A topological approach to mapping space signatures

A common approach for describing classes of functions and probability measures on a topological space X is to construct a suitable map Φ from X into a vector space, where linear methods can be applied to address both problems. The case where X is a space of paths [0,1]→Rn and Φ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized Φ for the case where X is a space of maps [0,1]d→Rn for any d∈N, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to d≥2. The key ingredient to our approach is topological; in particular, our starting point is a generalization of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.