order to do so we denote the branching times along the spine that took place when the spine was at the origin by $S_n^0$ and the number of particles produced at these times by $A_n^0$, $n \geq 1$. Similarly, we denote the branching times along the spine when it was not at the origin by $S_n'$and the number of particles produced at these times by $A_n'$, $n \geq 1$. We also denote the counting processes for $(S_n^0)_{n \geq 1}$ and $(S_n')_{n \geq 0}$ by $(n_t^0)_{t \geq 0}$ and $(n_t')_{t \geq 0}$ respectively.

Observe that conditional on the path of the spine $(\xi_t)_{t \geq 0}$, $(n_t^0)_{t \geq 0}$ and $(n_t')_{t \geq 0}$ are independent (inhomogeneous in the first case) Poisson processes (or Cox processes) with jump rates $\beta_0 \delta_0(\cdot)$ and $\beta$ respectively so that $$\tilde{P}\Big( n_t^0 = k \big\vert (\xi_s)_{0 \leq s \leq t}\Big) = \frac{(\beta_0 \tilde{L}_t)^k}{k!} \mathrm{e}^{- \beta_0 \tilde{L}_t}.$$ and $$\tilde{P}\Big( n_t' = k \big\vert (\xi_s)_{0 \leq s \leq t}\Big) = \frac{(\beta t)^k}{k!} \mathrm{e}^{- \beta t}$$ Finally, it is convinient to define several filtrations of the now extended probability space in order to take various conditional expectations.

\[filtrations\] $ $

-   $\mathcal{F}_t$ was defined in Subsection 1.2. It is the 