Normal derivative for bounded domains with general boundary

Abstract:

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