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

  • Amarjit Budhiraja
  • Dominik ReinholdEmail author
Conference paper
  • 792 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 38)

Abstract

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.

Keywords

Branching processes Catalyst-reactant branching processes Quasi stationary distributions Yaglom distributions Q-processes Near critical regime Chemical reaction networks Diffusion approximations Stochastic averaging Multiscale approximations Reflected diffusions Constrained martingale problems Echeverria criterion Invariant measure convergence 

Notes

Acknowledgements

We gratefully acknowledge the valuable feedback from the referee which, in particular, led to a simplification of Condition 2.1.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversity of North CarolinaChapel HillUSA
  2. 2.Department of Mathematics and Computer ScienceClark UniversityWorcesterUSA

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