Regularization

Abstract

In this chapter we present a general “local compactification” method that enables us to associate to a quasi-regular Dirichlet form on an arbitrary topological space a regular Dirichlet form on a locally compact separable metric space. This is done in such a way that we can transfer results obtained in the latter “classical” framework to our more general situation. The “local compactification” is constructed in Section 1 and it is shown that “without loss of generality” one can also restrict to Hunt processes. In Section 2 the “transfer method” is described in general and subsequently illustrated by several examples. In all of this chapter E is supposed to be a Hausdorff topological space with B(E) = σ(C(E)). We fix a σ-finite positive measure m on (E, B(E)) and a quasi-regular Dirichlet form (ε, D(ε)) on L2(E; m).