PhD Defense: Mads Christian Hansen

Title: Quasi-Stationary Distributions in Stochastic Reaction Networks

Abstract:
Stochastic reaction networks compose a broad class of applicable continuous-time Markov processes with a particularly rich structure defined through a corresponding graph. Many such systems are certain to go “extinct” eventually, yet appear to be stationary over any reasonable time scale. This phenomenon is termed quasi-stationarity. A stationary measure for the stochastic process conditioned on non-extinction, called a quasi-stationary distribution, assigns mass to states in a way that mirrors this observed quasi-stationarity.

In the first paper, we introduce for any reaction network the inferred notion of an absorbing set, and prove through the use of Foster-Lyapunov theory, sufficient conditions for the associated Markov process to have a globally attracting quasi-stationary distribution in the space of all probability measures on the complement of the absorbing set.

The second manuscript deals with connections to the corresponding deterministic system. Through the use of Morse-decompositions, we show that under the classical scaling, the limit of quasi-stationary measures converges weakly to a probability measure whose support is contained in the attractors of the deterministic system lying entirely within the strictly positive orthant. 

The final manuscript exploits a center manifold structure to provide an inductive procedure to analytically determine the quasi-stationary distribution. Using a linear perturbation, the stationary distribution of a certain sub-network may be viewed as a first approximation to the quasi-stationary distribution. We furthermore characterize such stationary distributions for one-species networks.

Supervisor: Prof. Carsten Wiuf, MATH, University of Copenhagen

Assessment Committee:
Prof. (Chairman) Jeffrey Collamore, University of Copenhagen
Prof. Enrico Bibbona, Politecnico di Torino
Senior Lecturer Simon Cotter, University of Manchester