Felix
Medwed
U Potsdam
Recovering the Length of a Path from its Signature
Abstract.
A central question in rough path theory is how geometric information about a path is reflected in its signature, the formal series of its iterated path integrals. In their seminal work, B. Hambly and T. Lyons (2010) showed that, for bounded variation paths in finite-dimensional spaces, the signature transformation determines the path up to a natural notion of equivalence, called tree-like equivalence. Moreover, every such equivalence class contains a unique representative that has no redundant backtracking, named the tree-reduced path. This result was later generalised by H. Boedihardjo, X. Geng, T. Lyons and D. Yang to weakly geometric rough paths in arbitrary real Banach spaces. In the same work, B. Hambly and T. Lyons proved, under additional regularity assumptions and for suitable tensor norms, that the length of the tree-reduced path can be recovered from the asymptotic behaviour of its normalised signature. Subsequently, J. Chang, T. Lyons and H. Ni (2018) established the corresponding limit to exist also for general bounded variation paths over arbitrary Banach spaces and conjectured the same recovery to hold for the choice of reasonable tensor algebra norms. This problem is now commonly referred to as the length conjecture and H. Boedihardjo and X. Geng (2022) later gave a more precise formulation as well as proving the conjecture for a class of planar tree-reduced paths using sl₂-developments. The main difficulty of the conjecture lies in establishing a suitable lower estimate for the normalised signature. Our approach, while inspired by the underlying initial strategy of H. Boedihardjo and X. Geng, differs in that we relate natural powers of polygonal approximations back to the signature, and identify a family of continuous functionals that allow for controlled growth. In this talk, I will explain the core ideas behind this approach and show how, together with the existence result of Chang–Lyons–Ni, they give the required lower bound. This allows us to prove the length conjecture in the general setting.