Titlebook: Classical Potential Theory and Its Probabilistic Counterpart; Joseph L. Doob Book 2001 Springer-Verlag Berlin Heidelberg 2001 31XX.Brownia

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Introduction to the Mathematical Background of Classical Potential TheoryIn this chapter some of the mathematical ideas of classical potential theory are introduced, under simplifying assumptions. The basic space is Euclidean . space ?.. For a ball .(ξ, δ) in ?.

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The Fundamental Convergence Theorem and the Reduction Operation.. Let Γ: {u., α ∈ I} be a family of superharmonic functions defined on an open subset of ?., locally uniformly bounded below, and define the lower envelope u by u(ξ) = ..u.(ξ). Then .u ≤ u, ..

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The Martin BoundaryLet . be an open subset of ?.. If . is a ball, its Euclidean boundary is so well adapted to it from a potential theoretic point of view that the following statements are true.

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978-3-540-41206-9Springer-Verlag Berlin Heidelberg 2001

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Basic Properties of Harmonic, Subharmonic, and Superharmonic Functions = δ.. To simplify the notation take ξ. = .. Then .., as defined by.with the understanding that ..(ξ, ξ)= +∞, satisfies items (ix′)–(ivx′) of Section 1.8, so that harmonic measure for . is given by.where .. here refers to surface area on ?. and