Simple Branching Processes

Consider a particle that can replicate itself instantaneously at random times. In general there is a distribution of times from which the particle draws in order to initiate replication. For simplicity, we assume a Poisson process, and a replication rate of λ. Consider the total integer number of particles N(t) at any time t. Let us now consider one realization of the continuous-time branching processes shown in Fig. 1. Given a infinitesimally small time after the initiation of the branching process (from a single progenitor), there will be a number N(ε) of first generation particles. Each of these particles wil be labeled with index 0 ≤ i ≤ N(ε). The number of particles existing at time t+ ε, N(t+ ε), will be the sum of all particles that originate from each of the N(ε) first generation offspring. For a particular realization then,