  It has recently been shown that complete Bernstein functions of the Laplace
operator map the Dirichlet boundary condition of a related elliptic PDE to the
Neumann boundary condition. The importance of this mapping consists in being
able to convert problems involving non-local operators, like fractional
Laplacians, into ones that only involve differential operators. We generalise
this result to diffusion operators associated with stochastic differential
equations, using a method which is entirely based on stochastic analysis.
