some new notation in relation with the spine process.

For any $t \geq 0$ we let $node_t(\xi) := u$ for the unique particle $u \in N_t \cap \xi$. That is, $node_t(\xi)$ is the label of the spine particle at time $t$.

For any $t \geq 0$ we let $\xi_t := X^u_t$ for the unique particle $u \in N_t \cap \xi$. So that $\xi_t$ is the spatial position of the spine particle at time $t$. It is not hard to check that the process $(\xi_t)_{t \geq 0}$ is a Brownian motion under $\tilde{P}$. We shall denote by $(\tilde{L}_t)_{t \geq 0}$ its local time at the origin.

For any $t \geq 0$ we let $n_t := n$ for the unique $n$ such that $\xi_n \in N_t$. So that $(n_t)_{t \geq 0}$ is the counting process of the number of fissions that have occurred along the path of the spine by time $t$. We denote the sequence of such fission times by $S_n$ and the number of particles produced at each such fission by $A_n$, $n \geq 1$.

Moreover, we would like to distinguish fissions along the spine that occurred due to catalytic branching from those that occurred due to homogeneous branching. In 