Brownian First-Visit Cell Lengths on the Circle

Definitions and mathematical statement

Let $k\geq 1$ be an integer and let $p_1,\ldots,p_k$ be chosen points on the unit circle \[ S^1=\{x\in{\mathbb{R}}^2:\|x\|=1\}. \] Interpret the chosen points as deterministic parameters. Let $X_1,\ldots,X_k$ be independent standard Brownian motions on $S^1$, with $X_i(0)=p_i$, all starting at the same time and using the same time scale. Here standard circle Brownian motion means Brownian motion with generator $\tfrac12\partial_\theta^2$ in angular coordinates modulo $2\pi$. These process and timing conventions make explicit the usual interpretation of the supplied wording; they are not specified separately in the record.

For $x\in S^1$, define the first hitting time \[ T_i(x)=\inf\{t\geq 0:X_i(t)=x\},\qquad \inf\varnothing=+\infty. \] Under the provisional convention that a point belongs to a cell only when its winning particle is unique, define \[ B_i^{<}=\{x\in S^1:T_i(x)<T_j(x)\text{ for every }j\ne i\}. \] Thus tied first visits are left unassigned under this convention. The supplied source does not prescribe a tie rule. Other interpretations include assigning tied points to every minimizing particle or using a specified rule to select one minimizer; equivalence of the resulting cell lengths is not asserted here.

Let $\ell$ denote arclength on $S^1$, so that $\ell(S^1)=2\pi$. Under the preceding conventions, the requested problem is: for every $k\geq1$ and every prescribed starting configuration $(p_1,\ldots,p_k)$, determine the joint probability law \[ \mathcal L\!\left(\ell(B_1^{<}),\ldots,\ell(B_k^{<})\right). \] The source writes $(|B_1|,\ldots,|B_k|)$ without defining the length normalization. If normalized circle length is intended, replace each $\ell(B_i^{<})$ by $\ell(B_i^{<})/(2\pi)$. The source also does not state whether the starting points must be pairwise distinct, or whether a random choice of starting points is intended. No distribution for random starting points is supplied. Consequently, the displayed target is an explicit fixed-configuration interpretation, and its conventions require mathematical review before being identified with the exact intended source formulation.

Short English statement

Independent Brownian particles start at chosen points on a circle. Each particle claims points that it reaches before the other particles. Determine the joint distribution of the resulting cell lengths as a function of the starting configuration. The source leaves the tie convention, length normalization, and some starting-point conventions unspecified. Its open-problem label is supplied curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30003818$, title Brownian First-Visit Cell Lengths on the Circle, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. No individual proposer is identified. The record gives proposed year $2018$.
• Supplied source citation: Enumerative Combinatorics (2019), Oberwolfach Reports, DOI 10.4171/owr/2018/23, pp. 1381--1464. Source URL and supplied literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2018/23. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. This citation is reproduced as supplied and has not been independently consulted.
• Supplied literature assessment, checked on 2026-08-21: status "open"; the curation reports that no paper resolving the exact joint distribution was found and attributes an "unpublished and unknown" description to the original report. The sole listed checked source is the original OWR report at https://doi.org/10.4171/owr/2018/23. This is supplied dated curation and literature triage, not an independently verified assessment; time-sensitive status and attributed claims require independent verification.
• The registry labels the statement corrected_verified, explaining that mathematical content was preserved while notation, typography, and sentence structure were repaired for publication. This supplied editorial designation does not establish mathematical correctness or independent source verification for this draft.