Branching Brownian motion conditioned on small maximum

TL;DR

This paper confirms Derrida and Shi's conjecture on the asymptotic probability of the maximum of branching Brownian motion being unusually small and characterizes the process conditioned on this rare event.

Contribution

It verifies the conjectured asymptotics for the lower deviation probability of the maximum and describes the conditioned process's law.

Findings

Confirmed the asymptotic behavior of the lower deviation probability.

Described the law of the process conditioned on a small maximum.

Validated conjecture by Derrida and Shi (2017).

Abstract

We consider a standard binary branching Brownian motion on the real line. It is known that the maximal position $M_t$ among all particles alive at time $t$, shifted by $m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t$ converges in law to a randomly shifted Gumbel variable. Derrida and Shi (2017) conjectured the precise asymptotic behaviour of the corresponding lower deviation probability $\mathbb{P}(M_t \leq \sqrt{2}\alpha t)$ for $\alpha < 1$. We verify their conjecture, and describe the law of the branching Brownian motion conditioned on having a small maximum.