Upper moderate deviation probabilities for the maximum of branching Brownian motion

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.