N-point motions in Random Dynamical Systems

随机动力系统结合了联合收割机经典的确定性数学模型，旨在捕捉系统的管理本质，即其主要驱动力，以及被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各种过程[1]。为了研究这类系统，从随机分析、遍历理论到最近的分岔理论都发展出了强大的分析工具，不仅可以对模型进行统计解释，还可以对其行为进行动态路径解释。在这种背景下，我们主要研究随机动力系统中的多个（n）粒子，我们称之为n点运动，超越了通常的单点，统计和概率模型的描述。从两点运动开始，在随机微分方程的情况下，特别是对于随机流的同态，它是由H。Kunita在1990年[2]指出，该过程的定律完全由两点运动来表征，或者换句话说，对系统内任何两个粒子的动力学的了解产生了对流动的完整描述。事实上，随机动力系统中两点运动的研究对于同步的描述也是至关重要的，并且与该领域的基本概念密切相关，例如李雅普诺夫指数或系统的熵等。洪堡等人观察到两个变量的不变测度上的分叉之间有密切的联系，点的运动和相变，提供了一个更丰富的了解基础系统及其动力学。然而，一个完整的理论能够描述这个主题还有待开发。这个项目的目的是识别和分析这种新的机制，因为我们访问两点运动动力学背后的隐藏信息，并继续建立对系统的n点运动的完整描述，揭示这种复杂模型的属性。这个项目属于EPSRC统计和应用概率，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.