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. Such models are motivated by problems in chemical kinetics
where one wants to keep the level of a catalyst above a certain threshold in
order to maintain a desired level of reaction activity. 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. Proofs rely on constrained martingale
problem characterizations, Lyapunov function constructions, moment estimates
that are uniform in time and the scaling parameter and occupation measure
techniques.Comment: Published in at http://dx.doi.org/10.1214/12-AAP894 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
