Global Asymptotic Stability of Solutions of Nonautonomous Master Equations

We provide various conditions on the transition rate matrix of the (possibly nonautonomous) master equation of a finite-state, continuous-time jump process under which every probability distribution solution is globally asymptotically stable in the set of such solutions, thereby extending van Kampen's theorem for constant transition rate matrices. We propose a classification of the set of continuous, uniformly bounded transition rate matrices and show how our results fit into this classification. By constructing counterexamples, we show that certain natural assumptions on the transition rate matrix do not in general ensure that the probability distribution solutions of the associated master equation are globally asymptotically stable.