MC 5501
Speaker
Title
Stabilization by rough signals via dynamical systems: SBR and occupation measures, and Rademacher chaos
Abstract
Rough path analysis has provided new tools to tackle problems of regularity and regularization of (singular) stochastic systems so far
investigated with methods of stochastic calculus and analysis, control theory, (backwards) stochastic differential equations, and stochastic partial differential equations. Stabilization by noise is a traditional area in which random dynamical systems investigated by mathematicians meet applied and engineering research directions that tackle concrete and complex stochastic control problems for instance for jet engines.
In the spirit of rough path analysis, the problem of stabilization by noise has been recast in much more general terms as a problem of regularization by rough signals. Research has been done to determine which properties of rough paths are able to stabilize a given singular ODE. It has been seen that the existence and regularity of occupation measures and their densities plays a decisive role in this area. In the research work we will follow we also bridge the gap between aspects of rough path analysis and (random) dynamical systems.
We investigate geometric properties of graphs of Takagi or Weierstrass type functions, represented by series based on smooth functions. For instance, Gaussian randomizations of Takagi curves just reproduce the individual trajectories of Brownian motion. The curves are H¨older continuous, and can be embedded into smooth dynamical systems, where their graphs emerge as pullback attractors. It turns out that occupation measures and Sinai-Bowen-Ruelle (SBR) measures on their stable manifolds are dual by ”time” reversal. In this sense, absolute continuity of the SBR measure is seen to be dual to the existence of occupation densities. The investigation of questions of smoothness both for SBR as for occupation measures surprisingly leads to the Rademacher version of Malliavin’s calculus, Bernoulli convolutions and into probabilistic number theory.