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A fundamental question in rough path theory is whether the expected signatureof a geometric rough path completely determines the law of signature. Onesufficient condition is that the expected signature has infinite radius ofconvergence, which is satisfied by various stochastic processes on a fixed timeinterval, including the Brownian motion. In contrast, for the Brownian motionstopped upon the first exit time from a bounded domain $\Omega$, it is onlyknown that the radius of convergence for the expected signature on sufficientlyregular $\Omega$ is strictly positive everywhere, and that the radius ofconvergence is finite at some point when $\Omega$ is the $2$-dimensional unitdisc ([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 thestopped Brownian motion has finite radius of convergence everywhere. A keyingredient of our proof is the introduction of a "domain-averaging hyperbolicdevelopment" (see Definition 4.1), which allows us to symmetrize the PDE systemfor the hyperbolic development of expected signature by averaging over rotateddomains.

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