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arxiv: 2310.10742 · v3 · submitted 2023-10-16 · 🧮 math.PR · math.AP

Mean-field limit of particle systems with absorption

Gaoyue Guo , Maxime Latypov , Milica Tomasevic This is my paper

Pith reviewed 2026-05-24 06:20 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords mean-field limitparticle systemsabsorptionpropagation of chaosFokker-Planck equationstochastic differential equationsGirsanov transformone-dimensional processes
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The pith

A system of particles interacting through a bounded kernel and absorbed at a barrier converges to a mean-field stochastic differential equation as their number tends to infinity.

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

The paper examines one-dimensional particles that interact via a mean-field kernel but are removed upon hitting a fixed barrier, with the interaction depending only on surviving particles. This creates a singular interaction through absorption times. The authors establish that the finite-particle system has a weak solution and that its empirical distribution converges to the solution of a limiting mean-field SDE. Under continuity of the kernel, they prove the nonlinear Fokker-Planck equation has a unique classical solution, which implies the limit SDE is strongly well-posed and the convergence is propagation of chaos.

Core claim

The particle system admits a weak solution. Using partial Girsanov transforms, the particles are related to independent stopped Brownian motions to prove tightness and convergence to a mean-field limit SDE as the number of particles goes to infinity. The limit is studied by showing existence and uniqueness of the classical solution to the nonlinear Fokker-Planck equation under continuity of the interacting kernel, which gives strong well-posedness of the mean-field SDE and confirms propagation of chaos.

What carries the argument

Partial Girsanov transforms relating the absorbed interacting particles to independent stopped Brownian motions, combined with tightness arguments for convergence to the mean-field limit.

If this is right

  • The convergence result constitutes a propagation of chaos for the particle system.
  • The mean-field limit SDE is strongly well-posed.
  • The nonlinear Fokker-Planck equation has a unique classical solution when the kernel is continuous.
  • The particle system has a weak solution despite the non-uniform ellipticity and singular interaction.

Where Pith is reading between the lines

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

  • If the kernel continuity fails, uniqueness of the Fokker-Planck solution may be lost, breaking the strong well-posedness.
  • The one-dimensional setting and fixed barrier are crucial; similar results in higher dimensions would require different techniques.
  • This framework could extend to other singular interactions driven by hitting times in applied models like population dynamics with extinction.
  • The non-uniform ellipticity of the diffusion does not prevent the mean-field convergence in this setup.

Load-bearing premise

The interacting kernel is continuous, which is required to obtain uniqueness for the classical solution of the nonlinear Fokker-Planck equation.

What would settle it

A counterexample in which the empirical measure of the N-particle system does not converge weakly to the law of the mean-field SDE, or a continuous kernel for which the nonlinear Fokker-Planck equation lacks a unique classical solution.

read the original abstract

In this work, we consider one-dimensional particles interacting in mean-field type through a bounded kernel. In addition, when particles hit some barrier (say zero), they are removed from the system. This absorption of particles is instantaneously felt by the others, as, contrary to the usual mean-field setting, particles interact only with other non-absorbed particles. This makes the interaction singular as it happens through hitting times of the given barrier. In addition, the diffusion coefficient of each particle is non uniformly elliptic. We show that the particle system admits a weak solution. Through Partial Girsanov transforms we are able to relate our particles with independent stopped Brownian motions, and prove tightness and convergence to a mean-fied limit stochastic differential equation when the number of particles tends to infinity. Further, we study the limit and establish the existence and uniqueness of the classical solution to the corresponding nonlinear Fokker-Planck equation under some continuity assumption on the interacting kernel. This yields the strong well-posedness of the mean-field limit SDE and confirms that our convergence result is indeed a propagation of chaos result.

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 / 3 minor

Summary. The manuscript studies a system of one-dimensional particles interacting via a bounded mean-field kernel, with absorption at a fixed barrier (e.g., zero) that instantaneously affects the interaction among remaining particles. It establishes weak existence of the finite-particle system, applies partial Girsanov transforms to relate the dynamics to independent stopped Brownian motions, proves tightness and convergence in law to a limiting mean-field SDE as the number of particles tends to infinity, and proves existence and uniqueness of classical solutions to the associated nonlinear Fokker-Planck equation under a continuity assumption on the kernel. This yields strong well-posedness of the limit SDE and confirms the convergence is a propagation-of-chaos result.

Significance. If the arguments are correct, the paper provides a rigorous propagation-of-chaos result for mean-field systems with singular interactions induced by absorption and removal, extending existing theory to the non-uniformly elliptic, one-dimensional setting. The combination of Girsanov-based tightness arguments with classical well-posedness analysis of the nonlinear Fokker-Planck equation is a technical contribution relevant to models with killing or default mechanisms.

minor comments (3)
  1. [Abstract] Abstract, line 8: 'mean-fied' is a typographical error and should read 'mean-field'.
  2. [Introduction] The precise definition of the interaction kernel K and the absorption barrier should be stated explicitly in the introduction before the main theorems, to improve readability of the subsequent Girsanov and tightness arguments.
  3. The continuity assumption on the kernel used for the Fokker-Planck well-posedness (mentioned in the abstract) should be cross-referenced to the exact hypothesis in the theorem statement for the nonlinear equation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary and recommendation of minor revision. No major comments were listed in the report, so we have no specific points to address point-by-point. We will incorporate any minor suggestions or corrections in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes weak existence for the particle system, applies partial Girsanov transforms to relate particles to stopped Brownian motions, proves tightness and convergence to a mean-field SDE as N→∞, and separately shows classical well-posedness of the nonlinear Fokker-Planck equation under an explicit continuity assumption on the kernel. These steps are direct analytic arguments resting on the stated hypotheses (one-dimensional setting, non-uniform ellipticity, absorption at a fixed barrier, kernel continuity) rather than any self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. No equation or claim reduces to its own input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review yields limited visibility into the precise background results invoked; the paper relies on standard stochastic-analysis tools whose independence from the target result cannot be verified here.

axioms (2)
  • standard math Existence of weak solutions for the finite-particle system with singular interaction
    Invoked to start the analysis before applying Girsanov transforms.
  • standard math Standard tightness criteria for processes in Skorokhod space
    Used to pass to the mean-field limit as number of particles tends to infinity.

pith-pipeline@v0.9.0 · 5717 in / 1526 out tokens · 45068 ms · 2026-05-24T06:20:28.838162+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Mean field control with absorption

    math.AP 2025-09 unverdicted novelty 7.0

    Proves comparison principle for Hamilton-Jacobi equation on sub-probability measures and mean-field convergence for N-particle control problem with absorption.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · cited by 1 Pith paper

  1. [1]

    Bayraktar, G

    E. Bayraktar, G. Guo, W. Tang and Y. Zhang : McKean-Vlasov equations involving hitting times: blow-ups and global solvability . Ann. Appl. Probab. , 34(1B), 1600-1622, 2020

    work page 2020

  2. [2]

    Bayraktar, G

    E. Bayraktar, G. Guo, W. Tang and Y. Zhang : Systemic robustness: a mean-field particle system approach. To appear in Math. Finance. , arXiv:2212.08518, 2022

    work page arXiv 2022

  3. [3]

    Benedetto, E

    D. Benedetto, E. Caglioti, J.A. Carrillo and M. Pulvirenti : A non-Maxwellian steady distri- bution for one dimensional granular media . J. Statist. Phys. , 91(5-6): 979-990, 1998

    work page 1998

  4. [4]

    Bolley, I

    F. Bolley, I. Gentil and A. Guillin : Uniform convergence to equilibrium for granular media . Arch.Ration. Mech.Anal., 208(2): 429-445, 2013

    work page 2013

  5. [5]

    Burger, V

    M. Burger, V. Capasso and D. Morale : On an aggregation model with long and short range interactions. Nonlinear Anal. Real World Appl. , 8(3): 939-958, 2007

    work page 2007

  6. [6]

    Carmona and F

    R. Carmona and F. Delarue : Probabilistic theory of mean field games with applications. I, Mean field FBSDEs, control, and games , volume 83 of Probability Theory and Stochastic Modelling Springer, Cham , 2018

    work page 2018

  7. [7]

    Carmona and F

    R. Carmona and F. Delarue : Probabilistic theory of mean field games with applications. II, Mean field games with common noise and master equations , volume 84 of Probability Theory and Stochastic Modelling Springer, Cham , 2018

    work page 2018

  8. [8]

    Carmona, F

    R. Carmona, F. Delarue and A. Lachapelle : Control of McKean-Vlasov dynamics versus mean field games . Math.Financ.Econ., 7(2): 131-166, 2013

    work page 2013

  9. [9]

    Chaintron and A

    L-P. Chaintron and A. Diez. Propagation of chaos: A review of models, methods and applic a- tions. I. Models and methods , Kinetic and Related Models. , 15(6), 895-1015, 2022

    work page 2022

  10. [10]

    Chaintron and A

    L-P. Chaintron and A. Diez. Propagation of chaos: A review of models, methods and applic a- tions. II. Applications , Kinetic and Related Models. , 15(6), 1017-1173, 2022

    work page 2022

  11. [11]

    Cuchiero, S

    C. Cuchiero, S. Rigger and S. Svaluto-Ferro. Propagation of minimality in the supercooled Stefan problem, Ann. Appl. Probab. , 2023

    work page 2023

  12. [12]

    Delarue, J

    F. Delarue, J. Inglis, S. Rubenthaler and E. Tanr´ e. Global solvability of a networked integrate- and-fire model of McKean–Vlasov type , Ann. Appl. Probab. , 4, 2096 – 2133, 2015

    work page 2096

  13. [13]

    Delarue, J

    F. Delarue, J. Inglis, S. Rubenthaler and E. Tanr´ e. Particle systems with a singular mean- field self-excitation. Application to neuronal networks , Stoch. Process. Their. Appl. , 125(6), 2451-2492, 2015

    work page 2015

  14. [14]

    Djete, G

    M. Djete, G. Guo and N. Touzi. Mean field game of mutual holding with defaultable agents, and systemic risk . Preprint, arXiv:2303.07996, 2023. 24

    work page arXiv 2023

  15. [15]

    A. Figalli. Existence and uniqueness of martingale solutions for SDEs w ith rough or degenerate coefficients. J. Funct. Anal. , 254(1), 109-153, 2008

    work page 2008

  16. [16]

    Fournier and B

    N. Fournier and B. Jourdain. Stochastic particle approximation of the Keller–Segel equ ation and two-dimensional generalization of Bessel processes .Ann. Appl. Probab. 27(5), 2807-2861, 2017

    work page 2017

  17. [17]

    Garroni and J

    M. Garroni and J. Menaldi. Green Functions for Second Order Parabolic Integro-Differe ntial Problems. Chapman & Hall/CRC Research Notes in Mathematics Series. 1992

    work page 1992

  18. [18]

    Hambly, S

    B. Hambly, S. Ledger and A. Søjmark. A Mckean-Vlasov equation with positive feedback and blow-ups. Ann. Appl. Proba. , 29(4), 2338–2373, 2019

    work page 2019

  19. [19]

    Huang, R.P

    M. Huang, R.P. Malham´ e and P.E. Caines :Large population stochastic dynamic games: closed- loop McKean-Vlasov systems and the Nash certainty equivale nce principle. Commun.Inf.Syst., 6(3): 221-251, 2006

    work page 2006

  20. [20]

    Ikeda and S

    N. Ikeda and S. Watanabe. Stochastic Differential Equations and Diffusion Processes, 2nd Edition. North-Holland Mathematical Library. 1989

    work page 1989

  21. [21]

    Kusuoka : Continuity and Gaussian two-sided bounds of the density fun ctions of the solutions to path-dependent stochastic differential equations via pe rturbation

    S. Kusuoka : Continuity and Gaussian two-sided bounds of the density fun ctions of the solutions to path-dependent stochastic differential equations via pe rturbation. Stoch. Process. Their Appl., 127(2): 359-384, 2017

    work page 2017

  22. [22]

    Konakov, A

    V. Konakov, A. Kozhina and S. Menozzi : Stability of densities for perturbed diffusions and Markov chains . SAIM: PS , 21: 88-112, 2017

    work page 2017

  23. [23]

    Keller and L.A

    E.F. Keller and L.A. Segel : Model for chemotaxis . Journal of Theoretical Biology , 30(2): 225-234, 1971

    work page 1971

  24. [24]

    S. Mei, A. Montanari and P.-M. Nguyen : A mean field view of the landscape of two-layer neural networks . Proc. Natl. Acad. Sci. USA , 115(33): E7665-E7671, 2018

    work page 2018

  25. [25]

    Nadtochiy and M

    S. Nadtochiy and M. Shkolnikov : Particle systems with singular interaction through hittin g times: application in systemic risk modeling . Ann.Appl.Probab., 29(1): 89-129, 2019

    work page 2019

  26. [26]

    R¨ ockner and X

    M. R¨ ockner and X. Zhang.Well-posedness of distribution dependent SDEs with singul ar drifts., Bernoulli, 27 (2) 1131 - 1158, 2021

    work page 2021

  27. [27]

    Rotskoff and E

    G.M. Rotskoff and E. Vanden-Eijnden : Trainability and accuracy of neural networks: an interacting particle system approach . Preprint, arXiv: 1805.00915, 2018

    work page arXiv 2018

  28. [28]

    Strook and S

    D. Strook and S. Varadhan Multidimensional Diffusion Processes

    work page

  29. [29]

    Sznitman

    A-S. Sznitman. Topics in propagation of chaos. In ´Ecole d’ ´Et´ e de Probabilit´ es de Saint- Flour XIX-1989. Lecture Notes in Math. 1464 165–251, 1991

    work page 1989

  30. [30]

    Tomaˇ sevi´ c.Propagation of chaos for stochastic particle systems with s ingular mean-field interaction of Lq − Lp type

    M. Tomaˇ sevi´ c.Propagation of chaos for stochastic particle systems with s ingular mean-field interaction of Lq − Lp type. Electron. Commun. Probab. , 28: 1-13, 2023. 25

    work page 2023