Nonstandard representation of the Dirichlet form and application to the comparison theorem

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 compare the Dirichlet forms of two general Markov processes by applying the transfer of the well-known comparison theorem for finite Markov processes.