Extensions of diffusion processes on intervals and Feller's boundary conditions

TL;DR

This paper characterizes how minimal diffusion processes on an interval can be extended to include boundary points, using excursion measures and Feller's boundary conditions to describe the generators of these extensions.

Contribution

It provides a complete characterization of all possible extensions of minimal diffusions on an interval via boundary measures and Feller's boundary conditions.

Findings

Extensions are characterized by boundary excursion measures.

Generators are described by Feller's boundary conditions.

The framework applies to standard diffusion processes on intervals.

Abstract

For a minimal diffusion process on $ (a,b) $, any possible extension of it to a standard process on $ [a,b] $ is characterized by the characteristic measures of excursions away from the boundary points $ a $ and $ b $. The generator of the extension is proved to be characterized by Feller's boundary condition.