Moment determinacy for Brownian signatures stopped at a domain boundary
Let $d \ge 2$, let $D \subset \mathbb{R}^d$ be a nonempty bounded connected open set with $C^\infty$ boundary, let $z \in D$, and let $B$ be a standard Brownian motion started at $z$, with exit time $\tau_D = \inf\{t \ge 0 : B_t \notin D\}$. The signature of $B$ on $[0, \tau_D]$ is the sequence of iterated Stratonovich integrals $$S^0(B) = 1, \qquad S^n(B) = \int_{0 < t_1 < \dots < t_n < \tau_D} \circ dB_{t_1} \otimes \dots \otimes \circ dB_{t_n} \in (\mathbb{R}^d)^{\otimes n},$$ each of which is integrable; $(\mathbb{E}_z S^n(B))_{n \ge 0}$ is its expected signature. Lyons and Ni showed that the expected signature solves an elliptic system with a boundary condition and is geometrically bounded when $D$ is strongly Lipschitz and of class $C^m$, $m = \lfloor d/2 \rfloor + 1$. Chevyrev and Lyons proved that the law of a random signature is determined by its expected signature when the latter has infinite radius of convergence. For the unit disc and $z$ its centre, Boedihardjo, Diehl, Mezzarobba and Ni proved that the radius of convergence is finite, and Li and Ni proved it finite at every $z$ for every bounded $C^{2,\alpha}$ domain when $2 \le d \le 8$. Liu, Lyu and Ni (2026) proved that a random continuous geometric rough path with the same expected signature that also satisfies their analyticity conditions (ROC) has the same signature law.
Problem. Let $Y$ be a random continuous geometric $q$-rough path in $\mathbb{R}^d$ on $[0,1]$, for some deterministic finite $q \ge 1$, all of whose signature levels are integrable, with $\mathbb{E}\, S^n(Y) = \mathbb{E}_z S^n(B)$ for every $n \ge 0$. Must $S(Y)$ and $S(B)$ have the same law on $\prod_{n \ge 0} (\mathbb{R}^d)^{\otimes n}$ with its product Borel $\sigma$-field, for every such $D$, $z$ and $Y$?
Source: Lyons and Ni (2015), abstract and introduction, who ask whether the expected signature of Brownian motion stopped at its exit time from a bounded domain determines the law of its signature. Stated by AIM — Open Applied Problems for bounded connected domains with smooth boundary in dimension d ≥ 2, with competitors among random geometric rough paths with integrable signatures.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Keywords
- AIM Open Applied Problems, expected signatures, moment problems, rough paths
- Literature checked
- Nov. 5, 2026 by clemens-admin
References
@article{lyons2015,
author = {Lyons, Terry and Ni, Hao},
title = {Expected signature of {B}rownian motion up to the first exit time from a bounded domain},
journal = {Ann. Probab.},
volume = {43},
number = {5},
year = {2015},
pages = {2729--2762},
doi = {10.1214/14-AOP949},
eprint = {1101.5902},
archivePrefix = {arXiv},
}
@article{chevyrev2016,
author = {Chevyrev, Ilya and Lyons, Terry},
title = {Characteristic functions of measures on geometric rough paths},
journal = {Ann. Probab.},
volume = {44},
number = {6},
year = {2016},
pages = {4049--4082},
doi = {10.1214/15-AOP1068},
eprint = {1307.3580},
archivePrefix = {arXiv},
}
@article{boedihardjo2021,
author = {Boedihardjo, Horatio and Diehl, Joscha and Mezzarobba, Marc and Ni, Hao},
title = {The expected signature of {B}rownian motion stopped on the boundary of a circle has finite radius of convergence},
journal = {Bull. Lond. Math. Soc.},
volume = {53},
number = {1},
year = {2021},
pages = {285--299},
doi = {10.1112/blms.12420},
eprint = {1905.13034},
archivePrefix = {arXiv},
}
@article{li2022,
author = {Li, Siran and Ni, Hao},
title = {Expected signature of stopped {B}rownian motion on {$d$}-dimensional {$C^{2,\alpha}$}-domains has finite radius of convergence everywhere: {$2 \le d \le 8$}},
journal = {J. Funct. Anal.},
volume = {282},
number = {12},
year = {2022},
pages = {109447},
doi = {10.1016/j.jfa.2022.109447},
eprint = {2011.07917},
archivePrefix = {arXiv},
}
@misc{liu2026,
author = {Liu, Chong and Lyu, Zijiu and Ni, Hao},
title = {The {$L^q$}-bounds for derivatives of unitary developments of random continuous geometric rough paths},
year = {2026},
eprint = {2609.06980},
archivePrefix = {arXiv},
}
@misc{aim-moment-determinacy-for-brownian-signatures-stopped-at-a-domain-boundary,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Moment determinacy for Brownian signatures stopped at a domain boundary}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/4f5b1659355cb61d194da94ef14615214e1363a8/problems/997-moment-determinacy-for-brownian-signatures-stopped-at-a-domain-boundary.md},
note = {Entry 997 at commit 4f5b165, 2026-11-04},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01159, "Moment determinacy for Brownian signatures stopped at a domain boundary", clopen.solutions.