TY - BOOK

T1 - Quasi-Stationary Distributions in Stochastic Reaction Networks

AU - Hansen, Mads Christian

PY - 2018

Y1 - 2018

N2 - Stochastic reaction networks compose a broad class of applicable continuous-time Markov processes with a particularly rich structure dened through a corresponding graph. As such, they pose a general and natural framework for representing non-linear stochastic dynamical systems where the interactions among types of entities are themselves of transformational form. Many such systems, in particular when they model real world phenomena, 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 paper (Hansen and Wiuf, 2018a), we are concerned with providing sucient conditions for the existence and uniqueness of quasi-stationary distributions in reaction networks. Specically, for any reaction network we introduce the inferred notion of an absorbing set, and prove through the use of Foster-Lyapunov theory, sucient 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 manuscript (Hansen and Schreiber, 2018) deals with connections to the corresponding deterministic system, where qualitative information about the dynamics is often much easier to obtain. 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. Having shown that a specic network at hand has a unique quasi-stationary distribution, the manuscript (Hansen and Wiuf, 2018b) provides an inductive procedure to analytically determine this. Exploiting a center manifold structure, we show that, when one considers the full system as a linear perturbation of a particular sub-network and the coupling to the absorbing set is suciently weak, the quasi-stationary distribution may be written as a formal sum with the stationary distribution of the sub-network as a rst approximation. We furthermore characterize such stationary distributions for one-species networks.

AB - Stochastic reaction networks compose a broad class of applicable continuous-time Markov processes with a particularly rich structure dened through a corresponding graph. As such, they pose a general and natural framework for representing non-linear stochastic dynamical systems where the interactions among types of entities are themselves of transformational form. Many such systems, in particular when they model real world phenomena, 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 paper (Hansen and Wiuf, 2018a), we are concerned with providing sucient conditions for the existence and uniqueness of quasi-stationary distributions in reaction networks. Specically, for any reaction network we introduce the inferred notion of an absorbing set, and prove through the use of Foster-Lyapunov theory, sucient 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 manuscript (Hansen and Schreiber, 2018) deals with connections to the corresponding deterministic system, where qualitative information about the dynamics is often much easier to obtain. 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. Having shown that a specic network at hand has a unique quasi-stationary distribution, the manuscript (Hansen and Wiuf, 2018b) provides an inductive procedure to analytically determine this. Exploiting a center manifold structure, we show that, when one considers the full system as a linear perturbation of a particular sub-network and the coupling to the absorbing set is suciently weak, the quasi-stationary distribution may be written as a formal sum with the stationary distribution of the sub-network as a rst approximation. We furthermore characterize such stationary distributions for one-species networks.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122522604105763

M3 - Ph.D. thesis

BT - Quasi-Stationary Distributions in Stochastic Reaction Networks

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -