The evolution of the state occupancy probabilities of a CTMC can be described by a linear system of differential equations of the form $\dot{\mathbf{x}} = A.\mathbf{x}$.
The rate matrix A usually has a 0 zero eigenvalue.
Therefore, A violates the Routh-Hurwitz Criterion (RHC) for asymptotic stability. RHC mandates all the eigenvalues of a matrix to have strictly negative real parts.
Does this mean that CTMCs are not asymptotically stable? To be precise, the equilibria of CTMCs is not asymptotically stable?
Thank you.
