  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.
