The Excessive Functions of a Continuous Time Markov Chain

Abstract

Boundary conditions in the form of equalities have been used by Feller, Dynkin, and others to characterize the range of the resolvent operator for certain continuous time Markov chains. Along similar lines Denzel, Kemeny, and Snell were able to establish a characterization and Riesz decomposition for the excessive functions of a more restricted class of Markov chains through the use of boundary conditions in the form of inequalities. The present paper sets out to clarify and build upon this work by reanalyzing these excessive functions in a more general setting. Here the boundary theory developed by Chung is brought to bear on the problem so that the results can be derived in canonical form for probabilistic interpretation.