Limiting Distribution of the Rightmost Particle in Catalytic Branching Brownian Motion

TL;DR

This paper investigates the asymptotic distribution of the rightmost particle in a catalytic branching Brownian motion with a point catalyst, revealing convergence to a mixture of Gumbel distributions.

Contribution

It extends previous work by Lalley and Sellke to the catalytic case, showing the limiting distribution of the rightmost particle in this inhomogeneous branching model.

Findings

Distribution converges to a mixture of Gumbel distributions.

Centering the rightmost particle at (β/2)t reveals the limiting behavior.

Results generalize known models to spatially-inhomogeneous branching processes.

Abstract

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.