Dirichlet Forms on Infinite-Dimensional ‘Manifold-Like’ State Spaces: A Survey of Recent Results and Some Prospects for the Future

Abstract

We give a (to some extent pedagogical) survey on recent results about Dirichlet forms on infinite-dimensional ‘manifold-like’ state spaces including path and loop spaces as well as spaces of measures. The latter are associated with interacting Fleming-Viot processes resp. infinite particle systems. Also some new results, further developing the Dirichlet form approach to infinite particle systems, are enclosed. Finally, a brief summary of other research activities in the theory of Dirichlet forms is given and some prospects for the future are indicated.