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

TL;DR

This paper introduces a new backward construction method for branching Brownian motion with large displacements, revealing a continuous family of extremal processes and providing applications to related stochastic processes.

Contribution

It offers a novel probabilistic representation of extremal measures in branching Brownian motions and extends results to branching Ornstein-Uhlenbeck processes.

Findings

Limiting extremal point measures form a one-parameter continuous family

New representation simplifies understanding of large deviation probabilities

Applications to variable speed and multitype branching processes

Abstract

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 (2015)'s results about variable speed branching Brownian motion also describe the extremal point process of branching Ornstein-Uhlenbeck processes.