[Paper Review] Criteria for dynamics of one-dimensional continuous time Markov chains with applications

Chuang Xu, Mads Christian Hansen+1 more|arXiv (Cornell University)|Jun 17, 2020

Gene Regulatory Network Analysis4 citations

TL;DR

This paper establishes computable threshold criteria for key dynamical properties—such as explosivity, recurrence, and ergodicity—in continuous-time Markov chains with unbounded jump sizes on non-negative integers and polynomial transition rates. It provides explicit conditions for exponential ergodicity of stationary and quasi-stationary distributions, with applications to stochastic reaction networks and gene expression models beyond birth-death processes.

ABSTRACT

This paper studies the dynamics of continuous time Markov chains with possibly unbounded jump sizes on the non-negative integers with polynomial transition rate functions. Such stochastic processes are abundant in applications, in particular in biology. We provide threshold criteria in terms of easily computable parameters for various dynamical properties such as explosivity, recurrence, transience, certain absorption, positive/null recurrence, implosivity, and existence and non-existence of moments of hitting times. In particular, simple sufficient conditions for exponential ergodicity of stationary distributions and quasi-stationary distributions are obtained, and the few gap cases are identified and well-illustrated by examples. Subtle differences in conditions for different dynamical properties are revealed in terms of examples. Finally, we apply our results to stochastic reaction networks, an extended class of branching processes, a general bursty single-cell stochastic gene expression model, and population processes, none of which are birth-death processes.

Motivation & Objective

• To develop easily computable criteria for fundamental dynamical behaviors in continuous-time Markov chains with polynomial transition rates and unbounded jumps.
• To address the lack of general criteria for properties like explosivity, recurrence, and quasi-stationarity in non-birth-death processes.
• To identify conditions for exponential ergodicity of stationary and quasi-stationary distributions in complex stochastic systems.
• To bridge theoretical gaps in understanding subtle differences between dynamical properties through illustrative examples.
• To apply the framework to real-world models in systems biology, including gene expression and population processes.

Proposed method

• Derives threshold conditions based on the degree and coefficients of polynomial transition rate functions to classify dynamical behaviors.
• Uses Lyapunov-type functionals and moment-generating function techniques to analyze hitting time moments and ergodicity.
• Applies spectral and pathwise analysis to distinguish between positive and null recurrence, and implosivity.
• Introduces a systematic classification of 'gap cases' where standard criteria fail, using counterexamples to clarify boundaries.
• Employs comparison and coupling arguments to extend results to non-linear, non-birth-death processes.
• Validates theoretical findings via explicit examples from stochastic reaction networks and bursty gene expression models.

Experimental results

Research questions

• RQ1 What conditions on polynomial transition rates ensure exponential ergodicity of stationary and quasi-stationary distributions in continuous-time Markov chains with unbounded jumps?
• RQ2 How can one distinguish between explosive, transient, and recurrent behavior in such chains using computable parameters?
• RQ3 What are the precise thresholds for the existence and non-existence of moments of hitting times in these processes?
• RQ4 In what ways do the conditions for different dynamical properties—such as recurrence and implosivity—differ, and how are these differences illustrated?
• RQ5 How do the derived criteria apply to extended branching processes and bursty single-cell gene expression models?

Key findings

• Simple sufficient conditions for exponential ergodicity of stationary and quasi-stationary distributions are derived based on the degree and growth rate of transition rate polynomials.
• Threshold criteria for explosivity, recurrence, and transience are formulated in terms of the leading-order coefficients and degrees of the polynomial rates.
• Conditions for the existence of moments of hitting times are characterized, with explicit dependence on the polynomial degree and sign of leading coefficients.
• The paper identifies and illustrates 'gap cases' where standard criteria fail, providing counterexamples that clarify subtle distinctions between dynamical behaviors.
• The framework successfully extends to non-birth-death models, including stochastic reaction networks and bursty gene expression processes, where prior methods are inapplicable.
• Positive/null recurrence and implosivity are shown to depend on distinct parameter regimes, with clear analytical separation in the derived conditions.