Rough path analysis and non-linear stochastic systems

Many phenomena in nature appear as chaotic complex evolutions in time. Mathematicians often study these random movements by means of differential equations. In general these equations are very complicated with many unknown variables, and very often include noise. The rough path analysis which has been developed over the last decade by the principle investigator, with the Co-investigator and other co-workers, is the machinery which accurately describes the evolutions governed by chaotic complex systems. A key perspective in this rough path theory is the observation that the net information embedded in a complex evolution system can be completely described by its signature. When compared with the use of moments, the signature represents a huge step forward as a means for describing non-linear chaotic systems. In contrast to moments, it easily captures the order of events, and hence measures bulk velocity, rotation and much more besides. In this project we further develop the analysis of rough paths and establish a theory of differential equations with inputs involving complicated ensembles of random, interacting particles. We intend to apply the theory to study the signatures of turbulent flows modelled by stochastic evolution systems. The research developed during the progress of the project is to provide cutting-edge technologies for the study of high dimensional, complicated chaotic systems, for example turbulent flows.