used as Radon-Nikodym derivative, $(\tilde{M}^{(2)}_t)_{t \geq 0}$ has the effect of changing the instantaneous jump rate of $(n_t')_{t \geq 0}$ from $\beta$ to $m \beta$ and the jump rate of $(n_t^0)_{t \geq 0}$ from $\beta_0 \delta_0(\cdot)$ to $m_0 \beta_0 \delta_0(\cdot)$. The effect of $(\tilde{M}^{(3)}_t)_{t \geq 0}$ is to change the distribution of random variables $(A_n')_{n \geq 1}$ from $(p_k)_{k \geq 1}$ to $(\frac{k}{m} p_k)_{k \geq 1}$ and the distribution of random variables $(A_n^0)_{n \geq 1}$ from $(q_k)_{k \geq 1}$ to $(\frac{k}{m_0} q_k)_{k \geq 1}$ (while keeping them all independent). If we now define a new probability measure $\tilde{Q}$ as $$\frac{\mathrm{d} \tilde{Q}}{\mathrm{d} \tilde{P}} \Big\vert_{\tilde{\mathcal{F}}_t} = 
\tilde{M}_t \text{ , } \qquad t \geq 0$$ then one can check that the effects of $\tilde{M}^{(1)}$, $\tilde{M}^{(2)}$ and $\tilde{M}^{(3)}$ superimpose so that under $\tilde{Q}$ the branching process has the following description.

-   The process starts with a single spine particle whose path $(\xi_t)_{t \geq 0}$ is distributed like a Brownian motion with drift imposed by $\tilde{M}^{(1)}$.

-   At instantaneous rate $m_0 \beta_0 \delta_0(\cdot) + m \beta$ along its path the spine particle splits into $A(\cdot)$ particles. If splitting took place at position $x$ then $$\tilde{Q}(A(x) = n) = \left\{
    \begin{array}{rl}
    \frac{n}{m_0}q_n& \text{if } x = 