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Large deviations for the maximum of a reducible two-type branching Brownian motion

Hui He

TL;DR

This work analyzes upper large deviations for the maximum $M_t$ of a two-type reducible BBM, where type-1 particles can birth into type-2 and both types diffuse with independent Brownian motions. Building on decomposition via the oldest type-2 ancestor and the many-to-one framework, the authors derive region-dependent exponential decay rates for ${\mathbb P}(M_t \geq \theta m_t)$ with $\theta>1$, revealing phase transitions across the three regions $\mathcal{C}_{\text{I}}$, $\mathcal{C}_{\text{II}}$, and $\mathcal{C}_{\text{III}}$ of the phase diagram. The main contributions are explicit rate functions that depend on $(\beta, \sigma^2)$ and $\theta$, with distinct expressions corresponding to whether the maximizer of the rate function occurs at an early, late, or interior birth-time of type-2 descendants; these results are supported by matching upper and lower bounds via first/second moment techniques and PPP arguments. Overall, the paper extends large deviation understanding for multi-type BBMs by showing how reducibility and cross-type branching modify the tail behavior of extreme positions and clarifies the role of phase transitions in this stochastic growth model.

Abstract

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type $1$ can give birth to particles of types $1$ and $2$, but particles of type $2$ only give birth to descendants of type $2$. Under some specific conditions, Belloum and Mallein in \cite{BeMa21} showed that the maximum position $M_t$ of all particles alive at time $t$, suitably centered by a deterministic function $m_t$, converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as $t\rightarrow\infty$, \[ {\mathbb P}(M_t\geq θm_t),\quad θ>1. \] We shall show that the decay rate function exhibits phase transitions depending on certain relations between $θ$, the variance of the underlying Brownian motion and the branching rate.

Large deviations for the maximum of a reducible two-type branching Brownian motion

TL;DR

This work analyzes upper large deviations for the maximum of a two-type reducible BBM, where type-1 particles can birth into type-2 and both types diffuse with independent Brownian motions. Building on decomposition via the oldest type-2 ancestor and the many-to-one framework, the authors derive region-dependent exponential decay rates for with , revealing phase transitions across the three regions , , and of the phase diagram. The main contributions are explicit rate functions that depend on and , with distinct expressions corresponding to whether the maximizer of the rate function occurs at an early, late, or interior birth-time of type-2 descendants; these results are supported by matching upper and lower bounds via first/second moment techniques and PPP arguments. Overall, the paper extends large deviation understanding for multi-type BBMs by showing how reducibility and cross-type branching modify the tail behavior of extreme positions and clarifies the role of phase transitions in this stochastic growth model.

Abstract

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type can give birth to particles of types and , but particles of type only give birth to descendants of type . Under some specific conditions, Belloum and Mallein in \cite{BeMa21} showed that the maximum position of all particles alive at time , suitably centered by a deterministic function , converge weakly. In this work, we are interested in the decay rate of the following upper large deviation probability, as , We shall show that the decay rate function exhibits phase transitions depending on certain relations between , the variance of the underlying Brownian motion and the branching rate.
Paper Structure (20 sections, 6 theorems, 129 equations, 1 figure)

This paper contains 20 sections, 6 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Technical lemmas
  3. Proof of Theorem \ref{['ThmcaseI']}
  4. Upper bound
  5. Lower bound for Theorem \ref{['ThmcaseI']}
  6. The case of $\max_{0\leq u\leq 1} f_1(u)=f_1(0)$.
  7. The case of $\max_{0\leq u\leq 1} f_1(u)=f_1(1)$.
  8. The case of $\max_{0\leq u\leq 1} f_1(u)=f_1(u_0)$.
  9. Proof of Theorem \ref{['ThmcaseII']}
  10. Upper bound for Theorem \ref{['ThmcaseII']}
  11. Lower bound for Theorem \ref{['ThmcaseII']}
  12. The case of $\max_{0\leq u\leq 1} f_2(u)=f_2(0)$.
  13. The case of $\max_{0\leq u\leq 1} f_2(u)=f_2(1)$.
  14. The case of $\max_{0\leq u\leq 1} f_2(u)=f_2(u_0)$ for $0<u_0<1$.
  15. Proof of Theorem \ref{['ThmcaseIII']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume $\left(\beta, \sigma^2\right) \in \mathcal{C}_\text{I}$. Then the following limit exists for any $\theta>1$, where if $\sigma ^2\geq 1$, then and if $\beta > 1$ and $\sigma ^2< 1$, then we have

Figures (1)

  • Figure 1: Phase diagram of the two-type reducible BBM

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • proof
  • proof
  • proof