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This chapter is devoted to a construction of a probabilistic model of balayage spaces. Having collected some basic material from probability theory (section 1) we define Markov processes (section 2) which are the stochastic processes corresponding to sub-Markov semigroups (section 3). To overcome measurability problems in later sections it is necessary to modify processes by completion of various a-algebras (section 4). Our applications in chapter VI will require a Markov property where the constant times are replaced by certain random times (e.g. hitting times). Therefore we introduce stopping times (section 5) and study the strong Markov property (section 6). It turns out that Markov processes associated with balayage spaces are equivalent to Hunt processes (section 7) having all the regularity properties needed later. A global presentation of the equivalence of our four views of potential theory (balayage space, family of harmonic kernels, sub- Markov semigroup, Hunt process) finishes this chapter.
KeywordsMarkov Process Transition Function Markov Property Semi Group Uniform Motion
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