Branching brownian motion conditioned on large level sets

TL;DR

This paper investigates the probabilities of large deviations in the sizes of intermediate level sets in branching Brownian motion and explores the typical behaviors of BBM under such conditions, improving previous results.

Contribution

It provides refined large deviation estimates for intermediate level sets in BBM and analyzes their typical behaviors conditioned on large level sets, using novel connections with martingale limits and transformed BBMs.

Findings

Improved large deviation probability estimates for BBM level sets.

Characterization of typical behaviors of BBM conditioned on large level sets.

Connections established between level sets, martingale limits, and transformed BBMs.

Abstract

We study the precise large deviation probabilities for the sizes of intermediate level sets in branching Brownian motion (BBM). Our conclusions improve a result of A\"{i}dekon, Hu and Shi in [J. Math. Sci. \textbf{238}(2019)]. Additionally, we analyze the typical behaviors of BBM conditioned on large level sets. Our approach relies on the connections between intermediate level sets, additive martingale limits of BBM, and the global minimum of linearly transformed BBMs.