A topological approach to mapping space signatures

Abstract

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 $[0,1]\to R^n$ 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 ${[0,1]}^d\to R^n$ for any $d\in N$, and show that the map Φ generalizes many of the desirable algebraic and analytic properties of the path signature to $d\ge 2$. 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.