A Topological Approach to Mapping Space Signatures

Abstract

A common approach for describing classes of functions and probability measures on a topological space $\mathcal{X}$ is to construct a suitable map $\Phi$ from $\mathcal{X}$ into a vector space, where linear methods can be applied to address both problems. The case where $\mathcal{X}$ is a space of paths $[0,1]\to {\mathbb{R}}^n$ and $\Phi$ is the path signature map has received much attention in stochastic analysis and related fields. In this article we develop a generalized $\Phi$ for the case where $\mathcal{X}$ is a space of maps $[0,1{]}^d\to {\mathbb{R}}^n$ for any $d\in \mathbb{N}$, and show that the map $\Phi$ 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 generalisation of K-T Chen's path space cochain construction to the setting of cubical mapping spaces.

58 pages, comments welcome!