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

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 $b(θ)$ which depends only on the angle $θ$ of the particle. We assume that $b$ is maximal when $θ=0$, which is the preferred direction for breeding. Furthermore we assume that $b(θ) = 1 - β\abs{θ}^α+ O(θ^2)$, as $θ\to 0$, for $α\in (2/3,2)$ and $β>0.$ 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 $$m(t) = \sqrt{2} t - \frac{\vartheta_1}{\sqrt{2}} t^{(2-α)/(2+α)} - \left(\frac{3}{2\sqrt{2}} - \fracα{2\sqrt{2}(2+α)}\right) \log t. $$ and $\vartheta_1$ is explicit in terms of the first eigenvalue of a certain operator.