Near critical catalyst reactant branching processes with controlled immigration

TL;DR

This paper analyzes near-critical catalyst-reactant branching processes with controlled immigration, deriving diffusion limit theorems and stochastic averaging principles to understand their long-term behavior under different time scale regimes.

Contribution

It introduces a new model of catalyst-reactant processes with threshold-based immigration and provides diffusion limit theorems and averaging principles for these processes.

Findings

Diffusion limit theorem with reflected diffusion for catalyst

Reactant described by SDE with coefficients depending on invariant distribution

Stochastic averaging principles established for fast catalyst dynamics

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. 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.