Stochastic analysis on configuration spaces: basic ideas and recent results

Abstract

The purpose of this paper is to provide a both comprehensive and summarizing account on recent results about analysis and geometry on configuration spaces $\Gamma_X$ over Riemannian manifolds $X$. Particular emphasis is given to a complete description of the so--called ``lifting--procedure'', Markov resp. strong resp. $L^1$--uniqueness results, the non--conservative case, the interpretation of the constructed diffusions as solutions of the respective classical ``heuristic'' stochastic differential equations, and a self--contained presentation of a general closability result for the corresponding pre--Dirichlet forms. The latter is presented in the general case of arbitrary (not necessarily pair) potentials describing the singular interactions. A support property for the diffusions, the intrinsic metric, and a Rademacher theorem on $\Gamma_X$, recently proved, are also discussed.