Refined Large Deviation Principle for Branching Brownian Motion Conditioned to Have a Low Maximum

TL;DR

This paper investigates the effects of conditioning a branching Brownian motion to have a low maximum, analyzing how early or late branching times and locations are affected, and deriving precise large deviation estimates.

Contribution

It provides a refined large deviation principle for branching Brownian motion conditioned on a low maximum, including optimal first branching time and location under additional constraints.

Findings

Large deviation estimates for low maximum conditioning

Optimal first branching time identified

Impact of initial branching location analyzed

Abstract

Conditioning a branching Brownian motion to have an atypically low maximum leads to a suppression of the branching mechanism. In this note, we consider a branching Brownian motion conditioned to have a maximum below $\sqrt{2}\alpha t$ ($\alpha<1$). We study the precise effects of an early/late first branching time and a low/high first branching location under this condition. We do so by imposing additional constraints on the first branching time and location. We obtain large deviation estimates, as well as the optimal first branching time and location given the additional constraints.