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The Master equation describes the time evolution of the probabilities of asystem with a discrete state space. This time evolution approaches for longtimes a stationary state that will in general depend on the initial probabilitydistribution. Conditions under which the stationary state is unique are usuallygiven as remarks appended to more comprehensive theories in the mathematicalliterature. We provide a direct and complete derivation of a necessary andsufficient criterion for when this steady state is unique. We translate thisproblem into the language of graph theory and show that there is a one-to-onecorrespondence between minimal absorbing sets within the state-transitionnetwork and linearly independent stationary states of the Master equation.

Obtaining the long-term behavior of Master equations with finite state space from the structure of the associated state transition network