Eigenfunctions and the Dirichlet problem for the Classical Kimura Diffusion Operator

TL;DR

This paper investigates the eigenfunctions and Dirichlet problem for the Kimura diffusion operator on the simplex, providing explicit solutions and basis constructions that facilitate numerical applications and boundary singularity analysis.

Contribution

It introduces novel basis constructions for eigenpolynomials and explicit solutions to the Dirichlet problem tailored for the Kimura operator on the simplex.

Findings

Explicit basis of eigenpolynomials constructed.

Solution to the Dirichlet problem is explicit and boundary-aware.

Provides detailed description of boundary singularities.

Abstract

We study the classical Kimura diffusion operator defined on the n-simplex, $$L^{Kim}=\sum_{1\leq i,j\leq n+1}x_ix_j\partial_{x_i}\partial_{x_j}$$ We give novel constructions for the basis of eigenpolynomials, and the solution to the inhomogeneous Dirichlet problem, which are well adapted to numerical applications. Our solution of the Dirichlet problem is quite explicit and provides a precise description of the singularities that arise along the boundary.