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.
Organisations
People |
ORCID iD |
Terry Lyons (Principal Investigator) | |
Zhongmin Qian (Co-Investigator) |
Publications
Boedihardjo H
(2016)
The signature of a rough path: Uniqueness
in Advances in Mathematics
Boedihardjo H
(2015)
Uniform Factorial Decay Estimates for Controlled Differential Equations
in Electronic Communications in Probability
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
(2015)
Evolving communities with individual preferences
in Proceedings of the London Mathematical Society
Cass T
(2013)
On The Error Estimate for Cubature on Wiener Space
in Proceedings of the Edinburgh Mathematical Society
Cass T
(2013)
Integrability and tail estimates for Gaussian rough differential equations
in The Annals of Probability
Chen G
(2010)
A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions
in Indiana University Mathematics Journal
Chen G
(2009)
The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non-flat boundaries
in Acta Mathematica Scientia
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 Analysis
Flint G
(2016)
Discretely sampled signals and the rough Hoff process
in Stochastic Processes and their Applications
Flint Guy
(2015)
Pathwise approximation of SDEs by coupling piecewise abelian rough paths
in arXiv e-prints
Geng X
(2016)
On an inversion theorem for Stratonovich's signatures of multidimensional diffusion paths
in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
Gyurkó L
(2010)
Mathematics in Finance
Hambly B
(2010)
Uniqueness for the signature of a path of bounded variation and the reduced path group
in Annals of Mathematics
Le Jan Y
(2012)
Stratonovich's signatures of Brownian motion determine Brownian sample paths
in Probability Theory and Related Fields
Liang G
(2011)
Backward stochastic dynamics on a filtered probability space
in The Annals of Probability
Liang G.
(2010)
A Functional Approach to FBSDEs and Its Application in Optimal Portfolios
in arXiv e-prints
Litterer C
(2012)
High order recombination and an application to cubature on Wiener space
in The Annals of Applied Probability
Lyons T
(2015)
Expected signature of Brownian motion up to the first exit time from a bounded domain
in The Annals of Probability
LYONS T
(2015)
The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations
in Journal of the Mathematical Society of Japan
Lyons T
(2013)
A uniform estimate for rough paths
in Bulletin des Sciences Mathématiques
Lyons T
Hyperbolic development and inversion of signature
in arXiv preprint, to appear in Journal of Functional Analysis
Lyons T
(2009)
A signed measure on rough paths associated to a PDE of high order: results and conjectures
in Revista Matemática Iberoamericana
Lyons Terry
(2014)
Inverting the signature of a path
in arXiv e-prints
Lyons Terry J.
(2014)
Integration of time-varying cocyclic one-forms against rough paths
in arXiv e-prints
Qian Z
(2018)
Reflected backward stochastic differential equations with resistance
in The Annals of Applied Probability
Qian Z
(2011)
Differential structure and flow equations on rough path space
in Bulletin des Sciences Mathématiques
Qian Z
(2012)
Séminaire de Probabilités XLIV
Description | 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 |