Classical Potential Theory
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In this chapter we shall see that classical potential theory and Brownian motion are mathematically equivalent. After a short description of basic properties of harmonic and hyperharmonic functions in section 1, we shall introduce the Brownian semigroup in section 2 and prove some important properties. In section 3 the identity of the set of excessive functions with respect to Brownian semigroup and the set of positive hyperharmonic functions on ℝn will be shown, which means that in a technical sense these two theories are identical. The importance of this result will become clear later on, especially in chapter VI when the probabilistic model of Brownian semigroup, namely Brownian motion, will be available.
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