Existence of bounded invariant probability densities for Markov chains (Q1922140)

The existence of invariant probability density is usually approached by finding an invariant probability measure and then showing that it has a density. Here the problem is approached directly and the basic idea is to pose it as a ``linear system'' problem and then to use the generalized Farkas theorem of \textit{B. D. Craven} and \textit{J. J. Koliha} [SIAM J. Math. Anal. 8, 983-997 (1977; Zbl 0408.52006)] in order to obtain necessary and sufficient conditions for the existence of solutions to the linear system. The resulting existence criterion turns out to be similar to the Lyapunov criterion for stochastic ``stability''.