Slowdown for time inhomogeneous branching Brownian motion

TL;DR

This paper investigates the maximal displacement in one-dimensional time-inhomogeneous branching Brownian motion, revealing a non-logarithmic correction proportional to T^{1/3} for decreasing variances, challenging previous assumptions.

Contribution

It introduces a novel correction term proportional to T^{1/3} for maximal displacement under decreasing variance profiles in branching Brownian motion.

Findings

Correction from linear displacement is proportional to T^{1/3}

Conjecture that this correction is the worst-case scenario

Challenges the assumption of logarithmic correction in such processes

Abstract

We consider the maximal displacement of one dimensional branching Brownian motion with (macroscopically) time varying profiles. For monotone decreasing variances, we show that the correction from linear displacement is not logarithmic but rather proportional to $T^{1/3}$. We conjecture that this is the worse case correction possible.