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Leo Bronstein1, Heinz Koeppl1

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This study introduces a new framework for analyzing complex Markov jump processes by focusing on specific components. It simplifies models for applications in fields like chemical kinetics and network dynamics.

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Area of Science:

  • Computational science
  • Statistical physics
  • Systems biology

Background:

  • Markov jump processes are widely used across scientific disciplines.
  • Models often feature product-form state spaces where only one component is of primary interest.
  • Analyzing fully coupled processes presents significant computational challenges.

Purpose of the Study:

  • To extend the marginal process framework to fully coupled Markov jump processes.
  • To develop a principled model reduction framework for complex systems.
  • To provide a simplified analytical and numerical approach for analyzing specific components of interest.

Main Methods:

  • Extension of the marginal process framework to fully coupled systems.
  • Application of entropic matching for finite-dimensional approximation of filtering equations.
  • Development of a model reduction framework based on projection operations on the master equation.
  • Utilizing product Poisson distributions for mass-action systems.

Main Results:

  • A principled model reduction framework is established for fully coupled Markov jump processes.
  • The framework simplifies the analysis of systems like the totally asymmetric exclusion process.
  • A particularly simple instantiation is derived for mass-action systems using product Poisson distributions.
  • The resulting approximate marginal process is investigated both analytically and numerically.

Conclusions:

  • The extended marginal process framework offers an effective method for analyzing complex, fully coupled Markov jump processes.
  • This approach facilitates model reduction and simplifies the study of specific system components.
  • The findings have broad implications for stochastic reaction networks and other applications in science and engineering.