Integral formulation of the Kimura equation for random genetic drift

Abstract The Kimura equation is a fundamental one-dimensional parabolic equation that arises as the continuum limit of the Wright-Fisher model, describing random genetic drift. The associated parabolic operator is degenerate at the boundary, causing singular behavior in solutions and making it difficult to formulate customary boundary conditions. In this paper, we introduce an integral formulation that transforms the Kimura equation into a Kolmogorov backward equation. This reformulation yields improved analytical properties, including convexity of solutions and a natural Dirichlet boundary condition. We establish the existence, uniqueness, and large-time asymptotics of viscosity solutions to the integral formulation. Under additional regularity assumptions, we prove the uniqueness of solutions without imposing any boundary conditions. Connections with the original Kimura equation and related approaches are also discussed.