Normal derivative for bounded domains with general boundary

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Abstract
Let D D be a general bounded domain in the Euclidean space R n {R^n} . A Brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for f f with f f , ∇ f \nabla f and Δ f \Delta f all in L 2 {L^2} on D D . The corresponding Neumann condition (normal derivative = 0 = 0 ) is an honest boundary condition for the L 2 {L^2} generator of reflected Brownian notion on D D . A conditioning argument shows that for D D and f f sufficiently smooth this general definition of the normal derivative agrees with the usual one.

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