Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

Jacob Mercer

arxiv: 2510.12701 · v2 · submitted 2025-10-14 · 🧮 math.PR · math.AP

Two-Sided Free Boundary Problems Arising From Branching-Selection Particle Systems

Jacob Mercer This is my paper

Pith reviewed 2026-05-18 07:28 UTC · model grok-4.3

classification 🧮 math.PR math.AP

keywords branching Brownian motionfree boundary problemhydrodynamic limittravelling waveparticle systemselection processinverse first passage problem

0 comments

The pith

As N tends to infinity, the empirical distribution of the (N,p)-BBM converges to the solution of a two-sided free boundary problem on a finite interval with Neumann and Dirichlet boundary conditions parametrized by p.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a two-sided branching-selection particle system called the (N,p)-BBM that generalizes the standard N-particle branching Brownian motion by deleting either the leftmost or rightmost particle at each branching event with probability p. It establishes that the empirical distribution of particle positions converges as N goes to infinity to the solution of a deterministic two-sided free boundary problem on a finite interval whose boundaries move and whose boundary conditions are a mixture of Neumann and Dirichlet types controlled by p. The paper also shows that the asymptotic velocity of the particle system converges to the unique travelling-wave speed of this free boundary problem. Existence and regularity of the limit problem are proved using a connection to inverse first-passage problems. A sympathetic reader would care because this provides a hydrodynamic limit for a new class of particle systems and links them to free boundary problems that arise in population dynamics and flame propagation.

Core claim

The paper proves that as N tends to infinity the empirical distribution of the (N,p)-BBM converges to the solution of a two-sided free boundary problem on a finite interval with Neumann and Dirichlet boundary conditions parametrized by p, and the asymptotic velocity v_{N,p} converges to the unique travelling-wave speed v_p of that problem. Existence and regularity of the free boundary problem is obtained by appealing to a connection with inverse first-passage problems.

What carries the argument

The two-sided free boundary problem with p-parametrized Neumann and Dirichlet boundary conditions on a finite interval with two moving boundaries, which describes the hydrodynamic limit and supplies the travelling-wave speed for the velocity.

If this is right

The standard one-sided N-BBM is recovered when p tends to 0 or 1.
The limiting velocity is the unique speed admitting a travelling wave solution to the free boundary problem.
The model connects particle systems to free boundary problems in evolutionary dynamics and flame propagation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same selection rule could be applied to other branching processes to generate additional families of free boundary problems.
The inverse first-passage link might be used to find explicit formulas for the speed v_p.
Large but finite N simulations could be used to test the rate at which the particle system approaches the free boundary description.

Load-bearing premise

The convergence of the empirical distribution and the velocity rest on the existence and regularity of the solution to the two-sided free boundary problem, which is justified by its connection to inverse first-passage problems.

What would settle it

A simulation of the (N,p)-BBM for successively larger N in which the empirical distribution does not approach the predicted free boundaries or the velocity does not approach v_p would disprove the claimed limit.

read the original abstract

We introduce and analyse a two-sided branching-selection particle system which generalises the well-known $N$-particle branching Brownian motion ($N$-BBM) model, which we call the $(N,p)$-BBM, where either the leftmost or rightmost particle is deleted at each branching event according to a parameter $p\in(0,1)$. We establish that, as $N\to\infty$, the empirical distribution of the $(N,p)$-BBM converges to a deterministic hydrodynamic limit described by a free boundary problem on a finite interval with two moving boundaries, and Neumann and Dirichlet boundary conditions parametrized by $p$. Again, this generalises the one-sided free boundary problem which characterises the hydrodynamic limit of the $N$-BBM. Existence and regularity of the free boundary problem is also proved, by appealing to a connection with inverse first passage problems. We further prove that the asymptotic velocity $v_{N,p}$ of the $(N,p)$-BBM converges, as $N\to\infty$, to $v_p$, the unique travelling wave speed of the limiting free boundary problem. These results generalize previous one-sided models and connect to broader classes of free boundary problems found in evolutionary dynamics and flame propagation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines the (N,p)-BBM with biased deletion at either end and derives its hydrodynamic limit as a two-sided free-boundary PDE with mixed Neumann-Dirichlet conditions plus velocity convergence.

read the letter

The main point is a generalization of the N-BBM to a two-sided version called (N,p)-BBM, where p decides whether to delete the leftmost or rightmost particle after branching. The empirical measure converges to the solution of a free boundary problem on a finite interval with one moving boundary having a Neumann condition and the other a Dirichlet condition, both depending on p. The asymptotic velocity of the particles converges to the traveling wave speed of that PDE. This is new because the one-sided models in the literature do not have this mixed boundary setup or the p-parameter controlling the bias. The paper connects the limit to inverse first-passage problems to prove existence and regularity of the free boundary, which avoids reinventing PDE theory for this case. The particle system is defined clearly and the hydrodynamic limit follows from the usual tightness and identification arguments for branching-selection models. The velocity result is a direct consequence once the limit PDE is in hand. That part looks solid. The softer part is the reliance on the inverse first-passage connection for the two-sided problem. Those results typically need the free boundary to be sufficiently regular, say Lipschitz, and the diffusion to satisfy monotonicity conditions. With mixed Neumann and Dirichlet boundaries, it is not immediate that everything carries over without extra work. The abstract does not spell out the verification, so if the paper does not check these conditions explicitly for the p-dependent case, that leaves a gap in the justification. It is a minor concern if the proofs address it, but worth a closer look. This paper is aimed at people working on branching particle systems and their PDE limits, especially those interested in free boundary problems from evolutionary dynamics or flame propagation. A reader who knows the one-sided N-BBM literature will see the extension right away and get value from the new model and the velocity convergence. I would recommend sending it to peer review. The model is original, the results are stated clearly, and the technical approach is reasonable. Referees can check the details on the boundary conditions and suggest any needed clarifications.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces the (N,p)-BBM, a two-sided branching-selection particle system that generalizes the standard N-BBM by deleting either the leftmost or rightmost particle with probability p at each branching event. It claims that as N tends to infinity the empirical distribution converges to the solution of a two-sided free boundary problem on a finite interval with p-dependent Neumann and Dirichlet boundary conditions, proves existence and regularity of this limiting PDE by appealing to a connection with inverse first-passage problems, and shows that the asymptotic velocity v_{N,p} converges to the unique travelling-wave speed v_p of the free-boundary problem. The results generalize one-sided models and link to free-boundary problems arising in evolutionary dynamics and flame propagation.

Significance. If the central claims hold, the work supplies a non-circular, first-principles derivation of a hydrodynamic limit for a new two-sided selection mechanism, extending the well-studied one-sided N-BBM case. The explicit connection between the particle system and a mixed-boundary free-boundary problem, together with the velocity convergence result, would strengthen the interface between branching particle systems and PDE models in evolutionary dynamics. The absence of reduction to previously fitted quantities is a clear methodological strength.

major comments (1)

[Section establishing existence and regularity of the free-boundary problem (via inverse first-passage connection)] The existence and regularity of the two-sided free-boundary problem (with p-dependent Neumann/Dirichlet conditions) are obtained by appealing to a known connection with inverse first-passage problems. This connection ordinarily requires explicit verification that the free boundary is at least Lipschitz (or C^{1,α}) and that the underlying diffusion satisfies the requisite monotonicity or regularity hypotheses on the moving boundaries. It is unclear whether these hypotheses are checked for the mixed-boundary, two-sided setting; without such verification the hydrodynamic limit and velocity convergence statements rest on an unconfirmed applicability condition.

minor comments (1)

The abstract states that the limiting problem is posed 'on a finite interval with two moving boundaries' but does not specify the initial data or the precise form of the p-dependent boundary conditions; adding a brief display of the limiting PDE would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to confirm applicability of the inverse first-passage connection in the two-sided mixed-boundary setting. We address this point below and commit to a targeted revision.

read point-by-point responses

Referee: The existence and regularity of the two-sided free-boundary problem (with p-dependent Neumann/Dirichlet conditions) are obtained by appealing to a known connection with inverse first-passage problems. This connection ordinarily requires explicit verification that the free boundary is at least Lipschitz (or C^{1,α}) and that the underlying diffusion satisfies the requisite monotonicity or regularity hypotheses on the moving boundaries. It is unclear whether these hypotheses are checked for the mixed-boundary, two-sided setting; without such verification the hydrodynamic limit and velocity convergence statements rest on an unconfirmed applicability condition.

Authors: We thank the referee for identifying this gap. The manuscript invokes the inverse first-passage connection to obtain existence and regularity but does not supply an explicit verification that the free boundaries are Lipschitz or that the diffusion meets the monotonicity hypotheses under the p-dependent mixed Neumann-Dirichlet conditions. In the revised manuscript we will add a dedicated subsection that carries out this verification: we will establish Lipschitz continuity of both moving boundaries from the uniform modulus of continuity inherited from the tightness of the rescaled particle system, and we will confirm the required monotonicity and regularity properties of the underlying Brownian motion with respect to the two-sided boundaries. These additions will place the hydrodynamic limit and velocity convergence on a fully rigorous footing. revision: yes

Circularity Check

0 steps flagged

Existence and regularity of two-sided FBP obtained via external connection to inverse first-passage problems; central hydrodynamic limit and velocity convergence derived independently from particle system

full rationale

The derivation proceeds from the (N,p)-BBM particle system to its empirical measure limit and asymptotic velocity via standard probabilistic and PDE techniques that generalize the one-sided N-BBM case. Existence and regularity of the limiting two-sided free-boundary problem are obtained by appealing to a connection with inverse first-passage problems, presented as an external result rather than a self-referential definition, fitted parameter, or self-citation chain internal to this paper. No equation or claim reduces the target limit or velocity v_p to a quantity defined in terms of itself or to a previously fitted input. The chain is therefore self-contained against the particle-system construction and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard existence theory for free-boundary problems and on the validity of the inverse-first-passage connection; no new free parameters or invented entities are introduced beyond the tunable probability p.

axioms (1)

domain assumption Existence and regularity of the two-sided free-boundary problem follow from its equivalence to an inverse first-passage problem for Brownian motion.
Abstract states that existence and regularity are proved by appealing to this connection; the precise conditions under which the equivalence holds are not given.

pith-pipeline@v0.9.0 · 5738 in / 1472 out tokens · 24978 ms · 2026-05-18T07:28:47.932503+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1 and Proposition 3: empirical distribution converges to solution of FBP (2) with ux(Lt,t)=2p, ux(Rt,t)=2(p-1); unique travelling wave speed c = ±√[2 log²(p/(1-p)) / (log²(p/(1-p))+π²)]

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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K. B. Athreya and P. Ney. A New Approach to the Limit Theory of Recurrent Markov Chains.Trans- actions of the American Mathematical Society, 245:493–501, 1978

work page 1978

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The Genealogy of branching Brownian motion with absorption.The Annals of Probability, 41(2):527–618, 2013

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Microscopic models of traveling wave equations.Computer physics communications, 121-122:376–381, 1999

Eric Brunet and Bernard Derrida. Microscopic models of traveling wave equations.Computer physics communications, 121-122:376–381, 1999

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Effect of microscopic noise on front propagation.Journal of statistical physics, 103(1-2):269–282, 2001

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SpringerBriefs in mathematical physics, volume 12

Gioia Carinci, Anna De Masi, Cristian Giardin` a, and Errico Presutti.Free boundary problems in PDEs and particle systems. SpringerBriefs in mathematical physics, volume 12. Springer, Switzerland, 2016

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Chang, J.-S

Y.-L. Chang, J.-S. Guo, and Y. Kohsaka. On a two-point free boundary problem for a quasilinear parabolic equation.Asymptotic Analysis, 34:333–358, 2003

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Higher-order regularity of the free boundary in the inverse first-passage problem.SIAM journal on mathematical analysis, 54(4):4695–4720, 2022

Xinfu Chen, John Chadam, and David Saunders. Higher-order regularity of the free boundary in the inverse first-passage problem.SIAM journal on mathematical analysis, 54(4):4695–4720, 2022

work page 2022

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Ferrari, Errico Pesutti, and Nahuel Soprano-Loto

Anna De Masi, Pablo A. Ferrari, Errico Pesutti, and Nahuel Soprano-Loto. Hydrodynamics of the N-BBM Process. InStochastic Dynamics Out of Equilibrium, volume 282 ofSpringer Proceedings in Mathematics & Statistics, pages 523–549. Springer International Publishing AG, Switzerland, 2019

work page 2019

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Peter J. Downey. Distribution-free bounds on the expectation of the maximum with scheduling appli- cations.Operations research letters, 9(3):189–201, 1990

work page 1990

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Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary.SIAM journal on mathematical analysis, 42(1):377–405, 2010

Yihong Du and Zhigui Lin. Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary.SIAM journal on mathematical analysis, 42(1):377–405, 2010

work page 2010

[12] [12]

On a two-point free boundary problem.Quarterly of Applied Mathematics, 64(3):413–431, 2006

Jong-Shenq Guo and Bei Hu. On a two-point free boundary problem.Quarterly of Applied Mathematics, 64(3):413–431, 2006

work page 2006

[13] [13]

S. C. Harris, M. Hesse, and A. E. Kyprianou. Branching brownian motion in a strip: Survival near criticality.The Annals of probability, 44(1):235–275, 2016

work page 2016

[14] [14]

Springer, Cham, third edition, [2021]©2021

Olav Kallenberg.Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling. Springer, Cham, third edition, [2021]©2021. 40

work page 2021

[15] [15]

Kim, Eyal Lubetzky, and Ofer Zeitouni

Yujin H. Kim, Eyal Lubetzky, and Ofer Zeitouni. The maximum of branching Brownian motion inR d. The Annals of applied probability, 33(2):1315–, 2023

work page 2023

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Subadditive Ergodic Theory.The Annals of Probability, 1(6):883–909, 1973

J.F.C Kingman. Subadditive Ergodic Theory.The Annals of Probability, 1(6):883–909, 1973

work page 1973

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S. P. Lalley and T. Sellke. A Conditional Limit Theorem for the Frontier of a Branching Brownian Motion.The Annals of probability, 15(3):1052–1061, 1987

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work page

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American Mathematical Society, United States, 2010

work page 2010

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Critical Drift for Brownian Bees and a Reflected Brownian Motion Invariance Principle

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work page arXiv 2024

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Todorovic.An introduction to stochastic processes and their applications

P. Todorovic.An introduction to stochastic processes and their applications. Springer series in statistics. Probability and its applications. Springer-Verlag, New York, 1992

work page 1992

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A free boundary problem for the predator–prey model with double free boundaries.Journal of dynamics and differential equations, 29(3):957–979, 2017

Mingxin Wang and Jingfu Zhao. A free boundary problem for the predator–prey model with double free boundaries.Journal of dynamics and differential equations, 29(3):957–979, 2017. 41

work page 2017