On Boundary Behaviour of One-Dimensional Diffusions: From Brown to Feller and Beyond

Feller’s greatest discovery in mathematics was sticky (or slowly reflecting) boundary behaviour of one-dimensional diffusion processes (characterised by the appearance of the second derivative at the boundary point). Before him (Dirichlet1, Neumann2, Robin3) it was not known that this was possible. A boundary condition containing the second derivative (in one dimension) should therefore be referred to as Feller boundary condition (in addition to Dirichlet condition containing the function itself, Neumann condition containing its first derivative, Robin condition containing a linear combination of the function and its first derivative) and extensions of this condition to higher dimensions should likewise contain the name of Feller. In this brief review I will address the relevance of this discovery and Feller’s work on boundary classification of one-dimensional diffusion processes within a general context of mathematics and physics. Feller’s motivating aim, often stated explicitly in his papers, was to disclose the ‘most general’ conditions, or to describe the ‘most general’ situations, and this attitude happened to be the key to unlock the mystery of the still unseen boundary behaviour. Although enduring the progress was slow and it took over 10 years (1951-1965) including help and insights of other people (Itô and McKean via Lévy and Volkonskii) to complete it at the level of a sample path. The general picture in one dimension is complete at present. After reading Feller’s papers time and again I wonder whether he ever thought that the ultimate goal of the ‘most general’ was achieved. The line had to be always drawn somewhere but only to be stretched further in a new paper. Likewise my review will fail to draw definite lines and at the end I will briefly indicate how to go beyond the ‘most general’ in hope that the suggested possibility would only please Feller in his never ending quest for the ‘most general’ conditions and situations. I will now return to the beginning of the story leaping forward in big jumps to get to Feller’s time as quickly as possible.