Quasi-Regular Dirichlet Forms and Applications

Since the celebrated result of Fukushima on the connection between regular Dirichlet forms and Hunt processes in 1971, the theory of Dirichlet forms has been rapidly developed and has brought a wide range of applications in various related areas of mathematics and physics (see e.g. the three new books [BH 91], [MR 92], [FOT 94] and references therein). In this survey paper I shall mainly discuss the development of quasi-regular Dirichlet forms and their applications. Roughly speaking, quasi-regular Dirichlet forms on general state spaces are those Dirichlet forms that are associated with right continuous strong Markov processes. Recently an analytic characterization of quasi-regular Dirichlet forms has been found [AM 91c], [AM92], [AMR92a,b,c] [MR 92], [AMR 93a,b], which has completed the solution of a long-standing open problem of this area. The characterization condition has been proved to be checkable in quite general situations [RS 93], and the framework of quasi-regular Dirichlet forms has been shown to be especially useful in dealing with very singular or infinite-dimensional problems. Applications are e.g. in the study of singular Schrödinger operators [AM 91a,b], loop or path spaces over Riemannian manifolds [ALR 93], [DR 92], infinitedimensional stochastic differential equations [AR 91], Quantum field theory [AR 90], large deviation theory [Mu 93], non-symmetric Ornstein-Uhlenbeck processes [Sch 93], measure valued processes [ORS 93], and Markov uniqueness for infinite dimensional operators [ARZ 93]. It was also proved that a Dirichlet form is quasi-regular if and only if it is quasi-homeomorphic to a regular Dirichlet form on a locally compact separable metric space [AMR 92c], [MR 92], [CMR 93]. Hence most of the results known for regular Dirichlet forms can be transferred to the quasi-regular case. This transfer method has been used e.g. in the study of absolute continuity of symmetric diffusions [Fi 94], transformation of local Dirichlet forms by supermartingale multiplicative functionals [Ta 94], and measures charging no exceptional sets and corresponding additive functionals [Kuw 94]. Concerning the history, it should be mentioned that the analytic part of the theory of Dirichlet forms goes back to the pioneering papers of Beurling and Deny