Exponential ergodicity of first order endotactic stochastic reaction systems

TL;DR

This paper proves that all first order endotactic stochastic reaction systems are essential and exponentially ergodic, and extends these results to systems dominated by such networks, with applications to higher order systems.

Contribution

It establishes exponential ergodicity and essentiality for all first order endotactic SMARTs and their dominated systems, advancing the understanding of their long-term behavior.

Findings

All first order endotactic SMARTs are essential and exponentially ergodic.

Systems dominated by first order endotactic SMARTs are also exponentially ergodic.

Examples include higher order SMARTs with endotactic asymptotic limits.

Abstract

Chemical reaction networks are a widely accepted modeling framework for diverse science phenomena stemming from all disciplines of science, such as biochemistry, ecology, epidemiology, social and political science. In this paper we prove that every first order endotactic stochastic mass-action reaction system (SMART) is essential (i.e., every state in the state space is within a closed communicating class of the underlying continuous time Markov chain model) and is exponentially ergodic. The proof is based on a recent result on first order endotactic reaction networks in a companion paper [C.X., First order endotactic reaction networks. arXiv:2409.01598v2]. Besides, we show that a stochastic reaction system (of possibly nonlinear propensities) dominated by a first order endotactic SMART is exponentially erogdic. To demonstrate the applicability of results, we provide various examples of higher order SMART, including e.g., (1) SMART with a first order endotactic asymptotic limit as well as, (2) joint of translations of first order endotactic SMART.