Brownian Motion and the PWB Method

Abstract

Let D be an open nonempty subset of ℝ^N, coupled with a boundary ∂D provided by a metric compactification. To avoid trivial complications, we shall assume that D is connected; if D is disconnected, the results are applicable to each open connected component of D. Let h be a strictly positive harmonic function on D. The PWB method of attacking the first boundary value (Dirichlet) problem for h-harmonic functions on D was detailed in Chapter VIII of Part 1. Recall that the σ algebra of μ ^h _D measurable boundary subsets is the σ algebra of boundary subsets A for which the boundary indicator function l _A is h-resolutive and that μ ^h _D (·, A) = H ^h_1A . The class of Borel boundary subsets for which l _A is h-resolutive is a σ algebra, and for each point ξ of D the restriction of μ ^h _D (ξ, ·) to this σ algebra, on completion, is the measure μ ^h _D (ξ, ·) on the σ algebra of μ ^h _D measurable sets. The class of μ ^h _D measurable boundary functions f which are μ ^h _D (ξ, ·) integrable does not depend on ξ and is the class of h-resolutive boundary functions, and H ^h _f = μ ^h _D (· f). This class of boundary functions will also be described as the class of μ ^h _D integrable boundary functions.