N-point motions in Random Dynamical Systems

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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.