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

Horatio Boedihardjo, Xi Geng, Xuan Liu, Zhongmin Qian

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.

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

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.

Paper Structure

This paper contains 7 sections, 11 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Formulation of Main Result
  3. Proof of the Main Theorem
  4. A Maximal Type Capacity Estimate
  5. A Small Ball Capacity Estimate
  6. Kakutani's Sub-division Argument
  7. Final Remarks

Key Result

Theorem 2.1

For $n\in\mathbb{N}$, define where $S_n(\mathbf{w})_{0,t}$ is the truncated Brownian signature path up to degree $n$. Then $\mathcal{O}_{n}$ has zero $(r,q)$-capacity provided In particular, the Brownian signature path is non-self-intersecting quasi-surely.

Theorems & Definitions (25)

  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 3.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 15 more