Cartan's Path Development, the Logarithmic Signature and a Conjecture of Lyons-Sidorova

Abstract

The signature transform, which is defined in terms of iterated path integrals of all orders, provides a faithful representation of the group of tree-reduced geometric rough paths. While the signature coefficients are known to decay factorially fast, the coefficients of the logarithimic signature generically only possess geometric decay. It was conjectured by T. Lyons and N. Sidorova that the only tree-reduced paths with bounded variation (BV) whose logarithmic signature can have infinite radius of convergence are straight lines. This conjecture was confirmed in the same work for certain types of paths and the general BV case remains unsolved. The aim of the present article is to develop a deeper understanding towards the Lyons-Sidorova conjecture. We prove that, if the logarithmic signature has infinite radius of convergence, the signature coefficients must satisfy an infinite system of rigid algebraic identities defined in terms of iterated integrals along complex exponential one-forms. These iterated integral identities impose strong geometric constraints on the underlying path, and in some special situations, confirm the conjecture. As a non-trivial application of our integral identities, we prove a strengthened version of the conjecture, which asserts that if the logarithmic signature of a BV path has infinite radius of convergence over all sub-intervals of time, the underlying path must be a straight line. Our methodology relies on Cartan's path development onto the complex semisimple Lie algebras $\mathfrak{sl}_m(\mathbb{C})$. The special root patterns of $\mathfrak{sl}_m(\mathbb{C})$ allow one to project the infinite-dimensional Baker-Campbell-Hausdorff (BCH) formula in a very special finite dimensional manner to yield meaningful quantitative relations between BCH-type singularities and the vanishing of certain iterated path integrals.