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A partition method for bounding continuous-time Markov chain models of general reaction network

Guillaume Ballif, Laurent Pfeiffer, Jakob Ruess

TL;DR

The paper addresses the challenge of analyzing multi-dimensional CTMCs arising from stochastic reaction networks by introducing a partition-based bounding framework. It constructs upper- and lower-bounding one-dimensional birth-death processes via coupling and transport theory, enabling tractable analysis of long-term behavior and truncation errors. The authors provide explicit procedures to obtain optimal bounding generators, prove key coupling results, and illustrate the approach on a toy chemical network, emphasizing how partition choice impacts bound quality. This framework offers practical tools for verifying stationary distributions and guiding finite-state projections in complex reaction networks.

Abstract

In this work, we present a general method to establish properties of multi-dimensional continuous-time Markov chains representing stochastic reaction networks. This method consists of grouping states together (via a partition of the state space), then constructing two one-dimensional birth and death processes that lower and upper bound the initial process under simple assumptions on the infinitesimal generators of the processes. The construction of these bounding processes is based on coupling arguments and transport theory. The bounding processes are easy to analyse analytically and numerically and allow us to derive properties on the initial continuous-time Markov chain. We focus on two important properties: the behavior of the process at infinity through the existence of a stationary distribution and the error in truncating the state space to numerically solve the master equation describing the time evolution of the probability distribution of the process. We derive explicit formulas for constructing the optimal bounding processes for a given partition, making the method easy to use in practice. We finally discuss the importance of the choice of the partition to obtain relevant results and illustrate the method on an example chemical reaction network.

A partition method for bounding continuous-time Markov chain models of general reaction network

TL;DR

The paper addresses the challenge of analyzing multi-dimensional CTMCs arising from stochastic reaction networks by introducing a partition-based bounding framework. It constructs upper- and lower-bounding one-dimensional birth-death processes via coupling and transport theory, enabling tractable analysis of long-term behavior and truncation errors. The authors provide explicit procedures to obtain optimal bounding generators, prove key coupling results, and illustrate the approach on a toy chemical network, emphasizing how partition choice impacts bound quality. This framework offers practical tools for verifying stationary distributions and guiding finite-state projections in complex reaction networks.

Abstract

In this work, we present a general method to establish properties of multi-dimensional continuous-time Markov chains representing stochastic reaction networks. This method consists of grouping states together (via a partition of the state space), then constructing two one-dimensional birth and death processes that lower and upper bound the initial process under simple assumptions on the infinitesimal generators of the processes. The construction of these bounding processes is based on coupling arguments and transport theory. The bounding processes are easy to analyse analytically and numerically and allow us to derive properties on the initial continuous-time Markov chain. We focus on two important properties: the behavior of the process at infinity through the existence of a stationary distribution and the error in truncating the state space to numerically solve the master equation describing the time evolution of the probability distribution of the process. We derive explicit formulas for constructing the optimal bounding processes for a given partition, making the method easy to use in practice. We finally discuss the importance of the choice of the partition to obtain relevant results and illustrate the method on an example chemical reaction network.

Paper Structure

This paper contains 25 sections, 15 theorems, 109 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Construction of an upper-bounding process
  4. Main result
  5. A preliminary result
  6. Proof of the main result
  7. Construction of an upper-bounding transition-rate matrix
  8. Uniform bounds of solutions of the master equation
  9. Construction of a lower-bounding process
  10. Main result
  11. Construction of a lower-bounding transition rate-matrix
  12. Applications
  13. Asymptotic behavior
  14. Theoretical results
  15. Application of the method to the toy model
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let $a$ and $b$ be in $\ell^1(\mathbb{N},\mathbb{R}_+){}$. Assume that $a_{0:k} \geq b_{0:k}$, for all $k \in \mathbb{N}$. Assume further that $a_{0:\infty}= b_{0:\infty}$. Then there exists $\pi \in \ell^1(\mathbb{N}^2,\mathbb{R}_+){}$ satisfying the following properties:

Figures (2)

  • Figure 1: Three trajectories of the coupling process for the two sets of classes $S^+$ (left panel) and $\tilde{S}^{+}$ (right panel). On each panel, we plot the trajectory of the upper-bounding process (second component of the coupling obtained with \ref{['thm:main_upper-bound']}) in solid lines and the class of the original process in dashed lines with respect to time.
  • Figure 2: Upper-bound $E^T(N)$ of the truncation error as a function of the final time $T$ and truncation parameter $N$ for the well-selected classes $S^+$. The $y$-axis represents the parameter $N$ chosen for truncation. The orange line (respectively the red line) represents the smallest $N$ obtained with \ref{['eq:esti_truncation']} that leads to an error smaller than $0.1$ (resp. $0.01$). Note that since $E^T(N)$ is an upper-bound of a probability, it could take values greater than $1$ even if we know that the estimated probability is lower than $1$. Knowing that values of $E^T(N)$ greater than 1 are set to 1 (yellow part of the graph).

Theorems & Definitions (36)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 26 more