A simple backward construction of branching Brownian motion with large displacement and applications

In this article, we study the extremal processes of branching Brownian motions conditioned on having an unusually large maximum. The limiting point measures form a one-parameter family and are the decoration point measures in the extremal processes of several branching processes, including branching Brownian motions with variable speed and multitype branching Brownian motions. We give a new, alternative representation of these point measures and we show that they form a continuous family. This also yields a simple probabilistic expression for the constant that appears in the large deviation probability of having a large displacement. As an application, we show that Bovier and Hartung’s (ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015) 261–291) results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein–Uhlenbeck processes.