Dirichlet form analysis of the Jacobi process

TL;DR

This paper employs Dirichlet form techniques to analyze the Jacobi process, also known as Wright-Fisher diffusion, characterizing its boundary behavior, functional inequalities, and connection to the Jacobi SDE.

Contribution

It introduces a Dirichlet form framework for the Jacobi process, providing new insights into its boundary behavior, Sobolev space structure, and explicit calculations using hypergeometric functions.

Findings

Characterization of boundary behavior of functions in the Dirichlet space

Derivation of Sobolev embeddings and Hardy-type inequalities

Establishment of the process as a solution to the Jacobi SDE

Abstract

We construct and analyze the Jacobi process - in mathematical biology referred to as Wright-Fisher diffusion - using a Dirichlet form. The corresponding Dirichlet space takes the form of a Sobolev space with different weights for the function itself and its derivative. Depending on the parameters we characterize the boundary behavior of the functions in the Dirichlet space, show density results, derive Sobolev embeddings and verify functional inequalities of Hardy type. Since the generator is a hypergeometric differential operator, many of the proofs can be carried out by explicit calculations involving hypergeometric functions. We deduce corresponding properties for the associated semigroup and Markov process and show that the latter is up to minor technical modifications a solution to the Jacobi SDE.