A Quasi-sure Non-degeneracy Property for the Brownian Rough Path

TL;DR

This paper proves that, except on a slim set, the signature of Brownian motion is non-self-intersecting and uniquely determines the path, highlighting a key non-degeneracy property in rough path theory.

Contribution

It establishes a quasi-sure non-degeneracy property for Brownian rough paths, linking signature uniqueness to non-self-intersecting behavior.

Findings

Brownian signature paths are non-self-intersecting quasi-surely

Brownian rough paths lack tree-like pieces quasi-surely

Paths are uniquely determined by their signatures up to reparametrization

Abstract

In the present paper, we are going to show that outside a slim set in the sense of Malliavin (or quasi-surely), the signature path (which consists of iterated path integrals in every degree) of Brownian motion is non-self-intersecting. This property relates closely to a non-degeneracy property for the Brownian rough path arising naturally from the uniqueness of signature problem in rough path theory. As an important consequence we conclude that quasi-surely, the Brownian rough path does not have any tree-like pieces and every sample path of Brownian motion is uniquely determined by its signature up to reparametrization.