arXiv subjects

Felix Medwed

Publications and source records attributed to Felix Medwed.

3 recordsLinked to original sources

arXiv2026

Recovering the Length of a Path from its Signature

We prove the length conjecture of J. Chang, T. Lyons and H. Ni for paths of bounded variation for arbitrary real Banach spaces and any reasonable tensor (algebra) norms. More precisely, we establish the missing lower bound and in this way show that the length of a tree-reduced path is recovered from the asymptotic behaviour of its normalised signature. The main technical argument uses the norm duality and shuffle algebra to express natural powers of polygonal length approximations as combinations of linear functionals acting on restricted signatures, which are then glued by suitable integrations into a linear functional of controlled growth acting on the full signature. In particular, the full proof combines tools of classical analysis and functional analysis, such as localisation by continuous cutoff functions, Stone-Weierstrass approximation of continuous functions by signature polynomials, the Hahn-Banach theorem and a support detection lemma for compactly supported real signed Borel measures.

math.FA↗

arXiv2026

A universal characterisation of the Bochner integral

We provide a universal characterisation of the Banach spaces of equivalence classes of $L^p$ Bochner integrable functions, under equality almost everywhere, as initial objects in appropriate categories of bifunctors, for all $p \in [1,\infty)$. As a by-product of our construction, we also obtain a unique characterisation of the Bochner integral.

math.FA↗

arXiv2026

Metric Geometry of the Signature Group for $p$-Variation Rough Paths

The signatures of $p$-rough paths form a subgroup of sufficiently high-level truncated tensor algebras, whose inverse limit is a subgroup of the full tensor algebra. For $p \geq 1$, we provide a top-down description of the signature group as the inverse limit of finite-dimensional Carnot--Carathéodory geometries in the $p$-variation setting. We show that every compatible choice of metrics induces a topological tree structure on the inverse-limit group, under which the signature group is not a topological group. This extends the results of Enrico Le Donne and Roland Züst from bounded variation to rough paths. We also characterise the dependence of the inverse-limit groups and their metric completions on the choice of metric, identifying them with the tree-reduced path group of Horatio Boedihardjo, Xiang Geng, Terry Lyons, and Danyu Yang.

math.MG↗