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\, = \,{h_1}\, + \,{h_2} , h 1 {h_1} , h 2 ∈ H {h_2} \in H implies that h 1 {h_1} , h 2 {h_2} are proportional to h. We give a limit procedure for computing minimal excessive measures for an arbitrary Markov semigroup T t {T_t} in a standard Borel space E. Analogous results for excessive functions are obtained assuming that an excessive measure γ \gamma on E exists such that T t f = 0 {T_t}f\, = 0 if f = 0 f\, = \,0 γ \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.