Pith. sign in

REVIEW 1 major objections 4 minor 71 references

A moment approach for the convergence of spatial branching processes to the Continuum Random Tree

T0 review · 1 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that the rescaled genealogy of a wide class of critical spatial branching processes converges to the Brownian Continuum Random Tree, with the limit determined only by a spatial variance parameter and the equilibrium…

desk verdict A genuinely new many-to-few formula and a broad Brownian CRT invariance principle, but the proof of Lemma 5.2 has a load-bearing gap that needs a fix or a stronger assumption. read the letter →

arxiv 2412.16035 v1 pith:ZM4DQEP3 submitted 2024-12-20 math.PR

Félix Foutel-Rodier This is my paper · ORCID
classification math.PR MSC 60J8060F1760B10
keywords spatialbranchingprocessesBrownianContinuumRandomTreemany-to-fewformulamethodofmomentsGromov-vaguetopologycoalescentpointprocessspinedecompositioncritical
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a universality theorem for the genealogical tree of a critical spatial branching process. Starting from any type $x$, the rescaled marked tree $\bar T_n$, with edge lengths divided by the number of generations and particle masses by $n^2$, converges in the marked Gromov-vague topology to $h(x)$ times the law of a free Brownian Continuum Random Tree whose leaves carry independent marks with equilibrium distribution $\pi$; the only quantity of the offspring law that survives is a spatial variance $\Sigma^2$ (Theorem 1.1(i)). The same moment computation yields a second limit: the reduced tree spanned by the $n$-th generation converges to the Brownian coalescent point process with the same parameters, and with a supplementary survival-probability assumption both limits strengthen to the Gromov-Hausdorff-Prohorov topology. The interest is that the tree shape and particle locations may be strongly dependent, so classical path-encoding techniques fail; the paper's moment approach applies to a general class of such processes. A new many-to-few formula expresses $k$-th moments of the tree through a Markov chain indexed by a uniform tree with $k$ leaves, reducing convergence of the genealogy to computing moments.

What carries the argument

The engine is the many-to-few formula (Theorem 3.1): for a weighting function $\psi$, the $k$-th moment measure of the branching tree equals $\psi(x)\sum_{\tau\in T_k} Q^\psi_{x,\tau}[\Delta^\psi_k F]$, a sum over planar trees with $k$ leaves of an expectation under a Markov chain indexed by that tree, with an explicit bias factor $\Delta^\psi_k$. With $\psi=h$, the harmonic function of Assumption 1, the chain becomes the spinal Markov chain, the $h$-transform of the mean semigroup, and the bias simplifies. Proposition 5.1 then shows that, under Assumption 1, the rescaled $k$-th moments converge to $h(x)(\Sigma^2/2)^{k-1}\int E[F(\theta,(X_i)_{i\le k})]\,d\Lambda_k(\theta)$, exactly the Brownian CRT moment formula of Proposition 4.2, with the $X_i$ i.i.d. under $\pi$; the method of moments (Proposition 4.1) converts this into vague Gromov-vague convergence.

What would settle it

Take a critical branching process satisfying Assumptions 1(i) and (iii) but not (ii), for example a branching diffusion on an unbounded domain where the mean semigroup converges only pointwise rather than uniformly. Compute the fourth moment of the rescaled tree and compare it with the Brownian CRT moment formula of Proposition 5.1: a mismatch would show that the claimed universality fails exactly where uniform ergodicity fails.

Watch

Extended reading notes

Core claim

The central claim, stated as Theorem 1.1(i), is that under Assumption 1 the law $n L_x(\bar T_n)$ converges vaguely in the marked Gromov-vague topology to $h(x)L(T_{b,\pi})$, where $T_{b,\pi}$ is the free Brownian CRT with variance $\Sigma^2$ and independent marks $\pi$. Equivalently, the rescaled genealogy of a critical spatial branching process is universal: the Brownian CRT emerges regardless of the detailed offspring law, provided the mean semigroup mixes uniformly and the offspring have finite moments. The same method gives Theorem 1.1(ii), convergence of the rescaled generation-$n$ reduced tree to the Brownian coalescent point process, and Theorem 1.2, which upgrades both convergences to the Gromov-Hausdorff-Prohorov topology when the Kolmogorov survival estimate $nP_x(Z_n>0)\to 2h(x)/\Sigma^2$ holds. The paper also deduces conditioned limits, such as survival for a long time or a forest started from many ancestors, as corollaries.

Load-bearing premise

The load-bearing premise is uniform ergodicity of the mean semigroup (Assumption 1(ii)): no matter where the process starts, the expected number of descendants of a given type approaches $h(x)\langle \pi,f\rangle$ at a rate independent of the starting type, and without that uniformity the spine chain need not converge to the equilibrium $h\pi$.

Editorial extensions

If this is right

  • Under Assumption 1, the full rescaled genealogy of a critical spatial branching process is universal: no matter the offspring law, the limit is the free Brownian CRT with variance $\Sigma^2$ and equilibrium marks $\pi$.
  • The reduced tree spanned by a single generation converges to the Brownian coalescent point process with the same parameters, giving a scaling limit for the ultrametric genealogy of the $n$-th generation.
  • Conditioning on survival to a long time $tn$, or starting from $n$ ancestors, yields the conditioned Brownian CRT and a Poisson forest of Brownian CRTs (Corollary 1.3), so the moment method covers conditioned and multi-root limits as corollaries.
  • Kolmogorov's survival estimate $nP_x(Z_n>0)\to 2h(x)/\Sigma^2$ (when Assumption 2 holds) upgrades Gromov-vague convergence to Gromov-Hausdorff-Prohorov convergence, making diameter and height functionals continuous in the limit.
  • Convergence holds uniformly in the initial type $x$, so initial conditions can vary with $n$ as long as the empirical measure of starting types converges.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct extension of the argument would replace the Brownian CRT moment formula with the moments of an $\alpha$-stable CRT; the many-to-few machinery could then prove $\alpha$-stable invariance principles for spatial branching processes, a case the paper only touches through examples and conjecture.
  • Because Assumption 1(ii) is a property of the one-particle mean semigroup rather than of tree shapes, the method should apply to processes whose genealogies are far from Galton-Watson trees as long as this uniform ergodicity holds; the paper proves this for critical processes, but the moment computation itself is not tied to a Galton-Watson structure.
  • The many-to-few formula gives an explicit biased representation of the subtree spanned by $k$ uniformly sampled particles, which could be used directly for coalescent inference or for simulating genealogies without constructing the whole population tree.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper proves an invariance principle for the genealogical tree of a critical spatial branching process. Under Assumption 1 (existence of a harmonic function and stationary measure for the mean semigroup, uniform ergodicity of the one-particle semigroup, and uniform moment bounds), Theorem 1.1 identifies the vague Gromov-vague limit of the rescaled marked tree n L_x(\bar T_n) as h(x) times the law of a free Brownian CRT with variance \Sigma^2 and independent marks distributed according to \pi. It also gives the corresponding limit for the generation-n ultrametric tree conditioned to have macroscopic size. Under the additional Kolmogorov-type estimate in Assumption 2, Theorem 1.2 reinforces the convergence to the marked Gromov-Hausdorff-Prohorov topology. The proof introduces a new many-to-few formula expressing k-th moments in terms of a Markov chain indexed by a uniform planar tree, then derives moment asymptotics via an induction on the number of leaves, and finally applies a method of moments for random metric measure spaces.

Significance. This is a substantial contribution to the scaling-limit theory of spatial branching processes. The many-to-few formula of Theorem 3.1 is a new and potentially reusable tool, and the moment approach provides a unified route to results that previously required model-specific encodings. The assumptions are explicit and cover multitype branching processes, branching diffusions in bounded domains, and other examples. The paper is careful with the infinite-measure formalism for Brownian CRT limits, and the statement of the limiting object in terms of \Sigma^2 and \pi is parameter-free. If the gap discussed below is repaired, the result would be an important step toward a general theory of genealogical convergence for spatially dependent branching mechanisms.

major comments (1)
  1. [Section 5.1, Lemma 5.2] The induction step of Lemma 5.2 has a load-bearing gap. After conditioning on the reduced tree eτ_n, the proof applies the k-leaf induction hypothesis to a functional of the leaves of eτ_n that includes B_n(X_w), where B_n(y) = E_y[Σ_{i≠j} h(ξ_i) E_{ξ_i}[f_k(ζ)/h(ζ)] h(ξ_j) E_{ξ_j}[f_{k+1}(ζ)/h(ζ)]]. The proof shows that B_n(y) converges uniformly to (1/2)⟨π,f_k⟩⟨π,f_{k+1}⟩ m_2(y), and then invokes the induction hypothesis. However, the induction hypothesis is stated only for continuous bounded leaf functions, and Assumption 1(iii) guarantees continuity of x ↦ E|Ξ_x|^2, not of m_2(x) = E_x[Σ_{i≠j} h(ξ_i)h(ξ_j)] or of B_n. Without an additional approximation argument, the induction step is not justified. Since Lemma 5.2 feeds directly into Proposition 5.1 and hence into the identification of the limit as the Brownian CRT moment, this affects the central claim of Theorem 1.1. The proof can be repaired by strengthening Assumption 1(iii) to require continuity of m_2 (or of the relevant joint offspring moments), or by supplying an approximation argument that avoids evaluating m_2 as a continuous test function.
minor comments (4)
  1. [Section 5.1, Lemma 5.2] The sentence 'Since the trees are converging deterministically, it is sufficient to prove the result for functionals that only depend on the types of the leaves' is terse; the passage from product functions to arbitrary continuous bounded F requires a monotone-class or tightness argument because the mark space E is not compact. The step is likely valid, but it deserves a short justification.
  2. [Section 5.2, Proposition 5.3] The induction proving the bound (36) is only sketched ('It will follow by an induction on k'). Since this bound is used to control the error terms B_{x_n,n} and eB_{x_n,n}, the induction should be written out or at least the base case and inductive step should be indicated explicitly.
  3. [Section 5.4, Corollary 5.4] The inequality n P_x(Σ_{m≥nR} Z_m ≥ ε n^2) ≤ n P_x(Z_{nR} > 0) is correct because the event on the left implies Z_{nR} > 0, but this implication should be stated for clarity, especially since the sum starts at m = nR rather than m > nR.
  4. [Section 5.3, Lemma 5.7] The notation T(2δn) and T_v^{2δn} is introduced quickly; the superscript convention (height truncation versus subtree truncation) could be made explicit to avoid confusion.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the CRT convergence is derived from the mean-semigroup assumption via the proved many-to-few formula and a general method-of-moments lemma; self-citations are technical, not reductions.

full rationale

No derivation step in the paper is equivalent by construction to its own inputs. The invariance principle is conditional on Assumption 1, which is a condition on the one-particle mean semigroup, not on tree shapes; the k=1 moment asymptotics use this assumption directly, but the k>=2 tree-structure moments are obtained through the many-to-few formula (Theorem 3.1), which is proved from the branching property and the many-to-one formula (Proposition 3.2). The limiting moments are then identified as those of the Brownian CRT via Proposition 4.2, an external reformulation of Le Gall's excursion theorem. The parameters h, pi, and Sigma^2 are defined directly from the model data (Assumption 1(i) and the displayed definition of Sigma^2), not tuned to the CRT limit. The method-of-moments reduction (Proposition 4.1) is cited from the author's own work [30]/[34], but it is a general topological lemma whose hypotheses are moment convergence plus Carleman's condition, independent of the branching process and of the CRT conclusion; it therefore functions as independent support rather than a circular reduction. A separate technical gap exists in Lemma 5.2's induction, where the continuity of m_2 is not implied by Assumption 1(iii); this is a correctness risk, not a circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on explicit domain assumptions about the branching mechanism: existence and normalization of a harmonic function h and stationary measure π, uniform convergence of the mean semigroup, uniform moment bounds, and, for the stronger topology, the Kolmogorov survival estimate. No parameters are fitted and no new entities are postulated.

assumptions (4)
  • domain assumption Assumption 1(i): existence of continuous bounded harmonic h and stationary probability π for the mean semigroup, normalized with ⟨π,h⟩=1.
    Introduced in Section 1.2; defines criticality and the asymptotic type distribution.
  • domain assumption Assumption 1(ii): uniform convergence (2) of the mean semigroup to h(x)⟨π,f⟩.
    Used in Lemma 5.2 to show asymptotic independence of types and convergence of the spine to hπ.
  • domain assumption Assumption 1(iii): uniform moment bounds sup_x E|Ξ_x|^k < ∞ for all k and continuity of x -> E|Ξ_x|^2.
    Controls the bias term Δ_k and justifies dominated convergence in Proposition 5.1.
  • domain assumption Assumption 2: Kolmogorov estimate n P_x(Z_n>0) -> 2h(x)/Σ² uniformly.
    Required only for Theorem 1.2 to reinforce convergence to Gromov-Hausdorff-Prohorov; the paper leaves it as a condition to verify.

how reviews work

0 comments
Cite this review

Pith. "Pith review of A moment approach for the convergence of spatial branching processes to the Continuum Random Tree." pith.science (2026). https://pith.science/paper/ZM4DQEP3

@misc{pith2026241216035,
  author       = {Pith},
  title        = {Pith review of: A moment approach for the convergence of spatial branching processes to the Continuum Random Tree},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZM4DQEP3}},
  note         = {Machine review of arXiv:2412.16035}
}
abstract

We consider a general class of branching processes in discrete time, where particles have types belonging to a Polish space and reproduce independently according to their type. If the process is critical and the mean distribution of types converges for large times, we prove that the tree structure of the process converges to the Brownian Continuum Random Tree, under a moment assumption. We provide a general approach to prove similar invariance principles for branching processes, which relies on deducing the convergence of the genealogy from computing its moments. These are obtained using a new many-to-few formula, which provides an expression for the moments of order $k$ of a branching process in terms of a Markov chain indexed by a uniform tree with $k$ leaves.

Figures

Figures reproduced from arXiv: 2412.16035 by the authors.

Figure 1
Figure 1. Left: A tree is decomposed at its first branch point into three subtrees S1, S2, and S3; the corresponding leaf partition is displayed on top. Right: Point process construction of the same tree. Recursive construction. We will need a few definitions, which are all illustrated in the left part of [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

71 extracted references · 65 canonical work pages

  1. [30]

    Vague convergence and method of moments for random metric measure spaces

    F´ elix Foutel-Rodier. Vague convergence and method of moments for random metric measure spaces. arXiv preprint arXiv:2402.05097 , 2024

    arXiv 2024

  2. [1]

    A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces

    Romain Abraham, Jean-Fran¸ cois Delmas, and Patrick Hoscheit. A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electronic Journal of Probability , 18:1–21, 2013

    work page 2013

  3. [2]

    The continuum limit of critical random graphs

    Louigi Addario-Berry, Nicolas Broutin, and Christina Goldschmidt. The continuum limit of critical random graphs. Probability Theory and Related Fields, 152(3):367–406, 2012

    work page 2012

  4. [3]

    The Continuum Random Tree III

    David Aldous. The Continuum Random Tree III. The Annals of Probability, 21(1):248– 289, 1993

    work page 1993

  5. [4]

    Strong limit theorems for general supercrit- ical branching processes with applications to branching diffusions

    Søren Asmussen and Heinrich Hering. Strong limit theorems for general supercrit- ical branching processes with applications to branching diffusions. Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete , 36(3):195–212, 1976

    work page 1976

  6. [5]

    Limit Theory for Subcritical and Critical Pro- cesses

    Søren Asmussen and Heinrich Hering. Limit Theory for Subcritical and Critical Pro- cesses. Birkh¨ auser Boston, Boston, MA, 1983

    work page 1983

  7. [6]

    Branching Processes

    Krishna Athreya and Peter Ney. Branching Processes. Springer Berlin Heidelberg, Berlin, Heidelberg, 1972

    work page 1972

  8. [7]

    The gap between Gromov-vague and Gromov–Hausdorff-vague topology

    Siva Athreya, Wolfgang L¨ ohr, and Anita Winter. The gap between Gromov-vague and Gromov–Hausdorff-vague topology. Stochastic Processes and their Applications , 126(9):2527–2553, 2016

    work page 2016

Show all 71 references
  1. [8]

    The distribution of surviving blocks of an ancestral genome

    Stuart Baird, Nicolas Barton, and Alison Etheridge. The distribution of surviving blocks of an ancestral genome. Theoretical Population Biology, 64(4):451–471, 2003

    2003

  2. [9]

    Spine for interacting populations and sampling

    Vincent Bansaye. Spine for interacting populations and sampling. Bernoulli, 30(2):1555–1585, 2024

    2024

  3. [10]

    A non- conservative harris ergodic theorem

    Vincent Bansaye, Bertrand Cloez, Pierre Gabriel, and Aline Marguet. A non- conservative harris ergodic theorem. Journal of the London Mathematical Society , 106(3):2459–2510, 2022

    2022

  4. [11]

    The genealogy of branching brownian motion with absorption

    Julien Berestycki, Nathana¨ el Berestycki, and Jason Schweinsberg. The genealogy of branching brownian motion with absorption. The Annals of Probability , 41(2):527– 618, 2013

    2013

  5. [12]

    Recent progress in coalescent theory

    Nathana¨ el Berestycki. Recent progress in coalescent theory. Ensaios Matem´ aticos, 16:1–193, 2009. 42

    2009

  6. [13]

    A probabilistic approach to spectral analysis of growth-fragmentation equations

    Jean Bertoin and Alexander Watson. A probabilistic approach to spectral analysis of growth-fragmentation equations. Journal of Functional Analysis , 274(8):2163–2204, 2018

    2018

  7. [14]

    Measure change in multitype branching

    John Biggins and Andreas Kyprianou. Measure change in multitype branching. Ad- vances in Applied Probability , 36(2):544–581, 2004

    2004

  8. [15]

    The genealogy of nearly critical branching processes in varying environment

    Florin Boenkost, F´ elix Foutel-Rodier, and Emmanuel Schertzer. The genealogy of nearly critical branching processes in varying environment. arXiv preprint arXiv:2207.11612, 2022

    2022

  9. [16]

    Estimation in emerging epidemics: biases and remedies

    Tom Britton and Gianpaolo Scalia Tomba. Estimation in emerging epidemics: biases and remedies. Journal of the Royal Society Interface , 16(150):20180670, 2019

    2019

  10. [17]

    Dynamics of lineages in adaptation to a gradual environmental change

    Vincent Calvez, Beno ˆ ıt Henry, Sylvie M´ el´ eard, and Viet Chi Tran. Dynamics of lineages in adaptation to a gradual environmental change. Annales Henri Lebesgue , 5:729–777, 2022

    2022

  11. [18]

    Random planar lattices and integrated su- perbrownian excursion

    Philippe Chassaing and Gilles Schaeffer. Random planar lattices and integrated su- perbrownian excursion. Probability Theory and Related Fields , 128(2):161–212, 2004

    2004

  12. [19]

    Growing conditioned trees

    Brigitte Chauvin, Alain Rouault, and Anton Wakolbinger. Growing conditioned trees. Stochastic Processes and their Applications , 39(1):117–130, 1991

    1991

  13. [20]

    Branching diffusion processes and spectral properties of Feynman-Kac semigroup

    Pierre Collet, Sylvie M´ el´ eard, and Jaime San Mart ´ ın. Branching diffusion processes and spectral properties of Feynman-Kac semigroup. arXiv preprint arXiv:2404.09568, 2024

    2024 arXiv

  14. [21]

    An introduction to the theory of point processes: Volume I: Elementary theory and methods

    Daryl Daley and David Vere-Jones. An introduction to the theory of point processes: Volume I: Elementary theory and methods . Springer New York, NY, 2 edition, 2003

    2003

  15. [22]

    An introduction to the theory of point processes: Volume II: General theory and structure

    Daryl Daley and David Vere-Jones. An introduction to the theory of point processes: Volume II: General theory and structure . Springer New York, NY, 2 edition, 2008

    2008

  16. [23]

    Measure-valued Markov processes

    Donald Dawson. Measure-valued Markov processes. In Ecole d’Et´ e de Probabilit´ es de Saint-Flour XXI - 1991 , pages 1–260, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg

    1991

  17. [24]

    On the stability of positive semi- groups

    Pierre Del Moral, Emma Horton, and Ajay Jasra. On the stability of positive semi- groups. The Annals of Applied Probability , 33(6A):4424–4490, 2023

    2023

  18. [25]

    Marked metric mea- sure spaces

    Andrej Depperschmidt, Andreas Greven, and Peter Pfaffelhuber. Marked metric mea- sure spaces. Electronic Communications in Probability, 16:174–188, 2011. 43

    2011

  19. [26]

    Soci´ et´ e math´ ematique de France Paris, France, 2002

    Thomas Duquesne and Jean-Fran¸ cois Le Gall.Random trees, L´ evy processes and spa- tial branching processes, volume 281. Soci´ et´ e math´ ematique de France Paris, France, 2002

    2002

  20. [27]

    Branching particle systems and superprocesses

    Eugene Dynkin. Branching particle systems and superprocesses. The Annals of Prob- ability, 19(3):1157–1194, 1991

    1991

  21. [28]

    An Introduction to Superprocesses , volume 20

    Alison Etheridge. An Introduction to Superprocesses , volume 20. American Mathe- matical Society Providence, 2000

    2000

  22. [29]

    Genealogies in bistable waves

    Alison Etheridge and Sarah Penington. Genealogies in bistable waves. Electronic Journal of Probability , 27:1–99, 2022

    2022

  23. [31]

    From individual-based epidemic models to McKendrick-von Foerster PDEs: a guide to modeling and inferring COVID- 19 dynamics

    F´ elix Foutel-Rodier, Fran¸ cois Blanquart, Philibert Courau, Peter Czuppon, Jean-Jil Duchamps, Jasmine Gamblin, ´Elise Kerdoncuff, Rob Kulathinal, L´ eo R´ egnier, Laura Vuduc, Amaury Lambert, and Emmanuel Schertzer. From individual-based epidemic models to McKendrick-von Fo...

    2022

  24. [32]

    Exchangeable co- alescents, ultrametric spaces, nested interval-partitions: A unifying approach

    F´ elix Foutel-Rodier, Amaury Lambert, and Emmanuel Schertzer. Exchangeable co- alescents, ultrametric spaces, nested interval-partitions: A unifying approach. The Annals of Applied Probability , 31(5):2046–2090, 2021

    2021

  25. [33]

    Convergence of genealogies through spinal decomposition with an application to population genetics

    F´ elix Foutel-Rodier and Emmanuel Schertzer. Convergence of genealogies through spinal decomposition with an application to population genetics. Probability Theory and Related Fields , 187(3):697–751, 2023

    2023

  26. [34]

    Convergence of spa- tial branching processes to α-stable CSBPs: Genealogy of semi-pushed fronts

    F´ elix Foutel-Rodier, Emmanuel Schertzer, and Julie Tourniaire. Convergence of spa- tial branching processes to α-stable CSBPs: Genealogy of semi-pushed fronts. arXiv preprint arXiv:2402.05096, 2024

    2024 arXiv

  27. [35]

    Asymptotic moments of spatial branching processes

    Isaac Gonzalez, Emma Horton, and Andreas Kyprianou. Asymptotic moments of spatial branching processes. Probability Theory and Related Fields , 184(3):805–858, 2022

    2022

  28. [36]

    Convergence in distribution of random metric measure spaces (λ-coalescent measure trees)

    Andreas Greven, Peter Pfaffelhuber, and Anita Winter. Convergence in distribution of random metric measure spaces (λ-coalescent measure trees). Probability Theory and Related Fields, 145(1):285–322, 2009

    2009

  29. [37]

    Many-to-few for non-local branching markov process

    Simon Harris, Emma Horton, Andreas Kyprianou, and Ellen Powell. Many-to-few for non-local branching markov process. Electronic Journal of Probability, 29:1–26, 2024. 44

    2024

  30. [38]

    Yaglom limit for critical nonlocal branching markov processes

    Simon Harris, Emma Horton, Andreas Kyprianou, and Minmin Wang. Yaglom limit for critical nonlocal branching markov processes. The Annals of Probability , 50(6):2373–2408, 2022

    2022

  31. [39]

    Universality classes for the coalescent structure of heavy-tailed Galton–Watson trees

    Simon Harris, Samuel Johnston, and Juan Carlos Pardo. Universality classes for the coalescent structure of heavy-tailed Galton–Watson trees. The Annals of Probability , 52(2):387–433, 2024

    2024

  32. [40]

    The coalescent structure of Galton-Watson trees in varying environments

    Simon Harris, Sandra Palau, and Juan Carlos Pardo. The coalescent structure of Galton-Watson trees in varying environments. arXiv preprint arXiv:2207.10923, 2022

    2022 arXiv

  33. [41]

    The many-to-few lemma and multiple spines

    Simon Harris and Matthew Roberts. The many-to-few lemma and multiple spines. Annales de l’Institut Henri Poincar´ e, Probabilit´ es et Statistiques, 53(1):226–242, 2017

    2017

  34. [42]

    Critical markov branching processes with general set of types

    Heinrich Hering. Critical markov branching processes with general set of types. Trans- actions of the American Mathematical Society , 160:185–202, 1971

    1971

  35. [43]

    Stochastic Neutron Transport

    Emma Horton and Andreas Kyprianou. Stochastic Neutron Transport . Birkh¨ auser Cham, 2023

    2023

  36. [44]

    Stability of (sub) critical non-local spatial branching processes with and without immigration

    Emma Horton, Andreas Kyprianou, Pedro Mart ´ ın-Ch´ avez, Ellen Powell, and Vic- tor Rivero. Stability of (sub) critical non-local spatial branching processes with and without immigration. arXiv preprint arXiv:2407.05472 , 2024

    2024 arXiv

  37. [45]

    The growth and composition of branching populations

    Peter Jagers and Olle Nerman. The growth and composition of branching populations. Advances in Applied Probability, 16(2):221–259, 1984

    1984

  38. [46]

    The genealogy of Galton-Watson trees

    Samuel Johnston. The genealogy of Galton-Watson trees. Electronic Journal of Prob- ability, 24:1–35, 2019

    2019

  39. [47]

    Random measures, theory and applications

    Olav Kallenberg. Random measures, theory and applications . Springer Cham, 2 edi- tion, 2017

    2017

  40. [48]

    The coalescent

    John Kingman. The coalescent. Stochastic processes and their applications, 13(3):235– 248, 1982

    1982

  41. [49]

    A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes

    Thomas Kurtz, Russell Lyons, Robin Pemantle, and Yuval Peres. A conceptual proof of the Kesten-Stigum theorem for multi-type branching processes. Classical and modern branching processes, pages 181–185, 1997

    1997

  42. [50]

    The contour of splitting trees is a L´ evy process

    Amaury Lambert. The contour of splitting trees is a L´ evy process. The Annals of Probability, 38(1):348–395, 2010

    2010

  43. [51]

    The coalescent point process of branching trees

    Amaury Lambert and Lea Popovic. The coalescent point process of branching trees. The Annals of Applied Probability , 23(1):99–144, 2013. 45

    2013

  44. [52]

    The comb representation of compact ultrametric spaces

    Amaury Lambert and Ger´ onimo Uribe Bravo. The comb representation of compact ultrametric spaces. p-Adic Numbers, Ultrametric Analysis and Applications , 9(1):22– 38, 2017

    2017

  45. [53]

    Brownian excursions, trees and measure-valued branching processes

    Jean-Fran¸ cois Le Gall. Brownian excursions, trees and measure-valued branching processes. The Annals of Probability , 19(4):1399–1439, 1991

    1991

  46. [54]

    A class of path-valued markov processes and its applications to superprocesses

    Jean-Fran¸ cois Le Gall. A class of path-valued markov processes and its applications to superprocesses. Probability Theory and Related Fields , 95(1):25–46, 1993

    1993

  47. [55]

    The uniform random tree in a brownian excursion

    Jean-Fran¸ cois Le Gall. The uniform random tree in a brownian excursion. Probability Theory and Related Fields , 96(3):369–383, 1993

    1993

  48. [56]

    Birkh¨ auser Basel, 1999

    Jean-Fran¸ cois Le Gall.Spatial branching processes, random snakes and partial differ- ential equations. Birkh¨ auser Basel, 1999

    1999

  49. [57]

    Random trees and applications.Probability Surveys, 2:245–311, 2005

    Jean-Fran¸ cois Le Gall. Random trees and applications.Probability Surveys, 2:245–311, 2005

    2005

  50. [58]

    Branching processes in L´ evy processes: Laplace functionals of snakes and superprocesses

    Jean-Fran¸ cois Le Gall and Yves Le Jan. Branching processes in L´ evy processes: Laplace functionals of snakes and superprocesses. The Annals of Probability , 26(4):1407–1432, 1998

    1998

  51. [59]

    Conceptual proofs of L log L criteria for mean behavior of branching processes

    Russell Lyons, Robin Pemantle, and Yuval Peres. Conceptual proofs of L log L criteria for mean behavior of branching processes. The Annals of Probability, pages 1125–1138, 1995

    1995

  52. [60]

    The lineage process in Galton–Watson trees and globally centered discrete snakes

    Jean-Fran¸ cois Marckert. The lineage process in Galton–Watson trees and globally centered discrete snakes. The Annals of Applied Probability , 18(1):209–244, 2008

    2008

  53. [61]

    Invariance principles for random bipartite planar maps

    Jean-Fran¸ cois Marckert and Gr´ egory Miermont. Invariance principles for random bipartite planar maps. The Annals of Probability , 35(5):1642–1705, 2007

    2007

  54. [62]

    Uniform sampling in a structured branching population

    Aline Marguet. Uniform sampling in a structured branching population. Bernoulli, 25(4A):2649–2695, 2019

    2019

  55. [63]

    Invariance principles for spatial multitype Galton-Watson trees

    Gr´ egory Miermont. Invariance principles for spatial multitype Galton-Watson trees. Annales de l’IHP Probabilit´ es et statistiques, 44(6):1128–1161, 2008

    2008

  56. [64]

    Tessellations of random maps of arbitrary genus

    Gr´ egory Miermont. Tessellations of random maps of arbitrary genus. Annales scien- tifiques de l’ ´Ecole Normale Sup´ erieure, 42(5):725–781, 2009

    2009

  57. [65]

    Dawson–Watanabe superprocesses and Measure-valued Diffusions

    Edwin Perkins. Dawson–Watanabe superprocesses and Measure-valued Diffusions . Lecture Notes in Math. Springer, Berlin, 2002. 46

    2002

  58. [66]

    Positive harmonic functions and diffusion , volume 45

    Ross Pinsky. Positive harmonic functions and diffusion , volume 45. Cambridge uni- versity press, 1995

    1995

  59. [67]

    Asymptotic genealogy of a critical branching process

    Lea Popovic. Asymptotic genealogy of a critical branching process. The Annals of Applied Probability, 14(4):2120–2148, 2004

    2004

  60. [68]

    An invariance principle for branching diffusions in bounded domains

    Ellen Powell. An invariance principle for branching diffusions in bounded domains. Probability Theory and Related Fields , 173(3):999–1062, 2019

    2019

  61. [69]

    Spectral analysis andk-spine decomposition of inhomogeneous branching Brownian motions

    Emmanuel Schertzer and Julie Tourniaire. Spectral analysis andk-spine decomposition of inhomogeneous branching Brownian motions. Genealogies in fully pushed fronts. arXiv preprint arXiv:2301.01697 , 2023

    2023 arXiv

  62. [70]

    Branching random walks

    Zhan Shi. Branching random walks. In ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour XLII – 2012 , volume 2151, pages 1–133. Springer Cham, 2015

    2012

  63. [71]

    A branching particle system as a model of semipushed fronts

    Julie Tourniaire. A branching particle system as a model of semipushed fronts. The Annals of Probability, 52(6):2104–2172, 2024. A Appendix This appendix contains several elementary results on discrete planar trees. Lemma A.1. The application τ 7→ |w|, Si(τ ) i≤|π(τ )| is a bi...

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.