Non-explosion of Brownian Fleming–Viot systems in arbitrary domains
Let $d \ge 2$, let $D \subset \mathbb{R}^d$ be a nonempty bounded connected open set, with no regularity assumed on $\partial D$, and let $N \ge 3$. The Brownian Fleming–Viot system of $N$ particles in $D$ starts from a deterministic configuration $x = (x_1, \dots, x_N) \in D^N$, coincident positions allowed. The particles move as independent standard Brownian motions (generator $\tfrac12 \Delta$); when a particle hits $\partial D$, it jumps instantly to the current position of one of the other $N - 1$ particles, chosen uniformly, and the motion continues with fresh independent increments and choices. Let $\tau_k$ be the time of the $k$th jump and $\tau_\infty = \lim_{k \to \infty} \tau_k$; the system is defined up to $\tau_\infty$. Burdzy, Hołyst and March stated that $\tau_\infty = \infty$ almost surely for every domain, but their proof contains an error. Grigorescu and Kang proved it for domains satisfying interior and exterior cone conditions with a sufficiently large interior aperture, and Bieniek, Burdzy and Finch for Lipschitz domains with a sufficiently small Lipschitz constant, depending on $d$ and $N$. Kwaśnicki proved it for two particles in every bounded open set, from every initial configuration.
Problem. Is $\mathbb{P}_x(\tau_\infty = \infty) = 1$ for every $d \ge 2$, every such $D$, every $N \ge 3$ and every $x \in D^N$, that is, are there almost surely only finitely many jumps in every bounded time interval?
Source: Burdzy, Hołyst and March (2000), Theorem 1.1, asserted for every domain with a proof that contains an error; posed for bounded connected open sets by Burdzy, My favorite open problems, Problem 6. Stated by AIM — Open Applied Problems for d ≥ 2 and N ≥ 3.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, Fleming–Viot particle systems, killed Brownian motion
References
@article{burdzy2000,
author = {Burdzy, Krzysztof and Ho{\l}yst, Robert and March, Peter},
title = {A {F}leming--{V}iot particle representation of the {D}irichlet {L}aplacian},
journal = {Comm. Math. Phys.},
volume = {214},
number = {3},
year = {2000},
pages = {679--703},
doi = {10.1007/s002200000294},
}
@misc{burdzyopen,
author = {Burdzy, Krzysztof},
title = {My favorite open problems},
howpublished = {Web page},
url = {https://sites.math.washington.edu/~burdzy/open_mathjax.php},
note = {Problem 6. Undated; accessed 14 October 2026},
}
@article{bieniek2012,
author = {Bieniek, Mariusz and Burdzy, Krzysztof and Finch, Sam},
title = {Non-extinction of a {F}leming--{V}iot particle model},
journal = {Probab. Theory Related Fields},
volume = {153},
number = {1-2},
year = {2012},
pages = {293--332},
doi = {10.1007/s00440-011-0372-5},
eprint = {0905.1999},
archivePrefix = {arXiv},
}
@article{grigorescu2012,
author = {Grigorescu, Ilie and Kang, Min},
title = {Immortal particle for a catalytic branching process},
journal = {Probab. Theory Related Fields},
volume = {153},
number = {1-2},
year = {2012},
pages = {333--361},
doi = {10.1007/s00440-011-0347-6},
}
@article{kwasnicki2024,
author = {Kwa{\'s}nicki, Mateusz},
title = {Fleming--{V}iot couples live forever},
journal = {Probab. Theory Related Fields},
volume = {188},
number = {3-4},
year = {2024},
pages = {1385--1408},
doi = {10.1007/s00440-023-01247-z},
eprint = {2307.04143},
archivePrefix = {arXiv},
}
@misc{aim-brownian-fleming-viot-nonexplosion,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Non-explosion of Brownian Fleming–Viot systems in arbitrary domains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/348-brownian-fleming-viot-nonexplosion.md},
note = {Entry 348 at commit a153f46, 2026-10-13},
}
Resolution claims
We resolve the long-standing non-extinction problem for Brownian Fleming-Viot systems in bounded domains. For every fixed N >= 3, we construct a bounded connected domain in R3 in which branching explosion occurs with positive probability before every prescribed positive time, from every interior initial configuration. On this event all particles converge to one boundary point. Together with two-particle non-extinction, this …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.18802v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00401, "Non-explosion of Brownian Fleming–Viot systems in arbitrary domains", clopen.solutions.