Stratonovich's Signatures of Brownian Motion Determine Brownian Sample Paths

Abstract

The signature of Brownian motion in ${\mathbb{R}}^d$ over a running time interval $[0,T]$ is the collection of all iterated Stratonovich path integrals along the Brownian motion. We show that, in dimension $d\ge 2$, almost all Brownian motion sample paths (running up to time $T$) are determined by its signature over $[0,T]$