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The leftmost particle of branching subordinators

Alexis Kagan, Grégoire Véchambre

TL;DR

The paper analyzes the leftmost particle in continuous-time branching subordinators—systems where particles move as subordinators and reproduce into random, potentially infinite offspring. It introduces truncated Laplace exponents and a truncation-based construction to handle finite and infinite birth intensities, and derives sharp criteria for whether the leftmost particle explodes to infinity or remains finite. By relaxing exponential moment assumptions and imposing stable-like and regularity conditions, it proves sub-linear growth of the leftmost position with exponent $\gamma$ given by $\gamma=\dfrac{1}{\alpha+(1-\alpha)(1+\sigma)}$, and provides explicit corollaries for two explicit models: rare infinite branchings and many binary branchings, yielding $\gamma$ in terms of $\alpha,\beta,\theta$. The results illuminate how population structure and branching dynamics shape the speed of the leftmost particle, revealing sub-linear growth in a broad class of no-exponential-moment regimes and connecting to fragmentation and age-dependent branching theory. These findings show that linear growth is not universal in such branching systems and quantify the impact of small-jump stability and branching regularity on propagation speeds.

Abstract

We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching Lévy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order $t^γ$ where $γ\in (0,1)$ depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements.

The leftmost particle of branching subordinators

TL;DR

The paper analyzes the leftmost particle in continuous-time branching subordinators—systems where particles move as subordinators and reproduce into random, potentially infinite offspring. It introduces truncated Laplace exponents and a truncation-based construction to handle finite and infinite birth intensities, and derives sharp criteria for whether the leftmost particle explodes to infinity or remains finite. By relaxing exponential moment assumptions and imposing stable-like and regularity conditions, it proves sub-linear growth of the leftmost position with exponent given by , and provides explicit corollaries for two explicit models: rare infinite branchings and many binary branchings, yielding in terms of . The results illuminate how population structure and branching dynamics shape the speed of the leftmost particle, revealing sub-linear growth in a broad class of no-exponential-moment regimes and connecting to fragmentation and age-dependent branching theory. These findings show that linear growth is not universal in such branching systems and quantify the impact of small-jump stability and branching regularity on propagation speeds.

Abstract

We define a family of continuous-time branching particle systems on the non-negative real line, called branching subordinators, where particles move as independent subordinators. Each particle can also split (at possibly infinite rate) into several children (possibly infinitely many) whose positions relative to the position of the parent are random. These particle systems are in the continuity of branching Lévy processes introduced by Bertoin and Mallein [Ann. Probab. 47(3): 1619-1652 (2019)]. We pay a particular attention to the asymptotic behavior of the leftmost particle of branching subordinators. It turns out that, under some assumptions, the rate of growth of the position of the leftmost particle is of order where depends explicitly on the parameters. This sub-linear growth is significantly different from the classical linear growth observed for regular branching random walks with non-negative displacements.
Paper Structure (26 sections, 15 theorems, 55 equations)

This paper contains 26 sections, 15 theorems, 55 equations.

Table of Contents

  1. Introduction
  2. Examples of branching subordinators
  3. A simple example: the branching Poisson process
  4. $\alpha$-stable trajectories with rare infinite branchings
  5. $\alpha$-stable trajectories with many binary branchings
  6. Some more definitions
  7. Main results
  8. Divergence of the leftmost particle
  9. Classical results and linear behavior
  10. Relaxing the exponential moments assumption, sub-linear behavior
  11. Organization of the paper
  12. Application of Theorem \ref{['generalcriteria']} on two examples
  13. $\alpha$-stable trajectories with rare infinite branchings: proof of Corollary \ref{['corri']}
  14. $\alpha$-stable trajectories with rare infinite branchings: proof of Corollary \ref{['corri']}
  15. $\alpha$-stable trajectories with many binary branchings: proof of Corollary \ref{['cormb']}
  16. ...and 11 more sections

Key Result

Proposition 1.3

For any $d\geq0$ and for any measure $\Lambda$ on $\mathcal{Q}$ satisfying condbranchinglp0 and condbranchinglplight, the branching subordinator $Y$ with characteristic couple $(d,\Lambda)$ satisfies $\mathbb{P}(\underline{Y}(\infty)=\infty)\in\{0,1\}$ and

Theorems & Definitions (28)

  • Remark 1.1
  • Remark 1.2
  • Proposition 1.3
  • Example I
  • Remark 1.4: $p$-ary branchings
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Lemma 2.1
  • Remark 2.2: $p$-ary branchings
  • ...and 18 more