In how far are rough paths characterized by their signature?

04.02.2022, 11:00  –  Hybrid, Online und 2.22
Arbeitsgruppenseminar Analysis

Rosa Preiss (Uni Potsdam)

The aim of this talk is to give a definite answer to the question posed (in its earliest form) by Chen some 65 years ago, whether or not the signature, i.e. the full time increment footprint in the group-like elements, characterizes the path. We show that the signature uniquely characterizes all reduced rough paths if and only if the primitive elements are free as a Lie algebra.

This is carried out in the general setting of finite p-variation rough paths living in the group-like elements of a connected graded cocommutative Hopf algebra of finite type (cgccft Hopf algebra). Using the notion of tree-like equivalence of continuous paths in a metric space originally introduced in 2005, which this talk will present both via visiual examples and rigorously, we may look at the following surprisingly concise precise statement: For a cgccft Hopf algebra, there exists a p>1 and two finite p-variation rough paths which share a common signature while not being tree-like equivalent, if and only if the Lie algebra of primitive elements of the Hopf algebra is NOT a free Lie algebra over a Lie generating subset.

After a short excursion into the theory of smooth rough paths, we may actually show the existence of counterexamples in the non-free-Lie setting within so-called genuinely smooth rough paths.

If time permits, we may briefly touch the connection of the uniqueness statement with the problem of introducing a notion of area between two arbitrary rough path components.

For those of you who can and would like to join us, please meet us in the seminar Room 2.22 of the maths institute, where we can follow the talk together. Please let us know in advance if you plan to come to the institute.

You are welcome to invite your friends and colleagues to join us! If you wish to attend the talks, please contact Sylvie Paycha paycha@math.uni-potsdam.de for the login details.