Mixing times for two classes of stochastically modeled reaction networks

David F. Anderson; Jinsu Kim; David F. Anderson; Jinsu Kim

doi:10.3934/mbe.2023217

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 3: 4690-4713. doi: 10.3934/mbe.2023217

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Mixing times for two classes of stochastically modeled reaction networks

David F. Anderson ¹,
Jinsu Kim ^{2
,
,}

1.
Department of Mathematics, University of Wisconsin, Madison, USA
2.
Department of Mathematics, Pohang University of Science Technology, Pohang 37673, Republic of Korea

Received: 15 September 2022 Revised: 12 December 2022 Accepted: 17 December 2022 Published: 28 December 2022

The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an important practical question is: what is the rate of convergence of the distribution of the process to the stationary distribution? With the exception of ^[1] pertaining to models whose state space is restricted to the non-negative integers, there has been a notable lack of results related to this rate of convergence in the reaction network literature. This paper begins the process of filling that hole in our understanding. In this paper, we characterize this rate of convergence, via the mixing times of the processes, for two classes of stochastically modeled reaction networks. Specifically, by applying a Foster-Lyapunov criteria we establish exponential ergodicity for two classes of reaction networks introduced in ^[2]. Moreover, we show that for one of the classes the convergence is uniform over the initial state.
- stochastically modeled reaction networks,
- Markov chains,
- mixing times,
- total variation distance,
- Foster-Lyapunov criteria,
- exponential ergodicity,
- tiers
Citation: David F. Anderson, Jinsu Kim. Mixing times for two classes of stochastically modeled reaction networks[J]. Mathematical Biosciences and Engineering, 2023, 20(3): 4690-4713. doi: 10.3934/mbe.2023217

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Abstract

The past few decades have seen robust research on questions regarding the existence, form, and properties of stationary distributions of stochastically modeled reaction networks. When a stochastic model admits a stationary distribution an important practical question is: what is the rate of convergence of the distribution of the process to the stationary distribution? With the exception of ^[1] pertaining to models whose state space is restricted to the non-negative integers, there has been a notable lack of results related to this rate of convergence in the reaction network literature. This paper begins the process of filling that hole in our understanding. In this paper, we characterize this rate of convergence, via the mixing times of the processes, for two classes of stochastically modeled reaction networks. Specifically, by applying a Foster-Lyapunov criteria we establish exponential ergodicity for two classes of reaction networks introduced in ^[2]. Moreover, we show that for one of the classes the convergence is uniform over the initial state.

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