QUENCHED STATISTICAL LIMIT LAWS FOR RANDOM DYNAMICAL SYSTEMS

The proposed project concerns the study of statistical properties of a broad class of discrete time random dynamical systems, which includes systems arising from mechanical and biological models. A random dynamical systems evolution is given by the compositions of maps randomly chosen from some family. In contrast to deterministic dynamical systems, where the rule is predetermined, in our situation the rule is changed randomly at each step of the evolution. This models the situations where random forces are present in the system. The systems of our interest are chaotic in nature, and in particular have sensitive depends on initial conditions. This means that the future of orbits that started from nearby points can be dramatically different, and this compromises the study of exact orbits, and requires a statistical approach. The quenched setting means that we investigate the evolution of almost every random realisation. We aim to create rigorous mathematical framework to study such systems. It turns out that chaotic systems have intimate relations with random events as coin tossing: outcomes are random but statistically predictable. For example, if one tosses a fair coin infinitely often then asymptotically half of time heads and half of the time tails are observed. But for practical purposes such a beautiful result is not sufficient, one wants to know more quantitative results on how fast these statistical results are approached as the number of experiments increase. These type of results allow us to give estimates for the deviations from expected values when the experiments are repeated finitely many times. The aim of this project is to provide statistical limit laws which implies various quantitative results for broad class of random dynamical systems. Since the 1960s the statistical analysis of dynamical systems has attracted enormous attention of mathematicians and physicists. This started with the study of toy models and moved towards more realistic models. The project suggests the next step in this direction. The main novelty of the proposed project is that it provides statistical limit laws for observations closer to what we see in real life. In reality most dynamical systems are not purely deterministic, but usually contaminated by noise, hence random. In practice, we observe finitely many realisations of the random system. Studying the statistics of almost every realisation of the dynamics is more useful for practical purposes. Thus we obtain statistical information about the dynamics of physically relevant dynamical systems.

In the project, we analyzed chaotic systems subject to random perturbations. In particular, we obtained mixing rates for systems that are closer to realistic models. Our results imply that in many cases observations mode over these models behaves like coin tossing: random but statistically predictable. This confirms our hypothesis in the project. The results are published in research papers associated with the project.