## B3

Project B3: Stochastic systems with delay

Research Team: P. Catuogno (UNICAMP), P. Ruffino (UNICAMP), M. Scheutzow (TUB)

Outline: Stochastic differential equations with delay or, more generally, stochastic functional differential equations (SFDEs) correspond to considering, at each time of the solution, the velocity (or directions of noise perturbation) in a direction determined by the recent past of this trajectory. It models many real-world phenomena in biology, engineering, climate, gravitation, mathematical finance etc. If one considers the scale of nanoseconds (for electronic systems, e.g.), strictly speaking, any real world phenomenon presents a delay.
In this functional context, interesting mathematical questions arise: conditions for existence and uniqueness of solutions, long-term behaviour of the infinite dimensional dynamics, existence of invariant probability measures, continuous dependence on initial conditions, stable sub-manifolds and others.
It has been observed that for a large class of SFDEs (called "regular"), the solution depends continuously upon the initial condition but there are many (even linear) SFDEs which are singular. While the dynamics of regular SFDEs has been studied in some detail (eg regarding stable/unstable manifolds) this is not at all the case for singular SFDEs. In ongoing joint work of P. Ruffino and M. Scheutzow, we study the solutions of a very simple one-dimensional model SFDE without drift term and try to establish a version of Oseledec's theorem on the existence of a deterministic Lyapunov spectrum.

Research within the German group (PhD Supervisor: M. Scheutzow): Building on the anticipated result of an Oseledec type theorem for the model SFDE, we will try to generalize the approach to more general (multi-dimensional) linear SFDEs and then to the linearization of a nonlinear SFDE. The ultimate aim is to obtain a stable manifold theorem for singular SFDEs. In the linear case, it is natural to consider the projection of the solution of the SFDE onto the unit sphere in a suitable function space and to show that the projected (Markov) process is ergodic with respect to a unique invariant measure. Establishing uniqueness of the invariant measure is the first challenge and requires newly established coupling methods.

Research within the Brazilian group (PhD Supervisor: P. Ruffino): Some results on the geometrical aspects of SFDEs in a differentiable manifold have been done by Catuogno and Ruffino [Cat-1]. Still many interesting open questions arise in this context: 1) dependence on the functional; e.g. for a Dirac functional, varying the interval of delay; 2) a model for an SFDE driven by a general semimartingale (with jumps), extending the so called "Markus equations".