Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Abstract

Processes of Ornstein-Uhlenbeck type on R d are analogues of the Ornstein-Uhlenbeck process on R d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on R 1.