We consider a probabilistic method for constructing certain transition functions which have a given conservative matrix as their initial derivative matrix. The technique originated in a conjecture of Reuter (see [9]). Kingman (see [5]) has considered similar problems but no proof of the original conjecture has appeared. The methods used are intended to confirm a remark made by Chung (see [1] page 158) to the effect that it should be possible to view sticky (regular) boundary points as a suitable limiting case of non-sticky (non-regular) boundary points. The probabilistic construction in Reuter's conjecture can be used to motivate some general analytical constructions of transition functions. These constructions and the connection with modern boundary theory are discussed in the last two sections.