Polynomial slowdown in an angle-dependent 2d branching Brownian motion

TL;DR

This paper analyzes the maximum displacement in a 2D angle-dependent branching Brownian motion, revealing polynomial slowdown in the maximum's growth rate influenced by the angular inhomogeneity.

Contribution

It establishes the asymptotic behavior of the maximum displacement in an angle-dependent branching Brownian motion with a specific polynomial slowdown.

Findings

The maximum displacement's centered process is tight over time.

The growth rate of the maximum includes a polynomial correction term depending on lpha.

The correction term's coefficient involves the first eigenvalue of a related operator.

Abstract

We consider a branching Brownian motion in $\mathbb{R}^2$ in which particles independently diffuse as standard Brownian motions and branch at an inhomogeneous rate $b(\theta)$ which depends only on the angle $\theta$ of the particle. We assume that $b$ is maximal when $\theta=0$, which is the preferred direction for breeding. Furthermore we assume that $b(\theta ) = 1 - \beta \abs{\theta }^\alpha + O(\theta ^2)$, as $\theta \to 0$, for $\alpha \in (2/3,2)$ and $\beta>0.$ We show that if $M_t$ is the maximum distance to the origin at time $t$, then $(M_t-m(t))_{t\ge 1}$ is tight where $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-\alpha)/(2+\alpha)} - \left(\frac{3}{2\sqrt{2}} - \frac{\alpha}{2\sqrt{2}(2+\alpha)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator.