Brownian motion has been constructed in different ways. Einstein was the most outstanding physicists involved in its construction. From a physical point of view a dynamical theory of Brownian motion was favorable. The Ornstein‐Uhlenbeck process models such a dynamical theory and E. Nelson amongst others derived Brownian motion from Ornstein‐Uhlenbeck theory via a scaling limit. In this paper we extend the scaling result to α‐stable Lévy processes.