Details of Grant 

EPSRC Reference:	GR/S18526/01
Title:	Partial Differential Equations - A rough path approach
Principal Investigator:	Lyons, Professor T
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematical Institute
Organisation:	University of Oxford
Scheme:	Standard Research (Pre-FEC)
Starts:	01 June 2003	Ends:	30 November 2006	Value (£):	140,503
EPSRC Research Topic Classifications:	Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
The connection between linear second order parabolic and elliptic partial differential equations and the theory of diffusion processes is well established and important. Gelfand asked if this connection extends beyond linear 2nd order operators. Krylov and Hochberg made an initial connection (between linear higher order operators and finitely additive signed measure on cylindrical sets of paths supported on paths of finite p variation for p > d.).More recently the PI has established a theory of differential equations driven by rough paths that has within its scope paths of finite p variation. The proposed RA has used this to develop effective numerical techniques for numerical solution of partial differential equations using quadrature on Wiener space and applying to sub-elliptic parabolic PDEs (e.g. those that occur in pricing of Asian options and convertible bonds in finance) where the core presumption of micro-local analysis (regularisation of solutions) is simply not satisfied. This project aims to substantially deepen the connection between rough path space, analysis and partial differential equations. We expect close connections between distributions on the rough paths and higher order equations and aim to constructing solutions to non-linear second order equations as solutions of control problems driven by the deterministic rough path known as the Brownian Ensemble .
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