A General Correspondence between Dirichlet Forms and Right Processes

The theory of Dirichlet forms as originated by Beurling-Deny and developed particularly by Fukushima and Silverstein, see e.g. [Fu3, Si], is a natural functional analytic extension of classical (and axiomatic) potential theory. Although some parts of it have abstract measure theoretic versions, see e.g. [BoH] and [ABrR], the basic general construction of a Hunt process properly associated with the form, obtained by Fukushima [Fu2] and Silverstein [Si] (see also [Fu3]), requires the form to be defined on a locally compact separable space with a Radon measure m and the form to be regular (in the sense of the continuous functions of compact support being dense in the domain of the form, both in the supremum norm and in the natural norm given by the form and the L2(m)-space). This setting excludes infinite dimensional situations. In this letter we announce that there exists an extension of FukushimaSilverstein's construction of the associated process to the case where the space is only supposed to be metrizable and the form is not required to be regular. We shall only summarize here results and techniques, for details we refer to [AMI, AM2]. Before we start describing our results let us mention that some work on associating strong Markov processes to nonregular Dirichlet forms had been done before, by finding a suitable representation of the given nonregular form as a regular Dirichlet form on a suitable compactification of the original space. In an abstract general setting this was done by Fukushima in [Ful]. The case of local Dirichlet forms in infinite dimensional spaces, leading to associated diffusion processes, was studied originally by Albeverio and Hoegh-Krohn in a rigged Hubert space setting [AH1-AH3], under a quasi-invariance and smoothness assumption on m . This work was extended by Kusuoka [Ku] who worked in a Banach space setting. Albeverio and Röckner [ARO 1-4] found a natural setting in a Souslin space, dropping the quasi-invariance assumption.