with external stochastic fluctuations known as noise, providing us with a significantly more realistic framework to describe a wide range of processes [1]. In order to study these systems, robust analytical tools have been developed from stochastic analysis, ergodic theory and more recently bifurcation theory, allowing not only for a statistical explanation of the model but also for a dynamical path-wise interpretation of its behavior.In this setting, we focus on the study of several (n) particles within a random dynamical system, which we refer to as the n-point motion, to go beyond the usual single-point, statistical, and probabilistic description of a model. Starting with the two-point motion, in the case of stochastic differential equations and particularly for stochastic flows of diffeomorphisms, it was shown by H. Kunita in 1990 [2] that the law of the process is fully characterized by the two-point motion, or in other words that knowledge of the dynamics of any two particles evolving within the system yields a full description of the flow. Indeed, the study of the two-point motion in random dynamical systems is also crucial for the description of synchronization and closely relates to fundamental notions in the field such as Lyapunov exponents or the system's entropy amongst others.More recently, Homburg et al. have observed a close link between the bifurcations on the invariant measure of the two-point motion and phase transitions that provide a much richer understanding of the underlying system and its dynamics. However, a complete theory able to describe this topic is yet to be developed.The aim of this project is to identify and analyze such novel mechanisms, as we access the hidden information behind the two-point motion's dynamics, and continue by building towards a full description of the system's n-point motion, uncovering the properties of such complex models.This project falls within the EPSRC statistics and applied probability, and non-linear systems research areas.