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A path method for non-exponential ergodicity of Markov chains and its application for chemical reaction systems

Minjoon Kim, Jinsu Kim

Abstract

In this paper, we present criteria for non-exponential ergodicity of continuous-time Markov chains on a countable state space. These criteria can be verified by examining the ratio of transition rates over certain paths. We applied this path method to explore the non-exponential convergence of microscopic biochemical interacting systems. Using reaction network descriptions, we identified special architectures of biochemical systems for non-exponential ergodicity. In essence, we found that reactions forming a cycle in the reaction network can induce non-exponential ergodicity when they significantly dominate other reactions across infinitely many regions of the state space. Interestingly, the special architectures allowed us to construct many detailed balanced and complex balanced biochemical systems that are non-exponentially ergodic. Some of these models are low-dimensional bimolecular systems with few reactions. Thus this work suggests the possibility of discovering or synthesizing stochastic systems arising in biochemistry that possess either detailed balancing or complex balancing and slowly converge to their stationary distribution.

A path method for non-exponential ergodicity of Markov chains and its application for chemical reaction systems

Abstract

In this paper, we present criteria for non-exponential ergodicity of continuous-time Markov chains on a countable state space. These criteria can be verified by examining the ratio of transition rates over certain paths. We applied this path method to explore the non-exponential convergence of microscopic biochemical interacting systems. Using reaction network descriptions, we identified special architectures of biochemical systems for non-exponential ergodicity. In essence, we found that reactions forming a cycle in the reaction network can induce non-exponential ergodicity when they significantly dominate other reactions across infinitely many regions of the state space. Interestingly, the special architectures allowed us to construct many detailed balanced and complex balanced biochemical systems that are non-exponentially ergodic. Some of these models are low-dimensional bimolecular systems with few reactions. Thus this work suggests the possibility of discovering or synthesizing stochastic systems arising in biochemistry that possess either detailed balancing or complex balancing and slowly converge to their stationary distribution.
Paper Structure (20 sections, 19 theorems, 75 equations, 3 figures)

This paper contains 20 sections, 19 theorems, 75 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main analytic results
  3. Notations
  4. The basic model
  5. The main theorem
  6. Main application: non-exponential ergodicity of stochastic reaction networks
  7. Stochastic models for reaction networks
  8. Strong tier-1 cycles
  9. Structural conditions for non-exponential ergodicity
  10. Extension via coupling
  11. Non-exponentially ergodic biochemical systems with detailed balance and complex balance
  12. Proofs of the results in Sections \ref{['sec:strong']} and \ref{['subsec:structural']}
  13. Proof of Proposition \ref{['prop:strong tier 1']}
  14. Proof of Theorem \ref{['thm: stong tier1 and irr is enough']}
  15. General structural conditions for exponential ergodicity
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $X$ be an ergodic continuous-time Markov chain with state space $\mathbb S$. Suppose that there exists $x\in \mathbb S$ such that for each positive constant $\rho$, where $\tau_x=\inf\{t> J_x: X(t)=x\}$ is the first return time of $x$, where $J_x=\inf\{t>0: X(t)\neq x\}$. Then $X$ is non-exponentially ergodic.

Figures (3)

  • Figure 1: Let $X$ be the Markov chains defined in Example \ref{['ex:the main example 1']}. Let $\bar{X}$ be the Markov chain with the transition rates $q_{z,z+(1,1)}=1, q_{z,z-(1,1)}=(z)_1(z)_2, q_{z,z+(0,1)}=1$ and $q_{z,z-(0,1)}=(z)_2$ for each state $z$. The graphs of log-scaled total variation norms of $X$ (left) and $\bar{X}$ (right) are given. Exponentially ergodic of $\bar{X}$ holds anderson2023new. Thus by \ref{['eq:expo ergo']} the convergence for the initial states $(10,0), (15,0)$ and $(20,0)$ can be approximated with almost the same straight lines as indicated by the dotted lines. However $X$ is non-exponentially ergodic so that the convergence cannot be approximated with the same straight line for the initial conditions.
  • Figure 2: Heat maps of trajectories of the associated Markov chains for \ref{['eq:main reaction network example']} (left) and another reaction network $B\rightleftharpoons \emptyset \rightleftharpoons A+B$ (right), respectively within a time interval $[0,100]$. The exponential ergodicity of the second model can be shown in the way provided in anderson2023new. The trajectory of the non-exponentially ergodic model \ref{['eq:main reaction network example']} is trapped on the path $((n,0),(n+1,1),(n,0))$ while the second model moves towards the origin quickly.
  • Figure 3: The red sold lines and the blue dotted lines are the graphs of $s=f(\ell)$ and $s=f(\ell+(\bar{y})_{i_1})$. The complexes $y\in \mathcal{C}\setminus\{\bar{y}_1, \dots, \bar{y}_M\}$ are in the shaded area.

Theorems & Definitions (67)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Definition 2.4
  • Example 2.1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • ...and 57 more