Wright-Fisher Diffusion in One Dimension

on the interval [0, 1], where a(x) > 0 on the interior and vanishes simply at the endpoints, and b(x)∂x is a vector field which is inward-pointing at both ends. We consider various aspects of this problem, motivated by their applications in biology, including a comparison of the natural boundary conditions from the probabilistic and analytic points of view, a sharp regularity theory for the “zero flux” boundary conditions, as well as a derivation of the precise asymptotics of solutions of this equation, both as t → 0,∞ and as x → 0, 1. This is a precursor to our more complicated analysis of these same questions for Wright-Fisher type problems in higher dimensions.