Asymptotic Results for Near Critical Bienaymé–Galton–Watson and Catalyst-Reactant Branching Processes

Near critical single-type Bienaymé–Galton–Watson (BGW) processes are considered. Results on convergence of Yaglom distributions of suitably scaled BGW processes to that of the corresponding diffusion approximation are given. Convergences of stationary distributions for Q-processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established. Similar results can be obtained in a multitype setting. To illustrate this, a result on convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is presented.

In the second part, near critical catalyst-reactant branching processes with controlled immigration are considered. The reactant population evolves according to a branching process whose branching rate is proportional to the total mass of the catalyst. The bulk catalyst evolution is that of a classical continuous-time branching process; in addition, there is a specific form of immigration. Immigration takes place exactly when the catalyst population falls below a certain threshold, in which case the population is instantaneously replenished to the threshold. A diffusion limit theorem for the scaled processes is presented, in which the catalyst limit is described through a reflected diffusion, while the reactant limit is a diffusion with coefficients that are functions of both the reactant and the catalyst. Stochastic averaging under fast catalyst dynamics is considered next. In the case where the catalyst evolves “much faster” than the reactant, a scaling limit, in which the reactant is described through a one-dimensional SDE with coefficients depending on the invariant distribution of the reflected diffusion, is obtained.