Brownian motion in a wedge with jump boundary conditions

We consider a stochastic process in a two-dimensional infinite wedge that behaves like a Brownian motion in the interior but instantaneously jumps to the diagonal upon hitting the boundary according to a function dependent on its location.  These functions have a discontinuity at the corner of the wedge. We construct the harmonic functions for a specific class of these processes and use these harmonic functions to get information about the behavior of the process relative to the corner of the wedge.