Limiting Distribution of the Rightmost Particle in Catalytic Branching Brownian Motion

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta {\delta }_0(\cdot )$, where ${\delta }_0(\cdot )$ is the Dirac delta function and $\beta$ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta }{2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [6] for the degenerate case of catalytic branching.