Inhalt des Dokuments
Differential equations are the common denominator of all modern exact sciences, may it be physics, chemistry, engineering or economics and social sciences: the modelling of an evolving system with many variables leads almost inevitably to (partial) differential equations. Since the central aspect of various applications is the intrinsic randomness, a novel type of mathematics, distinct from deterministic PDE theory, was came in the form of (martingale based) stochastic analysis, a theory that had already proved extremely useful in the classical analysis of stochastic differential equations, as developed by Ito, Stroock-Varadhan (and many others).
On the other hand, in the 90's, T. Lyons devised a revolutionary theory, known as rough path analysis, which fully overcame the gap between ordinary and stochastic differential equations. Tremendous progress has been made in the application of rough path ideas to the construction of solutions of stochastic partial differential equations. One may distinguish three main directions based on different methods:
- the method of rough flow transformation pursued by P. K. Friz et al.
- paracontrolled distributions by M. Gubinelli, P. Imkeller, N. Perkowski
- regularity structures by M. Hairer
Our research program aims to push forward the understanding in these directions. Various aspects are treated within five projects, whose description can be found below.