Upper moderate deviation probabilities for the maximum of branching Brownian motion

TL;DR

This paper provides an asymptotic estimate for the probability that the maximum of branching Brownian motion exceeds its typical value by a moderate amount, using a probabilistic second moment approach.

Contribution

It introduces a new asymptotic equivalent for upper moderate deviation probabilities of the maximum in branching Brownian motion.

Findings

Asymptotic equivalent for upper moderate deviation probability

Probabilistic approach using second moment method

Insights into particle behavior during deviations

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.