Constrained random phenomena using rough paths

Random process are ubiquitous throughout the natural and man-made world. These processes are often observed, measured or experienced as paths evolving over time in some state space. The effect of noise often makes the trajectory of these evolving paths unpredictable and highly erratic. In simple examples, such as the movement of a share price, the state space might be the non-negative real numbers. In more general examples the dynamics of the evolution may be constrained, such as the movement of a rigid body, or the evolution of particles in a section of the earth's atmosphere. This project will develop broad modelling framework for the analysis of highly-energetic random trajectories on curved spaces. A key ingredient of this will be the use Lyons' rough path analysis.

The precise study of Brownian motion over the last century has led to spectacular results in the modelling of natural phenomena. Our understanding of manifold-valued Brownian motion was given great impetus by the Eells-Elworthy-Malliavin global construction of Riemannian Brownian motion. It is now however increasingly well understood that model based on Brownian motion are not always appropriate; persistence, long-time dependence and momentum are long-observed features of behaviour in queueing networks for internet-traffic, in hydrology, and in the fluctuation of market prices. Brownian motion belongs to a fundamental class of random processes in statistics called Gaussian processes. This class is both simple enough to work with, and broad enough to capture random memory-effects in evolving systems.

In this project will will combine techniques from stochastic analysis, probability the theory of rough paths and stochastic differential geometry to study, in a precise and quantitative way, properties of a class of Gaussian processes on Riemannian manifolds. We expect there to be interesting interplay between the randomness and the geometry of the space which the process inhabits. A key objective of the project will be to furnish the wider scientific community with deeper understanding and techniques which they can utilise in their work.

The project will help to consolidate the established research strengths of the UK. By bringing together researchers from the stochastic analysis on manifolds and the rough path community it will give rise to new collaborations and thereby improve the efficiency of exchange between different branches of the subject. Academic channels and networks will be used to explore the implications of the projects key theoretical findings. We expect large impact down the line in the context of control theory and practical engineering problems; the project will provide new techniques in which practitioners in this field can formulate and analyse models. Visitors will be invited throughout the project with the aim of accelerating dissemination and hence impact. A workshop with participants from a variety of backgrounds (including industry) will further promote the impact objectives.