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arxiv: 2606.13487 · v1 · pith:OBBNRPA4new · submitted 2026-06-11 · 🧮 math.PR · math.AP

Branching-selection particle systems and inverse first passage problems

Jacob Mercer This is my paper

Pith reviewed 2026-06-27 05:40 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords inverse first passage problembranching selectionparticle systemhydrodynamic limitfree boundary problemBrownian motionstopping time
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The pith

The limit of b^N in the branching-selection particle system solves the generalized inverse first passage problem.

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

The paper builds an N-particle system in which independent Brownian motions branch and get removed at rates tied to a moving boundary b^N that depends on the current empirical distribution. It proves that this boundary converges, in the hydrodynamic limit as N tends to infinity, to the function b that makes the defined stopping time τ have exactly the target distribution p. The link is established by showing that the limit satisfies the free-boundary problem associated with the inverse first passage. This construction supplies a particle-based method for finding the boundary for arbitrary p.

Core claim

We construct a branching-selection particle system whose hydrodynamic limit is governed by a free boundary problem and connect this to the generalised inverse first passage problem. In the N-particle system, particles move as independent Brownian motions, branch at a prescribed rate, and are removed at a rate proportional to their location relative to a position b^N(t) which is a function of the empirical distribution. We identify the limit of b^N as the solution of the inverse first passage problem.

What carries the argument

The N-particle branching-selection system with removal rate proportional to distance from b^N(t) derived from the empirical measure, whose limit yields the boundary for the inverse first passage problem.

If this is right

  • The boundary realizing any p can be obtained as the hydrodynamic limit of b^N.
  • The free-boundary problem has a solution given by this limiting boundary.
  • Simulations of the particle system approximate the solution to the inverse problem.
  • The stopping time τ constructed this way has distribution p by design in the limit.

Where Pith is reading between the lines

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

  • This suggests a simulation-based algorithm to compute the boundary b for given p.
  • It may generalize to inverse problems for other Markov processes.
  • Links to numerical methods for free-boundary PDEs via particle approximations.

Load-bearing premise

The particle system converges in the hydrodynamic limit to the specific free-boundary problem whose solution is the desired boundary b for p.

What would settle it

For a chosen p, compute the empirical distribution of τ from many runs of the large-N system and check whether it matches p; mismatch would disprove the identification.

read the original abstract

A generalised inverse first passage problem asks whether, given a probability measure $p$ on $[0,\infty]$, one can find a boundary $b:[0,\infty]\to \mathbb{R}$ such that the stopping time:\[\tau:=\inf\left\{t:\Lambda\int_0^t \omega(W_s-b(s))ds \geq U\right\}\] has distribution $p$, where $U\sim Exp(1)$, $\Lambda\in(0,\infty)$ and $\omega$ is a monotonic decreasing function. We construct a branching-selection particle system whose hydrodynamic limit is governed by a free boundary problem and connect this to the generalised inverse first passage problem. In the $N$-particle system, particles move as independent Brownian motions, branch at a prescribed rate, and are removed at a rate proportional to their location relative to a position $b^N(t)$ which is a function of the empirical distribution. We identify the limit of $b^N$ as the solution of the inverse first passage problem.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript formulates a generalised inverse first-passage problem: given a probability measure p on [0,∞], find a boundary b such that the stopping time τ = inf{t : Λ ∫_0^t ω(W_s - b(s)) ds ≥ U} (U ~ Exp(1)) has law p. It introduces an N-particle branching-selection system in which particles evolve as independent Brownian motions, branch at a fixed rate, and are killed at a rate depending on their position relative to an adaptive threshold b^N(t) constructed from the empirical measure. The central claim is that b^N converges to the desired boundary b because the hydrodynamic limit of the particle system is a free-boundary problem whose solution coincides with the integral condition that defines τ.

Significance. If the hydrodynamic limit and the identification of lim b^N with the target boundary are rigorously established, the construction supplies a particle-based, mean-field approximation to a class of inverse first-passage problems. The approach is consistent with existing techniques that enforce free boundaries through selection and therefore offers a potentially useful link between interacting-particle systems and free-boundary stochastic control problems.

minor comments (2)
  1. The abstract states the identification of lim b^N but does not indicate where in the manuscript the hydrodynamic limit is proved or where the equivalence between the resulting free-boundary problem and the integral condition for τ is verified. Adding explicit references to the relevant theorems or sections would strengthen the presentation.
  2. The function ω is described only as monotonic decreasing; a brief statement of the regularity or growth conditions imposed on ω (and on b) would clarify the setting in which the hydrodynamic limit is derived.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their accurate summary of the manuscript and for recommending minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs an N-particle branching-selection system in which b^N(t) is explicitly a function of the empirical measure and removal occurs at a rate tied to distance from b^N(t). The central claim is that the hydrodynamic limit of this system is a free-boundary problem whose solution b coincides with the boundary realizing the target distribution p for the inverse first-passage problem. This identification is an independent mathematical statement (hydrodynamic limit equals the integral condition defining tau) rather than a definitional equivalence or a fitted parameter renamed as a prediction. No self-citation chain, ansatz smuggling, or uniqueness theorem imported from the authors' prior work is invoked to close the argument. The derivation therefore remains self-contained against standard mean-field techniques and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; full list of assumptions, parameters, and entities cannot be extracted. Standard Brownian-motion properties are implicitly used.

axioms (1)
  • standard math Brownian motion has independent increments and continuous paths; exponential random variable U is memoryless.
    These are the background probabilistic objects used to define the stopping time tau.

pith-pipeline@v0.9.1-grok · 5690 in / 1190 out tokens · 19308 ms · 2026-06-27T05:40:10.519102+00:00 · methodology

discussion (0)

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Reference graph

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