Extensions of diffusion processes on intervals and Feller's boundary   conditions
  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. 
Kouji Yano
math.PR