Eigenfunctions and the Dirichlet Problem for the Classical Kimura Diffusion Operator

Abstract

We study the classical Kimura diffusion operator defined on the $n$-simplex, $\operatorname{L^{Kim}}=\sum_{1\leq i,j\leq n+1}x_i(\delta_{ij}-x_j)\partial_{x_i}\partial_{x_j},$ which has important applications in population genetics. Because it is a degenerate elliptic operator acting on a singular space, special tools are required to analyze and construct solutions to elliptic and parabolic problems defined by this operator. The natural boundary value problems are the “regular” problem and the Dirichlet problem. For the regular problem, one can only specify the regularity of the solution at the boundary. For the Dirichlet problem, one can specify the boundary values, but the solution is then not smooth at the boundary. In this paper we give a computationally effective recursive method to construct the eigenfunctions of the regular operator in any dimension, and a recursive method to use them to solve the inhomogeneous equation. As noted, the Dirichlet problem does not have a regular solution. We give an e...