Geometry for the continuous field of controlled rough paths

28.10.2022, 11:00  –  Raum 2.22 + Zoom
Arbeitsgruppenseminar Analysis

Alexander Schmeding (Nord Universitet, Norway)

In rough path theory one seeks to solve differential equations driven by rough signals (i.e. Hölder paths of low regularity). To this end one needs to take a close look at the technical setup enabling to work with such rough signals. This leads to the concept of controlled paths over a given rough path. In this talk we present some recent results on these spaces of controlled paths over the space of all branched rough paths. It turn out that while there does not seem to exist a smooth vector bundle structure for these objects, one can construct a weaker structure of a so called continuous field of Banach spaces. This concept from non-commutative geometry can be used to phrase stability of rough equations and rough dynamical systems. We shall explain how this result follows from algebraic work by Foissy and a novel approximation result to rough paths. Time permitting we will highlight some perspectives as to how this geometric perspective could be exploited in applications.
This is based on joint work with S. Riedel, M.G. Varzaneh and N. Tapia.

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