Equivalence of Continuous-Time Markov Chains and Linear Dynamical Systems

2026-06-25

AI Summary

This study establishes a rigorous bidirectional equivalence between continuous-time Markov chains and linear dynamical systems. By employing a state-space embedding approach that integrates continuous-time Markov processes with linear systems theory, the authors demonstrate that the dynamics of any d-state continuous-time Markov chain can be exactly represented by a linear dynamical system of dimension at most (d−1), and that this correspondence is invertible in both directions. This work provides the first formal characterization of such an equivalence within a continuous-time framework, systematically extending previously known results from the discrete-time setting. The findings lay a theoretical foundation for cross-domain modeling and analysis, enabling the transfer of tools and insights between stochastic processes and deterministic linear systems.

Abstract

The purpose of this short note is to record that an analogue of the following result, which is known for discrete-time linear dynamical systems, also holds in the continuous-time setting. The dynamics of a $d$-state Markov chain is governed by that of a linear dynamical system of dimension at most $d-1$; conversely, a linear dynamical system of dimension $d-1$ can be "embedded" into a Markov chain with $d$ states.

Problem

• Markov chains
• linear dynamical systems
• continuous-time
• equivalence
• embedding

Innovation

• Continuous-Time Markov Chains
• Linear Dynamical Systems
• Equivalence
• Embedding
• Dimension Reduction