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Upper moderate deviation probabilities for the maximum of branching Brownian motion

Louis Chataignier

TL;DR

This work derives a sharp asymptotic for the upper moderate deviation probabilities of the maximum M_t in binary branching Brownian motion, in the regime 1 ≪ x_t ≪ t, by establishing that P(M_t > m_t + x_t) is asymptotically equivalent to C^* γ_t(x_t). The authors deploy a probabilistic second-moment framework built on many-to-one and many-to-two lemmas, coupled with precise Brownian barrier estimates and barrier-based path decompositions. The main result expresses the probability through a simple, explicit factor γ_t(x_t) multiplied by a universal constant C^*, derived from a late-time limiting integral. As a byproduct, the paper provides insights into the typical ancestry and trajectory of particles responsible for moderate deviations, and the methods are adaptable to related models such as branching random walk.

Abstract

It is known from Bramson (1983) that the maximum of branching Brownian motion at time $t$ is asymptotically around an explicit function $m_t$, which involves a first ballistic order and a logarithmic correction. In this paper, we give an asymptotic equivalent for its upper moderate deviation probability, that is, the probability that the maximum achieves $m_t + x_t$ at time $t$, where $1 \ll x_t \ll t$. We adopt a probabilistic approach that employs a modified version of the second moment method. As a byproduct, we obtain information about the typical behavior of particles contributing to such deviations.

Upper moderate deviation probabilities for the maximum of branching Brownian motion

TL;DR

This work derives a sharp asymptotic for the upper moderate deviation probabilities of the maximum M_t in binary branching Brownian motion, in the regime 1 ≪ x_t ≪ t, by establishing that P(M_t > m_t + x_t) is asymptotically equivalent to C^* γ_t(x_t). The authors deploy a probabilistic second-moment framework built on many-to-one and many-to-two lemmas, coupled with precise Brownian barrier estimates and barrier-based path decompositions. The main result expresses the probability through a simple, explicit factor γ_t(x_t) multiplied by a universal constant C^*, derived from a late-time limiting integral. As a byproduct, the paper provides insights into the typical ancestry and trajectory of particles responsible for moderate deviations, and the methods are adaptable to related models such as branching random walk.

Abstract

It is known from Bramson (1983) that the maximum of branching Brownian motion at time is asymptotically around an explicit function , which involves a first ballistic order and a logarithmic correction. In this paper, we give an asymptotic equivalent for its upper moderate deviation probability, that is, the probability that the maximum achieves at time , where . We adopt a probabilistic approach that employs a modified version of the second moment method. As a byproduct, we obtain information about the typical behavior of particles contributing to such deviations.
Paper Structure (8 sections, 16 theorems, 80 equations)

This paper contains 8 sections, 16 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Many-to-few lemmas
  3. Brownian estimates
  4. A first upper bound
  5. Non-contributing particles
  6. A first lower bound
  7. Reduction to an expectation
  8. Proof of the equivalent

Key Result

Theorem 1.1

For any $(x_t)_{t \geq 0}$ satisfying $1 \ll x_t \ll t$ as $t \to \infty$, we have the asymptotic equivalent $\mathbb{P}(M_t > m_t + x_t) \sim C^* \gamma_t(x_t)$ as $t \to \infty$, where, for $x > 0$,

Theorems & Definitions (27)

  • Theorem 1.1
  • Corollary 1.2
  • Remark 1.3
  • Lemma 2.1: Many-to-one
  • Lemma 2.2: Many-to-two
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 4.1
  • proof
  • Corollary 4.2
  • ...and 17 more