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abstract: 'We consider a model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are simultaneously present. We establish the almost sure growth rates of population in certain time-dependent regions and as a consequence the first-order asymptotic behaviour of the rightmost particle.'
author:
- 'Sergey Bocharov[^1], Li Wang[^2]'
title: 'Branching Brownian Motion with spatially-homogeneous and point-catalytic branching'
---

Introduction and main results
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Description of the model
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We are going to consider a model of Branching Brownian Motion in which the usual spatially-homogeneous and catalytic branching at a single point are present simultaneously.

Such a process is initiated with a single particle at time $0$ whose spatial position at time $t \geq 0$ up until the time it dies is given by $X_t$ where $(X_t)_{t \geq 0}$ is distributed like a standard Brownian motion. We let $T'$ and $T_0$ be two random times which are independent conditional on $(X_t)_{t \geq 0}$ and satisfy $P(T' > t | (X_s)_{s \geq 0}) = \mathrm{e}^{- \beta t}$ and $P(T_0 > t | (X_s)_{s \geq 0}) = \mathrm{e}^{- \beta_0 L_t}$, where $\beta > 0$ and $\beta_0 > 0$ are some constants and $(L_t)_{t \geq 0}$ 