Felix
Medwed
U Potsdam
A full characterisation of the signature group of bounded variation paths
Abstract.
The signature transformation of a continuous bounded variation path is the formal tensor series formed by its iterated integrals, already studied by K. T. Chen (1954) for piecewise regular paths in connection with their algebraic and topological properties. The signature has since, through its connection to controlled differential equations, played a fundamental role in the development of rough path theory by T. Lyons (1998) and appears in many areas of mathematics such as control theory, sub-Riemannian geometry, combinatorics, algebra and probability theory. Satisfying certain algebraic identities as well as universal factorial growth bounds, a fundamental question for such a transformation is then to understand both the faithfulness of the signature and its image. The first was answered by B. Hambly and T. Lyons (2010), who established uniqueness up to tree-like equivalence and explicitly formulated the image problem (Problem 1.12). Since signatures are group-like elements satisfying factorial decay, but the group-like property alone is insufficient, as illustrated by the root obstruction by T. Lyons (2014), the image question has remained present in the literature, e.g. W. Xu and H. Boedihardjo (2020); I. Chevyrev et al. (2026). A first characterisation of the image was obtained geometrically, through the connection with sub-Riemannian geometry (P. K. Friz and N. Victoir (2006, 2010)) by E. Le Donne and R. Züst (2021). They identified the image using the limiting Carnot–Carathéodory metric on an inverse system of free nilpotent Lie groups. In joint work with S. Paycha and A. Schmeding (2026), we extended this metric perspective to higher variation and raised the question of whether group-likeness together with uniform factorial growth already characterises the bounded variation signature group. We refer to this question as the G1 conjecture. Following my recent presentation at TU Berlin (01.10.2026) on the length conjecture (J. Chang, T. Lyons and H. Ni (2018); H. Boedihardjo and X. Geng (2019, 2023)), I will present a proof of the G1 conjecture. In this way we establish that every group-like tensor series satisfying the universal factorial estimate is the signature of a bounded variation path, whose minimal length is exactly the optimal growth bound. The proof uses finite-dimensional unitary developments (I. Chevyrev and T. Lyons (2016)) together with the connection to entire functions of exponential type, already noticed by T. Lyons and N. Sidorova (2006). Applying Bernstein's inequality to complex perturbations of unitary developments yields a commutator estimate, which, through the Peter–Weyl theorem and Carnot–Carathéodory duality, provides sharp geometric endpoint bounds. Controllability and weak-star compactness of bounded controls, using the Banach–Alaoglu theorem and the finite-intersection property, then provide one common control realizing all unitary developments, and hence the prescribed signature. The result also extends to signatures over free higher-step nilpotent Lie algebras. Finally, I will also present a natural Gp generalisation as well as possible next steps. In particular, the characterisation of smooth signature subgroups using the differentiable parametrization results of J. Duda, which provides a connection to the smooth rough paths introduced by C. Bellingeri, P. K. Friz, S. Paycha and R. Preiß. Another question raised by A. Schmeding of whether the signature group admits a suitable infinite-dimensional Lie group structure with a possible Silva topology while respecting the root obstruction will be commented on as well (Joint work in progress with Rosa Preiß and Alexander Schmeding).