α-selfdecomposable distributions and related Ornstein–Uhlenbeck type processes

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