Minimal excessive measures and functions

Abstract:

Let H be a class of measures or functions. An element h of H is minimal if the relation $h = {h_1} + {h_2}$, ${h_1}$, ${h_2} \in H$ implies that ${h_1}$, ${h_2}$ are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup ${T_t}$ in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure $\gamma$ on E exists such that ${T_t}f = 0$ if $f = 0$ $\gamma$-a.e. In the Appendix, we prove that each excessive element can be decomposed into minimal elements and that such a decomposition is unique.