[alpha]-selfdecomposable distributions and related Ornstein-Uhlenbeck type processes

The concept of selfdecomposability has been generalized to that of [alpha]-selfdecomposability, , by many authors. We first mention the existing results on the class of [alpha]-selfdecomposable distributions and investigate the remaining problems. We give complete characterizations by stochastic integrals with respect to Lévy processes for the case 1<=[alpha]<2. The main topic of this paper is Langevin type equations and the corresponding Ornstein-Uhlenbeck type processes related to [alpha]-selfdecomposable distributions.