[For the entire collection see Zbl 0626.00026.]
This is a review article on the theory of (r,p)-capacities introduced by the author for Dirichlet spaces and recently for the Ornstein-Uhlenbeck semigroup in Wiener spaces. Slim sets are defined as subsets of the state space X whose (r,p)-capacities are null for any \(r>0\) and \(p>0\). Many interesting results concerning P-a.e. versions in the classical case of the Brownian motion could be refined for ‘slim set’-versions in the infinite-dimensional case.
The new result here is an answer of the author to a quation raised by K. Itô concerning the existence of a continuous embedding of a dense open subset Y into X such that \[ C_ X(A)=C_ Y(A^ Y), \forall A\subset X, \] where \(C_ X,\) \(C_ Y\) denote respectively the (1,2)-capacities in X and Y. Applications to the study of diffusion processes in a separable Banach space are given.