Traveling Waves in Inhomogeneous Branching Brownian Motions. I

Consider a branching Brownian motion for which the instantaneous branching rate of a particle at position $x$ is given by $\beta(x)$. We assume that $\beta$ is an integrable continuous function converging to 0 as $x \rightarrow \pm \infty$. Let $R(t)$ be the position of the rightmost descendant at the time $t$ of a simple particle starting from position 0 at time 0. We show that there exists a constant $\lambda_0 > 0$ such that $R(t) - \sqrt{\lambda_0/2} t$ converges in distribution as $t \rightarrow \infty$ to a location mixture of the extreme value distribution $\exp (e^{-\sqrt{2\lambda_0 x}})$.