Alec Kriebel

An explicit joint Laplace transform for Brownian first-visit cell lengths on a circle

By Alec Kriebel, independent researcher.
Manuscript and declared publication date · Zenodo metadata version 1.0 · Unrefereed preprint

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Original abstract

We give a complete analytic-transform answer to Georgakopoulos’s question about the lengths of the sets first visited by independent Brownian particles on a circle, with uniform random or equally spaced starting points. For every finite number of particles, the joint Laplace transform is an absolutely convergent series whose coefficients are finite sums of ordinary integrals of explicit interval-exit kernels. The integration regions are specified linear inequalities in cumulative physical hitting times. The outer series has a uniform factorial remainder bound and uniquely determines the probability measure. The result also applies to arbitrary fixed distinct starting points. The construction uses classical heat kernels, strong Markov iteration, and bounded-support moments. Earlier competing-walk painting results are credited and distinguished. No compact density, efficient numerical algorithm, or absolute historical priority is asserted.

The idea in plain English

Imagine several independent particles moving randomly around a circle. Each point belongs to the particle that reaches it first. The question is how all their territory lengths vary together. The particles keep moving through encounters and already visited territory.

This paper describes the entire joint distribution through a Laplace transform. Its coefficients are finite sums of ordinary deterministic integrals built from classical Brownian interval-exit kernels. Tracking a particle’s cumulative elapsed time preserves the correlations between its visits to different points.

The formula covers every finite number of particles, arbitrary fixed distinct labeled starting points, equally spaced starts and independent uniform labeled starts. A uniform factorial bound controls truncation of the outer series. The coefficient integrals can be high-dimensional; no fast evaluator or compact density is claimed.

The precise result

Let k≥1 be a finite integer and let p₁,…,p_k be distinct fixed labeled points on the unit circle 𝕋=ℝ/ℤ. The particles are independent standard Brownian motions with generator ½d²/dx² and continue through every encounter and prior visit. Write T_i(x) for particle i’s first physical hitting time of x, assign x to the least label attaining min_i T_i(x), and let L_i be the Lebesgue length of its territory. Almost surely, L lies in the compact simplex {l_i≥0:∑_i l_i=1}.

For real θ_i≥0, set θ_*=max_i θ_i. The joint Laplace transform is ℒ_p(θ)=E_p exp(−∑_i θ_iL_i)=∑_{m=0}∞(−1)^m M_m(θ;p)/m!, with M₀=1. Every coefficient is completely specified by finite permutation and label sums and ordinary spatial/time integrals of explicit scalar interval-exit kernels in equations (2)–(9). No unknown Brownian hitting probability or conditional law remains inside those definitions.

For m queries, a particle’s ordered density is the product of its successive remaining-target kernels. If its discovery order is π_i and elapsed increments are u_ir, its physical hitting time of query j is h_ij=∑_{r≤π_i⁻¹(j)}u_ir. The integration region for owner labels a_j is given by h_(a_j,j)<h_ij for every competing i. These are explicit strict linear inequalities in cumulative clocks, retaining dependence among one walker’s query visits.

For every m≥0, 0≤M_m≤θ_*^m, and the series converges absolutely. For every integer N≥0, |ℒ_p(θ)−∑_{m=0}^N(−1)^m M_m(θ;p)/m!|≤θ_*^(N+1)/(N+1)!. This outer-series bound is uniform over all admissible seed vectors. The entire family of homogeneous coefficient polynomials determines all mixed moments and hence the full joint Borel probability measure on the simplex.

Equally spaced starts use p_i=(i−1)/k. For independent uniform labeled seeds, integrate each fixed-p coefficient over 𝕋^k; the same outer bound holds. Sorting the seeds before assigning labels instead requires the corresponding ordered seed law.

Why the result holds

Before reaching the next unvisited query point, one walker stays in an arc whose lift is an interval. The classical killed heat kernel gives its two endpoint exit-time densities. With a single query left, both lifted endpoints represent the same target and their fluxes are added, with total mass one.

The strong Markov property at successive finite discovery times gives each walker’s joint order-and-increment density. Removing a reached query changes only the auxiliary stopping targets: the actual particle continues and can cross earlier queries. Products across independent walkers and comparisons of their cumulative physical clocks give every finite-query ownership probability.

Joint measurability and Fubini show that tie locations have zero spatial measure almost surely; the least-label convention still specifies an owner. Spatial integration gives M_m=E_p Z^m for Z=∑_i θ_iL_i. Since 0≤Z≤θ_*, the exponential expansion is absolutely convergent and Taylor’s theorem gives the exact factorial outer remainder.

All homogeneous moment polynomials recover every mixed moment. Polynomial density on the compact simplex then makes the joint measure unique. The uniform bounds also justify averaging over independent uniform labeled seeds. These are classical heat-kernel, strong-Markov, moment and determinacy tools applied to the explicit cumulative-clock construction.

Scope and limitations

The model uses finitely many independent continuing Brownian particles with a common diffusivity. Stopping, coalescing or absorption at painted territory changes the model. Common positive diffusivity changes rescale all clocks equally; invariance under different per-particle diffusivities is not asserted.

Fixed starting points must be distinct and labeled. No extension to deterministically coincident seeds is claimed. Random-seed collisions, repeated query points and query/seed coincidences are discarded only on their appropriate Lebesgue-null sets.

Ties have zero spatial measure almost surely. This does not say that ties are absent simultaneously at every point. One particle’s different query hitting times remain correlated throughout.

The scalar spectral kernels are summed at each positive time before improper time integration. The factorial bound controls only the outer series for real nonnegative θ; it supplies no inner Fourier-tail, numerical quadrature or coefficient-evaluation certificate.

No compact or named density, efficient numerical algorithm, large-k asymptotic or discrete-to-Brownian convergence theorem is claimed. The Brownian result follows directly from the analytic construction; finite-model and high-precision diagnostics are supplementary evidence.

Independent uniform labeled starts have full permutation exchangeability. Equally spaced labeled starts have cyclic and dihedral symmetry; unrestricted permutation symmetry is not asserted. Both have mean territory length 1/k. A circle of circumference c has lengths cL_i, so its transform uses cθ.

First-arrival painting, interval kernels, strong Markov iteration, moment expansion and compact-support determinacy are established prior methods. Earlier lattice painting, interface, competitive-range and multitime span results receive credit. Historical priority, full novelty, independent discovery and worldwide current-open status remain unverified; incomplete coverage of the Gomes article and Dicker thesis does not support an exhaustive absence claim.

This preprint was prepared with extensive AI assistance and is unrefereed. Internal AI-assisted checks are neither independent human or journal peer review nor formal proof-assistant certification.

Status and AI use

AI tools were used extensively in mathematical development, research, checking and writing. Internal AI-assisted review does not constitute independent human or journal peer review. This preprint is unrefereed and has no formal proof-assistant certification. The named author is responsible for its claims, materials and citations.

The manuscript and Zenodo’s declared publication date are 7 October 2026. Version 1.0 is recorded in Zenodo and support metadata; the manuscript prints the date without a version number. Zenodo created the record on 7 October 2026 UTC. Its later metadata update does not identify a different manuscript edition. These edition dates do not establish historical priority.

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Cite this paper

Alec Kriebel. (2026). An explicit joint Laplace transform for Brownian first-visit cell lengths on a circle (Version 1.0, 7 October 2026) [Preprint]. Zenodo. https://doi.org/10.5281/zenodo.23224103

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Background and prior work