The concentration function problem for locally compact groups revisited

Abstract

The concentration function problem for locally compact groups, i.e., the structure of groups admitting adapted nondissipating random walks, is closely related to relatively compact M- or skew semigroups and corresponding space-time random walks, resp. to tau-decomposable laws, where tau denotes an automorphism. Analogous results are obtained in the case of continuous time: Non-dissipating Lévy processes are related to relatively compact distributions of generalized Ornstein Uhlenbeck processes and corresponding space-time processes, resp. T-decomposable laws, T =(tau_t) denoting a continuous group of automorphisms acting on groups of the form N = C_K(T).