Global Asymptotic Stability Of Solutions Of Nonautonomous Master Equations

We provide various conditions on the transition rate matrix of the master equation of a finite-state, continuous-time jump process under which every prob-ability distribution solution is globally asymptotically stable in the set of such solutions, thereby extending van Kampen’s theorem for constant transition rate matrices. The set of continuous, uniformly bounded transition rate matrices can be classified into two types based on the results. Certain natural assumptions on the transition rate matrix do not in general ensure that the probability dis-tribution solutions of the associated master equation are globally asymptotically stable.