Differential Equations and Rough Path Theory

An ordinary differential equation is often used to model the movement of a particle. Similarly, partial differential equation can be used to described the evolution of an entire distribution of particles. It is natural to add randomness to such models: sometimes because this is a more realistic description which takes into account random noise, sometimes because this randomness is fundamental to the model itself as is the case for the stock market. A good model of random noise, such as the celebrated Brownian motion, cannot evolve smoothly in time (otherwise the noise would be predictable on a small scale!). As a result, stochastic perturbations of differential equation are intrinsically irregular and require fundamentally new methods and theories. The ground-breaking contributions of It, which allowed to make all this possible, is one of the great achievements of 20th century mathematics. With all its benefits It's theory can be fragile and some questions that arise naturally in applications require major effort or cannot be treated at all. In essence, this restrictions come from the fact that stochastic integrals are transforms of fair games (think of Brownian motion as the limit of an unbiased or fair random walk) and the integrand (= the gambling strategy ). Life would be much easier if one could fix a noise-scenario and then apply a standard theory of (non-random) differential equations. It was only realized in 1998 that this is indeed possible and the corresponding theory has been labelled rough path theory . There is a conceptual price to pay: Brownian motion on the familiar Euclidean space has to be replaced by a stochastic process with values in a so-called Lie group. The first part of the proposed research aims to understand to what extent Brownian motion (on the Lie-group) may be replaced by other Gaussian processes and to give a full characterization of such Gaussian rough paths . There is a real chance for a unified theory which brings together many particular cases such as fractional Brownian motion and so-called Volterra processes. Potential applications include models of mathematical Finance. The second part of the research is aimed at partial differential equations with noise. (Such equations arise in many fields of applied science.) A deterministic theory of such equations with noise in the rough sense would provide a robust and constructive approach to stochastic partial differential equations, improving both our understanding of these equations as well as how to treat them numerically.