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We identify the linear space spanned by the real-valued excessive functionsof a Markov process with the set of those functions which are quasimartingaleswhen we compose them with the process. Applications to semi-Dirichlet forms aregiven. We provide a unifying result which clarifies the relations betweenharmonic, co-harmonic, invariant, co-invariant, martingale and co-martingalefunctions, showing that in the conservative case they are all the same.Finally, using the co-excessive functions, we present a two-step approach tothe existence of invariant probability measures.

Invariant, super and quasi-martingale functions of a Markov process