Rough Path Theory (RPT)

Minisymposium at the Fifth European Congress of Mathematics

Place  Time	Amsterdam July 17, 13:25 – 14:55 and 15:25 – 16:55
Speakers:	(Plenary) Prof. Sandy M Davie (University of Edinburgh) http://www.maths.ed.ac.uk/people/show/person/59 Discrete approximation to solutions of rough path equations. We consider 'rough path' differential equations of the form dy(t)=f(y(t))dx(t), where the (vector) driving path x(t) is non-differentiable, as introduced by Lyons. We develop an approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. Results on the convergence of discrete approximations are deduced. Christian Littererer (University of Oxford, email: litterer@maths.ox.ac.uk ) Numerical Analysis on Wiener Space: Cubature Methods We will survey recent progress related to cubature on Wiener space. Cubature on Wiener space is a high order method for the approximation of weak solutions to Stratonovich stochastic differential equations and was developed by T. Lyons and N. Victoir following the work of S. Kusuoka. In addition we will describe how recent work by D. Levin and T. Lyons on fourth order differential operators may lead to extensions of these methods. Massimiliano Gubinelli (University of Paris, Orsay) http://www.math.u-psud.fr/~gubinell/ Some infinite dimensional rough-paths. Lyons' theory or rough paths is by now a well-established technique to study ODE driven by irregular signals like paths of stochastic processes. In this talk we will review recent attempts to use the fundamental ideas of the theory to analyze infinite-dimensional differential equations like (nonlinear) stochastic partial differential equations (SPDEs) but also deterministic examples like the Korteweg-de-Vries (KdV) equation or the Navier-Stokes (NS) equation. In each example we will show the existence of an appropriate notion of rough-path which controls the time-evolution of the solution and which is not necessarily related to an external driving noise but can be generated by the intrinsic non-linearity of the equation like in the KdV or NS examples. To treat the SPDE and NS examples we are led to reformulate the integration theory in a form suitable to the convolutional integrals appearing the the mild formulation of the problems. In the context of the KdV equation we will show how conserved quantities reflect on the rough path expansion and can be used to prove global existence results. The exact form of the rough path depends on the equation considered and moreover in these settings the rough paths are indexed by (rooted, planar) trees and not by the integers (as it happens in the finite-dimensional theory). We argue that this is the right setting to study general, non-geometric, rough paths. We will also discuss the algebra associated to such tree-indexed rough paths and its relation with the Hopf-algebra on rooted trees of Connes and Kreimer. Euler-like finite difference methods to approximate the solutions of the rough equations will also be introduced. Peter Friz (University of Cambridge) http://www.statslab.cam.ac.uk/~peter/ On some properties of SDEs and linear SPDEs with Gaussian rough noise A stochastic differential equations can be viewed as solution to a rough differential equation driven by Brownian motion + Levy's area. Using rough path theory, this extends to many Gaussian signals other than Brownian motion. Under natural conditions, the solution flow is Frechet-smooth in (and beyond ...) Cameron-Martin directions. Assuming furthermore a Hoermander condition on the vector fields and some technical conditions on the signal, we show that the solution admits a density. The novelty in the argument is to replace the classical Doob-Meyer decomposition by a variation of the Stroock-Varadhan support theorem which is valid in absence of a semi-martingale setting. (Joint work with T. Cass.) We then consider a class of linear PDEs which involve a (smooth) driving signal. By a Limiting procedure, we are led to "rough PDEs" which depend continuously on a driving rough path. When applied in a Gaussian context, we obtain a robust theory of SPDEs with Gaussian noise for which a support theorem and large deviations are readily established. Subject to non-degeneracy, we also have a density result. (Joint work with M. Caruana.)
Organizer	Peter Friz (University of Cambridge)
Links	The   Fifth European Congress of Mathematics  (14–18 July 2008) Other  Minisymposia at the 5ecm