Convergence Rates for Stationary Distributions of Semistochastic Processes

Abstract

The primary objects of study in this dissertation are semistochastic processes. The types of semistochastic processes we consider are continuous-time and continuous-state processes consisting of intervals of deterministic evolution punctuated by random disturbances of random severity. A natural question regarding such processes is whether they admit stationary distributions. While partial answers to this question exist in the literature, the primary aim of this dissertation is to supplement the criteria for existence with bounds on convergence rates. This requires careful analysis of the associated Markov semigroups and infinitesimal generators. We obtain our bounds on convergence rates by establishing minorization and drift conditions. Specific examples are considered in cases of bounded and unbounded state spaces. We also discuss a method of exact computation for the stationary distributions of a certain class of semistochastic processes. An important example to which we can apply our work concerns the modelling of the carbon content of an ecosystem.