Nonstandard Representation of the Dirichlet Form

TL;DR

This paper introduces a nonstandard representation theorem for the Dirichlet form, approximating it by hyperfinite sums to facilitate transferring results from finite to general state spaces.

Contribution

It provides a novel nonstandard representation of the Dirichlet form, enabling direct translation of results across different state space types.

Findings

Approximation of Dirichlet form by hyperfinite sums

Generalization of a comparison theorem for Markov chains

Relation to previous generalization approaches

Abstract

The Dirichlet form is a generalization of the Laplacian, heavily used in the study of many diffusion-like processes. In this paper we present a nonstandard representation theorem for the Dirichlet form, showing that the usual Dirichlet form can be well-approximated by a hyperfinite sum. One of the main motivations for such a result is to provide a tool for directly translating results about Dirichlet forms on finite or countable state spaces to results on more general state spaces, without having to translate the details of the proofs. As an application, we prove a generalization of a well-known comparison theorem for Markov chains on finite state spaces, and also relate our results to previous generalization attempts.