Local regularity of set-indexed processes.

Abstract: we present shortly the theory of set-indexed processes as it was introduced by Ivanoff and Merzbach. We study the Hölder regularity of such processes. The first key result is a Hölder-continuity Theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections. Hölder-continuity based on the increment definition for set-indexed processes is also considered. Then, the localization of these properties leads to various definitions of Hölder exponents. In the case of Gaussian processes, almost sure values are proved for the Hölder exponents. As an application, the local regularity of the set-indexed fractional Brownian motion and the Ornstein-Uhlenbeck process are proved to be constant along the sample paths, with probability one. Finally, a weak continuity property which only considers single point jumps is presented. We shall present how this property helps caracterising certain classes of Lévy processes.