Large Deviations Theory for Chemical Reaction Networks

Abstract

In this thesis, we study the occurrence of rare events and large fluctuations around the expected behavior in stochastic dynamical models of chemical reaction networks. This is done through the application of results from large deviations theory and Wentzell-Freidlin theory to the framework of jump Markov processes for mass action kinetics. We start by introducing the class of models under consideration and the set of tail estimates that we aim to obtain. Proceeding to establish such estimates, we encounter obstructions resulting from the possible divergence of the process in finite time intervals, and from its possible degeneracy in some regions of phase space. We bypass such obstructions by designing topological conditions on chemical reaction networks preventing the occurrence of the scenarios mentioned above. We then prove the sufficiency of such conditions for the applicability of a large deviations principle in two alternative ways: by contradiction and by construction.