# Rough path analysis and non-linear stochastic systems

Lead Research Organisation:
University of Oxford

Department Name: Mathematical Institute

### Abstract

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.

## People |
## ORCID iD |

Terry Lyons (Principal Investigator) | |

Zhongmin Qian (Co-Investigator) |

### Publications

Boedihardjo H
(2015)

*Uniform Factorial Decay Estimates for Controlled Differential Equations*in Electronic Communications in Probability
Boedihardjo H
(2016)

*The signature of a rough path: Uniqueness*in Advances in Mathematics
Boutaib Y
(2014)

*Dimension-free Euler estimates of rough differential equations*in Rev. Roumaine Math. Pures Appl.
Cass T
(2015)

*Smoothness of the density for solutions to Gaussian rough differential equations*in The Annals of Probability
Cass T
(2013)

*On The Error Estimate for Cubature on Wiener Space*in Proceedings of the Edinburgh Mathematical Society
Cass T
(2015)

*Evolving communities with individual preferences*in Proceedings of the London Mathematical Society
Cass T
(2013)

*Integrability and tail estimates for Gaussian rough differential equations*in The Annals of Probability
Chevyrev Ilya
(2013)

*Characteristic functions of measures on geometric rough paths*in arXiv e-prints
Crisan D
(2015)

*Kusuoka-Stroock gradient bounds for the solution of the filtering equation*in Journal of Functional AnalysisDescription | The article "Rough Paths on Manifolds" develops the appropriate theory for the rough paths to exist on manifolds. This is strtegically neccesary, and actually quite technically challenging as it needs a development of Lipchitz manifold. This paper will take time for people to see why it is essential; one interestng observation that comes out of it is the way that a rough current can be locally invisible but globally active. |

Exploitation Route | This grant gave fundamental support to the development of Rough Path theory. This area is now seen to be a central one in mathematics as witnessed by the award of a fields medal, to Martin Hairer - whose citation mentioned the earlier work of the PI. In addition the work is having a modest but novel impact on data science. |

Sectors | Digital/Communication/Information Technologies (including Software),Financial Services, and Management Consultancy,Security and Diplomacy,Other |

Description | This award and its predecessor, which was the first to apply rough path theory to data, played an important part in migrating the theory of rough paths more towards applications and impact. Strategic understandings of how ensembles of paths might interact and the fundamental nature of the signature both began to emerge into our understanding. This foundational support played an essential role in allowing rough path theory to continue its development. The current state can honestly be described as exceptional - and three external highlights of the overal development should be noted: 1) The fields medal for Martin Hairer - 2) the use of rough path techniques in the main stream software for allowing entry of Chinese handwriting into mobile devices, (4 billion characters translated) and 3) The award of an advanced grant by the ERC to continue this area of research. The methods are also mainstream in the financial sector for computations (Ninomya and Victoir) |

First Year Of Impact | 2008 |

Sector | Digital/Communication/Information Technologies (including Software),Financial Services, and Management Consultancy,Other |

Impact Types | Cultural,Economic |