UC San Diego

This dissertation studies three aspects of a class of continuous-time Markov chain models frequently used to describe the stochastic dynamics of a chemical system undergoing a series of reactions which change the numbers of molecules of a finite set of species over time. We call these models Stochastic Chemical Reaction Networks (SCRNs). First, some stochastic ordering results for SCRNs are developed; these are called comparison theorems. These results provide sufficient conditions for establishing monotonic dependence on parameters for SCRNs and associated quantities such as mean first passage times and stationary distributions. Second, some results on singular perturbations for continuous-time Markov chains are developed, motivated by SCRN models for epigenetic cell memory. The focus here is on studying the behavior of the stationary distributions as functions of the singular perturbation parameter. Finally, a reflected diffusion approximation called the Constrained Langevin Approximation (CLA) is studied. Proposed by Leite and Williams, the CLA extends the traditional Langevin approximation beyond the first time some species becomes zero in number. In this dissertation, the approximation of Leite and Williams is extended to (nearly) density dependent Markov chains when the diffusion state space is one-dimensional. In this context, error bounds for the CLA are provided in the form of strong approximations.