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The aim of this chapter is an intrinsic characterization of the set of excessive functions of a sub-Markov semigroup. In section 1 we introduce kernels on X. In section 2 we derive basic properties of functions which are supermedian with respect to a kernel on X. On the one hand these functions form one of our standard examples (discrete potential theory), on the other hand their study allows us to obtain fundamental properties of excessive functions of sub-Markov semigroups and resolvents (section 3) which constitute one major motivation for the notion of balayage spaces (section 4). Introducing the cone p of continuous potentials on a balayage space (section 5) it is possible to construct associated potential kernels (section 6), resolvents (section 7), and semigroups (section 8). As a consequence we obtain a correspondence between balayage spaces and sub-Markov semigroups having “many” continuous excessive functions.
KeywordsFunction Cone Semi Group Excessive Function Kernel Versus Potential Cone
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