Branching Brownian motion conditioned on small maximum

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 $\mathbb{P}(M_t\le \sqrt{2}\alpha t)$ for $\alpha <1$. We verify their conjecture, and describe the law of the branching Brownian motion conditioned on having a small maximum.

42 pages, 2 figures