0, \\
p_n& \text{if } x \neq 0 
\end{array} \right.$$ and the new particles then stochastically repeat the behaviour of their parent starting from $x$. Thus the model can be thought of as the BBM model with spatially-inhomogeneous branching rate $\beta_0 \delta_0(\cdot) + \beta$, where $\delta_0(\cdot)$ is the Dirac delta function, and spatially-inhomogeneous offspring distribution given by since informally we may say that $L_t = \int_0^t \delta_0(X_s) \mathrm{d}s$ (this can be made formal via the theory of additive functionals of Brownian motion).

Our model combines in a natural way the classical BBM model with constant branching and the BBM model with a single catalytic point. The first one has been studied for many decades and numerous asymptotic results are available for it (see e.g. [@M75], [@B78], [@LS87], [@R13]). The catalytic model has been given less attention (one may for example look at [@K99] for a general review of the topic).

Some notation
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Using common practice we label the initial particle by $\varnothing$ and all the other particles according to the Ullam-Harris convention. So that for example particle “$\varnothing32$" is the second child of the third child of the initial particle.

For two particles $u$ and $v$ we shall write 