Wright–Fisher Diffusion in One Dimension

We analyze the diffusion processes associated to equations of Wright–Fisher type in one spatial dimension. These are associated to the degenerate heat equation $\partial_{t}u=a(x)\partial_{x}^{2}u+b(x)\partial_{x}u$ on the interval $[0,1]$, where $a(x)>0$ on the interior and vanishes simply at the end points and $b(x)\partial_{x}$ 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 sharp regularity theory for the “zero flux” boundary conditions, as well as an analysis of the infinitesimal generators of the $\mathcal{C}^{m}$-semigroups and their adjoints. Using these results we obtain precise asymptotics of solutions of this equation, both as $t\to0,\infty$ and as $x\to0,1$.