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

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Heretical

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 ℝ.. For a ball .(ξ, δ) in ℝ.

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名义上

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 ℝ., 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 ℝ.. 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