Dirichlet form analysis of the Jacobi process

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 and can be rewritten in a canonical form for strongly local Dirichlet forms in one dimension. Additionally to the statements following from the general theory on these forms, we obtain orthogonal decompositions of the Dirichlet space, derive Sobolev embeddings, verify functional inequalities of Hardy type and analyze the long time behavior of the associated semigroup. We deduce corresponding properties of the Markov process and show that it is up to minor technical modifications a solution to the Jacobi SDE. We also provide uniqueness statements for this SDE, such that properties of general solutions follow.