Near Critical Catalyst Reactant Branching Processes with Controlled Immigration

• Activity Number: 620
• Type: Contributed
• Date/Time: Thursday, August 2, 2012 : 8:30 AM to 10:20 AM
• Sponsor: IMS
• Author(s): Dominik Reinhold*+ and Amarjit Budhiraja
• Companies: Clark University and The University of North Carolina at Chapel Hill
• Keywords: Catalyst-reactant dynamics ; near critical branching processes ; diffusion approximations ; stochastic averaging ; multiscale approximations ; reflected diffusions

Abstract

Near critical catalyst-reactant branching processes with controlled immigration are studied. 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 principles under fast catalyst dynamics are established. 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.