is the local time at $0$ of $(X_t)_{t \geq 0}$. Note that almost surely $X_{T_0} = 0$ and $X_{T'} \neq 0$.

At the time $T' \wedge T_0$ the initial particle dies and is replaced with a random number of new particles. If $T_0 < T'$ then the number of new particles follows some given distribution $(q_n)_{n \geq 1}$. Otherwise it follows a different distribution $(p_n)_{n \geq 1}$.

All the new particles then independently of each other and of the past stochastically repeat the behaviour of their parent starting from $X_{T' \wedge T_0}$. That is, they move like Brownian motions, die after random times giving births to new particles, etc.

Note that all particles always produce at least one child upon their death ruling out the possibility of population extinction.

An equivalent (up to indistinguishability) description would be to say that after a random time $T$ such that $P(T > t | (X_s)_{s \geq 0}) = \mathrm{e}^{- \beta_0 L_t - \beta t}$ the initial particle dies, and at position $x$ where it died it is replaced with a random number $A(x)$ of new particles where for $n \geq 1$ $$\label{Ax}
P(A(x) = n) = \left\{
\begin{array}{rl}
q_n& \text{if } x = 