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arxiv: 2506.10623 · v2 · submitted 2025-06-12 · 🧮 math.PR · math.AP

Polynomial slowdown in an angle-dependent 2d branching Brownian motion

Julien Berestycki , David Geldbach , Michel Pain This is my paper

Pith reviewed 2026-05-19 09:47 UTC · model grok-4.3

classification 🧮 math.PR math.AP
keywords branching Brownian motionmaximum positionangle-dependent branchingtightnessasymptotic frontpolynomial correctioneigenvalue operator2D diffusion
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The pith

In a 2D branching Brownian motion with angle-dependent branching rate that peaks at direction zero, the maximum distance M_t from the origin stays tight around an explicit function m(t) containing a polynomial correction of order t to the (

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

The paper studies branching Brownian motion in the plane where each particle moves as a standard Brownian motion but reproduces at a rate b that depends only on its angular position and reaches its maximum at angle zero. Near that preferred direction the rate is assumed to satisfy a power-law expansion b(θ) = 1 - β|θ|^α plus higher-order terms, with α between 2/3 and 2. The main result establishes that the centered maximum process M_t - m(t) remains tight for large t, where m(t) combines the usual √2 t leading term, a sublinear correction whose exponent is exactly (2-α)/(2+α), and a logarithmic term whose coefficient is written explicitly in terms of α. A reader would care because the directional bias in reproduction produces a quantifiable slowdown whose precise power depends on how sharply the rate falls away from the optimum, giving a concrete link between local angular behavior and global front position.

Core claim

We consider a branching Brownian motion in R^2 in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate b(θ) which depends only on the angle θ of the particle. We assume that b is maximal when θ=0, which is the preferred direction for breeding. Furthermore we assume that b(θ) = 1 - β|θ|^α + O(θ²), as θ → 0, for α ∈ (2/3,2) and β>0. We show that if M_t is the maximum distance to the origin at time t, then (M_t - m(t))_{t ≥ 1} is tight where m(t) = √2 t - (ϑ₁/√2) t^{(2-α)/(2+α)} - ((3/(2√2)) - (α/(2√2(2+α)))) log t and ϑ₁ is explicit in terms of the first eigenvalue of a certain operator.

What carries the argument

Tightness of the centered maximum process (M_t - m(t)), proved via the first eigenvalue of an operator constructed from the local power-law expansion of the branching rate b(θ) near θ = 0.

If this is right

  • The leading growth of the maximum remains √2 t, identical to the homogeneous case.
  • A polynomial correction of order t raised to the exact power (2-α)/(2+α) appears because of the angular decay rate α.
  • An explicit logarithmic correction whose prefactor depends on α also contributes to m(t).
  • The amplitude ϑ₁ of the polynomial term is determined by the principal eigenvalue of an operator tied to the local expansion of b.
  • The result applies uniformly for all α in the open interval (2/3, 2).

Where Pith is reading between the lines

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

  • The same tightness statement might hold for branching random walks on lattices once the local angular expansion is matched.
  • The limiting distribution of the tight fluctuations M_t - m(t), left open here, could be identified by a separate analysis of the eigenvalue problem.
  • Analogous polynomial slowdowns are likely to appear in higher-dimensional or non-Brownian particle systems that carry a directional reproduction bias.
  • The explicit form of m(t) supplies a testable prediction for numerical or experimental realizations of directionally biased branching processes.

Load-bearing premise

The branching rate admits the local expansion b(θ) = 1 - β|θ|^α + O(θ²) as θ approaches zero, for α between 2/3 and 2.

What would settle it

A direct simulation of the particle system for large t and a concrete value such as α = 1 showing that M_t - m(t) escapes any fixed interval with positive probability bounded away from zero would disprove the claimed tightness.

Figures

Figures reproduced from arXiv: 2506.10623 by David Geldbach, Julien Berestycki, Michel Pain.

Figure 1
Figure 1. Figure 1: On the left, a realisation at time t = 16 of space–inhomogenous BBM with α ∈ {1/2, 1, 2, 4} and branching rate b(θ) = 1 − |sin(θ/2)| α , as well as homogenous BBM in black (which can be seen as α = ∞). The processes are coupled in such a way that every particle that exists for a smaller α also exists for a greater value of α. On the right, a realisation of a version of the model with discrete time and posi… view at source ↗
Figure 2
Figure 2. Figure 2: The exponent of the polynomial slowdown as function of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A plot of φ0 for different values of α. By integrating these bounds in the y coordinate and bounding them uniformly in the x coordinate we obtain the following corollary. Corollary 2.2. Let α ∈ (0, 2) and β > 0. There exist K, C, c > 0 such that the following holds for any s ≥ K and t ≥ s + Ksκ , the following holds. (i) For any x ∈ R, Z R G(s, x;t, y) dy ≤ C(t/s) κ/4 exp ϑ1(s 1−κ − t 1−κ )  . (ii) For a… view at source ↗
Figure 4
Figure 4. Figure 4: Construction of q∗ and q ∗ . For Part (v), consider t ∈ [0, 2ε1]. Then, q∗(t) (1 − t)−α ≥ 1 (1 − 2ε1)−α = (1 − 2ε1) α ≥ ( 1 − 2αε1, if α ≥ 1, 1 − 2ε1, if α < 1. This proves the lower bound. For the upper bound, using (1 − t) −α ≥ 1, we have q∗(t) (1 − t)−α ≤ (1 − ε1) −α ≤ 1 + ε1α(1 − ε1) −α−1 ≤ 1 + 2ε1(9/10)−3 , (2.23) using ε1 ≤ 1/10 and α ≤ 2. This proves Part (v). For Part (vi), consider t ∈ [T − 2ε2, T… view at source ↗
read the original abstract

We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta)$ which depends only on the angle $\theta$ of the particle. We assume that $b$ is maximal when $\theta=0$, which is the preferred direction for breeding. Furthermore we assume that $b(\theta ) = 1 - \beta \abs{\theta }^\alpha + O(\theta ^2)$, as $\theta \to 0$, for $\alpha \in (2/3,2)$ and $\beta>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-\alpha)/(2+\alpha)} - \left(\frac{3}{2\sqrt{2}} - \frac{\alpha}{2\sqrt{2}(2+\alpha)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator.

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 paper studies a 2D branching Brownian motion in which particles perform standard Brownian motion but branch at an angle-dependent rate b(θ) that attains its maximum at θ=0. Under the local expansion b(θ)=1−β|θ|^α+O(θ²) as θ→0 with α∈(2/3,2) and β>0, the authors establish that the maximum distance M_t to the origin satisfies tightness of the centered process (M_t−m(t))_{t≥1}, where m(t)=√2 t−(ϑ₁/√2) t^{(2−α)/(2+α)}−(3/(2√2)−α/(2√2(2+α)))log t and ϑ₁ is given explicitly in terms of the principal eigenvalue of a scaled angular operator obtained by balancing diffusion against the branching deficit.

Significance. If the central tightness result holds, the work is significant for the theory of inhomogeneous branching diffusions and front propagation. It supplies a precise asymptotic expansion that augments the classical √2 t−(3/(2√2))log t law by a polynomial correction whose exponent (2−α)/(2+α) is determined by the local angular deficit; the eigenvalue characterization of the prefactor ϑ₁ yields an explicit, parameter-dependent prediction. The proof architecture—scaling the angular variable by t^{−1/(2+α)}, controlling large-angle excursions by the global bound b≤1, and verifying integrability for α>2/3—adapts standard comparison techniques to this setting and produces a falsifiable correction term.

minor comments (3)
  1. The precise definition of the operator whose principal eigenvalue yields ϑ₁ should be stated explicitly (with domain and boundary conditions) already in the introduction or in a dedicated preliminary section, rather than deferred to the technical core.
  2. In the error analysis for the O(θ²) remainder under the scaling θ∼t^{−1/(2+α)}, an explicit bound on the perturbation of the eigenvalue (e.g., via the variational characterization or Kato perturbation) would make the lower-order claim fully transparent.
  3. The derivation of the logarithmic coefficient should include a short remark confirming that it reduces exactly to the classical 3/(2√2) value when β=0 (or α→∞), to facilitate comparison with the homogeneous case.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work on angle-dependent 2D branching Brownian motion and for recommending minor revision. We appreciate the recognition of the significance of the tightness result and the explicit polynomial correction term. Since no specific major comments were raised in the report, we provide a brief overall response below and confirm that the manuscript is ready for minor adjustments if any editorial suggestions arise.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's derivation of the tightness of (M_t - m(t)) proceeds by constructing matching upper and lower bounds on the maximum position. The leading correction term arises from a scaled eigenvalue problem whose principal eigenvalue ϑ₁ is defined directly from the local expansion of the branching rate b(θ) and the angular diffusion; this is a standard first-principles construction rather than a fit or self-referential definition. The exponent (2-α)/(2+α) is obtained by balancing the angular diffusion scale against the β|θ|^α deficit under the ansatz θ ~ t^{-1/(2+α)}, with the O(θ²) remainder shown to be negligible and large-angle excursions controlled by the global bound b ≤ 1. No step reduces by construction to a fitted parameter, a prior self-citation, or a renaming of an input; the argument is self-contained against the stated assumptions on b.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the local angular expansion of the branching rate and on the existence and properties of the first eigenvalue of an auxiliary operator; these are the main non-standard inputs.

axioms (1)
  • domain assumption b(θ) = 1 - β|θ|^α + O(θ²) as θ → 0 for α ∈ (2/3, 2) and β > 0
    This expansion is used to determine the exponent (2-α)/(2+α) in the correction term of m(t).

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Works this paper leans on

46 extracted references · 46 canonical work pages

  1. [1]

    Aïdékon, J

    E. Aïdékon, J. Berestycki, E. Brunet, and Z. Shi. Branching Brownian motion seen from its tip. Probab. Theory Related Fields, 157(1-2):405–451, 2013

    work page 2013

  2. [2]

    Alban, A

    A. Alban, A. Bovier, A. Gros, and L. Hartung. From 1 to infinity: The log-correction for the maximum of variable speed branching Brownian motion.Electronic Journal of Probability, 30:1–46, 2025

    work page 2025

  3. [3]

    Arguin, A

    L.-P. Arguin, A. Bovier, and N. Kistler. Poissonian statistics in the extremal process of branching Brownian motion.Ann. Appl. Probab., 22(4):1693–1711, 2012

    work page 2012

  4. [4]

    Arguin, A

    L.-P. Arguin, A. Bovier, and N. Kistler. The extremal process of branching Brownian motion. Probab. Theory Related Fields, 157(3-4):535–574, 2013. 51

    work page 2013

  5. [5]

    D. G. Aronson and P. Besala. Parabolic equations with unbounded coefficients.J. Differ- ential Equations, 3:1–14, 1967

    work page 1967

  6. [6]

    Bennewitz, M

    C. Bennewitz, M. Brown, and R. Weikard. Spectral and scattering theory for ordinary differential equations. Vol. I: Sturm-Liouville equations. Universitext. Springer, Cham, 2020

    work page 2020

  7. [7]

    Berestycki, E

    J. Berestycki, E. Brunet, C. Graham, L. Mytnik, J.-M. Roquejoffre, and L. Ryzhik. The distance between the two BBM leaders.Nonlinearity, 35(4):1558–1609, 2022

    work page 2022

  8. [8]

    Berestycki, E

    J. Berestycki, E. Brunet, J. W. Harris, and S. C. Harris. The almost-sure population growth rate in branching Brownian motion with a quadratic breeding potential.Statist. Probab. Lett., 80(17-18):1442–1446, 2010

    work page 2010

  9. [9]

    Berestycki, E

    J. Berestycki, E. Brunet, J. W. Harris, S. C. Harris, and M. I. Roberts. Growth rates of the population in a branching Brownian motion with an inhomogeneous breeding potential. Stochastic Process. Appl., 125(5):2096–2145, 2015

    work page 2096

  10. [10]

    Berestycki, Y

    J. Berestycki, Y. H. Kim, E. Lubetzky, B. Mallein, and O. Zeitouni. The extremal point process of branching Brownian motion inRd. Ann. Probab., 52(3):955–982, 2024

    work page 2024

  11. [11]

    Bocharov and S

    S. Bocharov and S. C. Harris. Branching Brownian motion with catalytic branching at the origin. Acta Appl. Math., 134:201–228, 2014

    work page 2014

  12. [12]

    Bocharov and S

    S. Bocharov and S. C. Harris. Limiting distribution of the rightmost particle in catalytic branching Brownian motion.Electron. Commun. Probab., 21:Paper No. 70, 12, 2016

    work page 2016

  13. [13]

    Bocharov and L

    S. Bocharov and L. Wang. Branching Brownian motion with spatially homogeneous and point-catalytic branching.J. Appl. Probab., 56(3):891–917, 2019

    work page 2019

  14. [14]

    A. N. Borodin and P. Salminen.Handbook of Brownian motion—facts and formulae. Prob- ability and its Applications. Birkhäuser Verlag, Basel, second edition, 2002

    work page 2002

  15. [15]

    Bovier and L

    A. Bovier and L. Hartung. The extremal process of two-speed branching Brownian motion. Electron. J. Probab., 19:no. 18, 28, 2014

    work page 2014

  16. [16]

    VariablespeedbranchingBrownianmotion1.Extremalprocesses in the weak correlation regime.ALEA Lat

    A.BovierandL.Hartung. VariablespeedbranchingBrownianmotion1.Extremalprocesses in the weak correlation regime.ALEA Lat. Am. J. Probab. Math. Stat., 12(1):261–291, 2015

    work page 2015

  17. [17]

    Bovier and L

    A. Bovier and L. Hartung. From 1 to 6: a finer analysis of perturbed branching Brownian motion. Comm. Pure Appl. Math., 73(7):1490–1525, 2020

    work page 2020

  18. [18]

    Bovier and I

    A. Bovier and I. Kurkova. Derrida’s generalized random energy models. II. Models with continuous hierarchies.Ann. Inst. H. Poincaré Probab. Statist., 40(4):481–495, 2004

    work page 2004

  19. [19]

    M. Bramson. Convergence of solutions of the Kolmogorov equation to travelling waves. Mem. Amer. Math. Soc., 44(285):iv+190, 1983

    work page 1983

  20. [20]

    M. D. Bramson. Maximal displacement of branching Brownian motion.Comm. Pure Appl. Math., 31(5):531–581, 1978

    work page 1978

  21. [21]

    Carmona and Y

    P. Carmona and Y. Hu. The spread of a catalytic branching random walk.Ann. Inst. Henri Poincaré Probab. Stat., 50(2):327–351, 2014

    work page 2014

  22. [22]

    Chaumont and G

    L. Chaumont and G. Uribe Bravo. Markovian bridges: weak continuity and pathwise constructions. Ann. Probab., 39(2):609–647, 2011

    work page 2011

  23. [23]

    Cortines, L

    A. Cortines, L. Hartung, and O. Louidor. The structure of extreme level sets in branching Brownian motion.Ann. Probab., 47(4):2257–2302, 2019

    work page 2019

  24. [24]

    Fang and O

    M. Fang and O. Zeitouni. Branching random walks in time inhomogeneous environments. Electron. J. Probab., 17:no. 67, 18, 2012

    work page 2012

  25. [25]

    Fang and O

    M. Fang and O. Zeitouni. Slowdown for time inhomogeneous branching Brownian motion. J. Stat. Phys., 149(1):1–9, 2012

    work page 2012

  26. [26]

    F. Gao, J. Hannig, T.-Y. Lee, and F. Torcaso. Laplace transforms via Hadamard factoriza- tion. Electron. J. Probab., 8:no. 13, 20, 2003. 52

    work page 2003

  27. [27]

    Hardy and S

    R. Hardy and S. C. Harris. A spine approach to branching diffusions with applications to Lp-convergence of martingales. InSéminaire de Probabilités XLII, volume 1979 ofLecture Notes in Math., pages 281–330. Springer, Berlin, 2009

    work page 1979

  28. [28]

    J. W. Harris and S. C. Harris. Branching Brownian motion with an inhomogeneous breeding potential. Ann. Inst. Henri Poincaré Probab. Stat., 45(3):793–801, 2009

    work page 2009

  29. [29]

    S. C. Harris, M. Hesse, and A. E. Kyprianou. Branching Brownian motion in a strip: survival near criticality.Ann. Probab., 44(1):235–275, 2016

    work page 2016

  30. [30]

    S. C. Harris and M. I. Roberts. The many-to-few lemma and multiple spines.Ann. Inst. Henri Poincaré Probab. Stat., 53(1):226–242, 2017

    work page 2017

  31. [31]

    Hartung, O

    L. Hartung, O. Louidor, and T. Wu. On the growth of the extremal and cluster level sets in branching Brownian motion. 2024. arXiv:2405.17634

    work page arXiv 2024

  32. [32]

    Kallenberg

    O. Kallenberg. Foundations of modern probability, volume 99 ofProbability Theory and Stochastic Modelling. Springer, Cham, 2021. Third edition

    work page 2021

  33. [33]

    Karatzas and S

    I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991

    work page 1991

  34. [34]

    Y. H. Kim, E. Lubetzky, and O. Zeitouni. The maximum of branching Brownian motion in Rd. Ann. Appl. Probab., 33(2):1315–1368, 2023

    work page 2023

  35. [35]

    S. P. Lalley and T. Sellke. A conditional limit theorem for the frontier of a branching Brownian motion.Ann. Probab., 15(3):1052–1061, 1987

    work page 1987

  36. [36]

    Liu and J

    J. Liu and J. Schweinsberg. Particle configurations for branching Brownian motion with an inhomogeneous branching rate.ALEA Lat. Am. J. Probab. Math. Stat., 20(1):731–803, 2023

    work page 2023

  37. [37]

    Maillard and O

    P. Maillard and O. Zeitouni. Slowdown in branching Brownian motion with inhomogeneous variance. Ann. Inst. Henri Poincaré Probab. Stat., 52(3):1144–1160, 2016

    work page 2016

  38. [38]

    B. Mallein. Maximal displacement of a branching random walk in time-inhomogeneous environment. Stochastic Process. Appl., 125(10):3958–4019, 2015

    work page 2015

  39. [39]

    B. Mallein. Maximal displacement ofd-dimensional branching Brownian motion.Electron. Commun. Probab., 20:no. 76, 12, 2015

    work page 2015

  40. [40]

    Mytnik, J.-M

    L. Mytnik, J.-M. Roquejoffre, and L. Ryzhik. Fisher-KPP equation with small data and the extremal process of branching Brownian motion.Adv. Math., 396:Paper No. 108106, 58, 2022

    work page 2022

  41. [41]

    M. I. Roberts and J. Schweinsberg. A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate.Electron. J. Probab., 26:Paper No. 103, 76, 2021

    work page 2021

  42. [42]

    B. A. Sevastyanov. Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries.Teor. Veroyatnost. i Primenen., 3:121–136, 1958

    work page 1958

  43. [43]

    DerivativemartingaleofthebranchingBrownian motion in dimensiond≥1

    R.Stasiński, J.Berestycki, andB.Mallein. DerivativemartingaleofthebranchingBrownian motion in dimensiond≥1. Ann. Inst. Henri Poincaré Probab. Stat., 57(3):1786–1810, 2021

    work page 2021

  44. [44]

    G. Teschl. Mathematical methods in quantum mechanics, volume 157 ofGraduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2014. With applications to Schrödinger operators

    work page 2014

  45. [45]

    Watanabe

    S. Watanabe. On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ., 4:385–398, 1965

    work page 1965

  46. [46]

    A. Zettl. Sturm-Liouville theory, volume 121 ofMathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. 53

    work page 2005