Foster-Lyapunov criteria with stopping times and applications to Stochastic Reaction Networks

The main objective in this thesis is to obtain a unified approach to the stability classification of continuous-time Markov chains defined on discrete state space via Foster-Lyapunov criteria. These criteria are tipically stated in terms of the generator of the process. Aleksandr Lyapunov first introduced these techniques for the study of ordinary differential equations and F. Gordon Foster first adapted them to a stochastic setting. After an introduction of the preliminaries about the main setting of work, a version of Dynkin's formula and its proof are provided. Ruling out unstable behaviours of the Markov chains such as explosivity or transience and establishing recurrence and positive recurrence is a complex task. In these regards, the second chapter analyzes the Foster-Lyapunov criteria that imply these properties and the results are proved by the systematic application of Dynkin's formula. Consider, for istance, the classical Foster-Lyapunov criterion for verifying the positive recurrence: it assumes that the Markov chain tends to drift in unit steps towards some finite subset of the state space, and it does not wander too far when it makes a one-step transition out of this set. The third chapter deals with establishing analogous drift criteria that are defined on random stopping times of the Markov chain. The first study that addressed such issues was Filonov, who enunciated a sufficient drift condition for a discrete time Markov chain on a countable space to be positive recurrent. The last chapter is devoted to analyzing some interesting examples of stochastic reaction networks and to studying their limit behaviour by Foster-Lyapunov criteria. In particular, stochastic reaction networks are a family of continuous time Markov chains used to model biochemical systems and intracellular processes. The idea is quite simple: the species react by a finite number of possible biochemical transformations and the state of the system, which is the count of the available species, changes by the occurrence of a reaction. Traditionally, the dynamics of the concentration of each species are modelled by means of an ordinary differential equation, however this type of models are inaccurate if the number of constituents of at least one species is extremely low, something common in biological setting. This makes the stochastic descriptions of reaction networks essential.