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 $b(\hat{I}¸)$ which depends only on the angle $\hat{I}¸$ of the particle. We assume that $b$ is maximal when $\hat{I}¸=0$, which is the preferred direction for breeding. Furthermore we assume that $b(θ) = 1 - β\abs{θ}^α+ O(θ^2)$, as $\hat{I}¸\to 0$, for $\hat{I}\pm \in (2/3,2)$ and ${\hat{I}}^2>0.$ 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 $$m(t)=\sqrt{2}t-\frac{{\vartheta }_1}{\sqrt{2}}{t}^{(2-\hat{I}\pm )/(2+\hat{I}\pm )}-\left(\frac{3}{2\sqrt{2}}-\frac{\hat{I}}{\pm }2\sqrt{2}(2+\hat{I}\pm )\right)\log t.$$ and ${\vartheta }_1$ is explicit in terms of the first eigenvalue of a certain operator.

57 pages, 4 figures