A peculiar two point boundary value problem

TL;DR

This paper investigates a one-dimensional diffusion equation with non-Feller boundary conditions, revealing unique challenges in solution properties and highlighting the role of probability theory in understanding these solutions.

Contribution

It introduces analysis of diffusion equations with non-Feller boundaries, demonstrating how probability theory aids in understanding their solutions despite non-standard boundary behavior.

Findings

Initial value problem fails to preserve nonnegativity

Solutions exhibit non-standard boundary behavior

Probability theory provides insights into solution structure

Abstract

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.