Lyapunov exponent

This project aims to get a better understanding of the bifurcation scenarios for noise-dependent random dynamical system which display a transition from trivial to emergence of chaos, as the noise strength increases. It is known that this transition is observed by the change of the sign of a statistical quantity called Lyapunov exponent, but little is known of the geometric mechanism generating this transition. It is known that in the deterministic system this transition to chaos is accompanied with a stretching and folding mechanism and the emergence of chaotic sets called horseshoes. In this project, we aim to check whether similar objects exist for the random settings or to establish the existence of analogous ones.
The importance of understanding the mechanism that rule chaotic dynamical systems is important in all sciences as numerous phenomena, for example climatic and medical ones, display chaotic behaviour in the sense that they are very sensitive to initial condition: a small modification somewhere leads to a big change in the outcome.
The first aim of this project is to characterize topologically the behaviour of random systems with positive Lyapunov exponent, in particular trying to establish the existence of random version of horseshoes generated by homoclinic intersections. Another goal is to establish a qualitative characterization of the phenomenon of transient chaos, which means that trajectories still converge to an equilibrium, but the convergence is slow and non uniform in the state space. Very little is known about the phenomenon of transient chaos, so the aim is to get a better understanding at it.
Currently, there is almost no literature on the existence of horseshoe for random dynamical systems, except for some very specific cases, and the results are much weaker than the deterministic analogous. The main issue is that current technique fail at estimating return times when randomness kicks in.
Also the phenomenon of transient chaos is poorly understood, and the main known results are related to prototypical examples and it is impossible to develop some general theory. Indeed, current techniques are privileged of random systems generated by stochastic differential equation, in which the property of Brownian motion allows to compute some statistics via Kolmogorov equations, which are not in general computable.
In order to overcome this issue, we developed the tool of random hyperbolic times and adopt large deviations technique in order to control the measure of the points in the state space which slower expansion rates, and used probabilistic techniques to establish a measure one existence of a random version of an horseshoe for a class of non uniformly hyperbolic random systems with positive Lyapunov exponents.
To our knowledge, we are the first to obtain this sort of result and we expect this kind of result to be generalized to a larger class of systems.
This project has collaborators, Dr Jeroen Lamb and Dr Dimitry Turaev.
This project falls within the EPSRC statistics and applied probability research area.

Probabilistic modelling permeates the Financial services, healthcare, technology and other Service industries crucial to the UK's continuing social and economic prosperity, which are major users of stochastic algorithms for data analysis, simulation, systems design and optimisation. There is a major and growing skills shortage of experts in this area, and the success of the UK in addressing this shortage in cross-disciplinary research and industry expertise in computing, analytics and finance will directly impact the international competitiveness of UK companies and the quality of services delivered by government institutions.
By training highly skilled experts equipped to build, analyse and deploy probabilistic models, the CDT in Mathematics of Random Systems will contribute to
- sharpening the UK's research lead in this area and
- meeting the needs of industry across the technology, finance, government and healthcare sectors

MATHEMATICS, THEORETICAL PHYSICS and MATHEMATICAL BIOLOGY

The explosion of novel research areas in stochastic analysis requires the training of young researchers capable of facing the new scientific challenges and maintaining the UK's lead in this area. The partners are at the forefront of many recent developments and ideally positioned to successfully train the next generation of UK scientists for tackling these exciting challenges.
The theory of regularity structures, pioneered by Hairer (Imperial), has generated a ground-breaking approach to singular stochastic partial differential equations (SPDEs) and opened the way to solve longstanding problems in physics of random interface growth and quantum field theory, spearheaded by Hairer's group at Imperial. The theory of rough paths, initiated by TJ Lyons (Oxford), is undergoing a renewal spurred by applications in Data Science and systems control, led by the Oxford group in conjunction with Cass (Imperial). Pathwise methods and infinite dimensional methods in stochastic analysis with applications to robust modelling in finance and control have been developed by both groups.
Applications of probabilistic modelling in population genetics, mathematical ecology and precision healthcare, are active areas in which our groups have recognized expertise.

FINANCIAL SERVICES and GOVERNMENT

The large-scale computerisation of financial markets and retail finance and the advent of massive financial data sets are radically changing the landscape of financial services, requiring new profiles of experts with strong analytical and computing skills as well as familiarity with Big Data analysis and data-driven modelling, not matched by current MSc and PhD programs. Financial regulators (Bank of England, FCA, ECB) are investing in analytics and modelling to face this challenge. We will develop a novel training and research agenda adapted to these needs by leveraging the considerable expertise of our teams in quantitative modelling in finance and our extensive experience in partnerships with the financial institutions and regulators.

DATA SCIENCE:

Probabilistic algorithms, such as Stochastic gradient descent and Monte Carlo Tree Search, underlie the impressive achievements of Deep Learning methods. Stochastic control provides the theoretical framework for understanding and designing Reinforcement Learning algorithms. Deeper understanding of these algorithms can pave the way to designing improved algorithms with higher predictability and 'explainable' results, crucial for applications.
We will train experts who can blend a deeper understanding of algorithms with knowledge of the application at hand to go beyond pure data analysis and develop data-driven models and decision aid tools
There is a high demand for such expertise in technology, healthcare and finance sectors and great enthusiasm from our industry partners. Knowledge transfer will be enhanced through internships, co-funded studentships and paths to entrepreneurs