title: Lyapunov Functions, Stationary Distributions, and Non-equilibrium Potential for Reaction Networks.
abstract: We consider the relationship between stationary distributions for stochastic models of reaction systems and Lyapunov functions for their deterministic counterparts. Specifically, we derive the well-known Lyapunov function of reaction network theory as a scaling limit of the non-equilibrium potential of the stationary distribution of stochastically modeled complex balanced systems. We extend this result to general birth-death models and demonstrate via example that similar scaling limits can yield Lyapunov functions even for models that are not complex or detailed balanced, and may even have multiple equilibria.
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA. anderson@math.wisc.edu.
authors.forename: David F
authors.initials: DF
authors.lastname: Anderson
authors.affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, WI, USA. craciun@math.wisc.edu.|Department of Biomolecular Chemistry, University of Wisconsin-Madison, Madison, WI, USA. craciun@math.wisc.edu.
authors.forename: Gheorghe
authors.initials: G
authors.lastname: Craciun
authors.affiliation: School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India. manoj.gopalkrishnan@gmail.com.
authors.forename: Manoj
authors.initials: M
authors.lastname: Gopalkrishnan
authors.affiliation: Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark. wiuf@math.ku.dk.
authors.forename: Carsten
authors.initials: C
authors.lastname: Wiuf
mesh_terms: D007700:Kinetics; D008390:Markov Chains; D055641:Mathematical Concepts; D053858:Metabolic Networks and Pathways; D008954:Models, Biological; D011157:Population Dynamics / Q000706:statistics & numerical data; D013269:Stochastic Processes
keywords: Birth-death process; Complex balanced; Continuous time Markov chain; Dynamical system; Long-term dynamics