The aim of this talk is to present and explain to non-specialists in this subject an extension theorem for regular solutions of systems of PDE's with constant coefficients. The theorem was obtained by Kaneko [2,3,4] in the real analytic case and was then generalised by the author [1] to quasi-analytic solutions.

Positive eigenfunctions for elliptic operators with oblique derivative boundary conditions arise as the stationary densities of reflected Brownian motions, briefly RBM's. An RBM is a diffusion process that behaves like Brownian motion with a constant drift velocity ~ in the interior of a d-dimensional domain and is instantaneously reflected at the boundary in a direction specified by a non-tangential vector field v on the boundary. When the domain is bounded and smooth and the vector field is smooth, it is shown that the stationary density is of a simple exponential form for all ~ if and only if the vector field v satisfies a certain skew-symmetry condition. A formal analogue of this result for polyhedral domains, where v is constant on each face, will also be given. Consequences for the existence and uniqueness of an RBM with such non-smooth data will be drawn from this.

Details will appear elsewhere.