In this paper we consider a one-dimensional diffusion equation on the interval [0, 1] satisfying non-Feller boundary conditions. As a consequence, the initial value Cauchy problem fails to preserve nonnegativity or boundedness. Nonetheless, probability theory plays an interesting role in our analysis and understanding of solutions to this equation.
"A peculiar two point boundary value problem." Ann. Probab. 35 (5) 1623 - 1641, September 2007. https://doi.org/10.1214/009117906000000818