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using Brownian motion, started a new era in the very fruitful interplay between probability theory and analysis. Here D is a bounded connected open subset of R. Since then, in place of the Laplacian Δ, more general second order elliptic differential operators have been studied in connection with the probabilistic approach. In another direction, other boundary conditions (Nuemann and mixed boundary) have been investigated using probabilistic methods. For example, in [19], Hsu studied solutions to the classical Neumann boundary value problems on a bounded domain D in R with C boundary: { Δu = 0 in D, ∂u ∂n = φ on ∂D,

A probabilistic approach to mixed boundary value problems for elliptic operators with singular coefficients