Equivalence of Continuous-Time Markov Chains and Linear Dynamical Systems
This provides a theoretical equivalence for researchers in stochastic processes and dynamical systems, but it is an incremental extension of existing work.
The authors prove that continuous-time Markov chains with d states are equivalent to linear dynamical systems of dimension at most d-1, extending a known discrete-time result to the continuous-time setting.
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.