Expected signature of stopped Brownian motion on $d$-dimensional $C^{2, \alpha}$-domains has finite radius of convergence everywhere: $2\leq d \leq 8$

TL;DR

This paper proves that the expected signature of stopped Brownian motion on certain regular domains in dimensions 2 to 8 has a finite radius of convergence everywhere, contrasting with previous results on simpler domains.

Contribution

It introduces a domain-averaging hyperbolic development technique to analyze the expected signature's convergence properties on complex domains.

Findings

Expected signature has finite radius of convergence everywhere on $C^{2,\alpha}$-domains.

The method applies to dimensions 2 through 8.

Introduces a new symmetrization technique via domain-averaging hyperbolic development.

Abstract

A fundamental question in rough path theory is whether the expected signature of a geometric rough path completely determines the law of signature. One sufficient condition is that the expected signature has infinite radius of convergence, which is satisfied by various stochastic processes on a fixed time interval, including the Brownian motion. In contrast, for the Brownian motion stopped upon the first exit time from a bounded domain $\Omega$, it is only known that the radius of convergence for the expected signature on sufficiently regular $\Omega$ is strictly positive everywhere, and that the radius of convergence is finite at some point when $\Omega$ is the $2$-dimensional unit disc ([1]). In this paper, we prove that on any bounded $C^{2,\alpha}$-domain $\Omega \subset \mathbb{R}^d$ with $2\leq d \leq 8$, the expected signature of the stopped Brownian motion has finite radius of convergence everywhere. A key ingredient of our proof is the introduction of a "domain-averaging hyperbolic development" (see Definition 4.1), which allows us to symmetrize the PDE system for the hyperbolic development of expected signature by averaging over rotated domains.