The Lamperti transformation for self-similar processes

In this paper we establish the uniqueness of the Lamperti transformation leading from self-similar to stationary processes, and conversely. We discuss alpha-stable processes, which allow to understand better the difference between the Gaussian and non-Gaussian cases. As a by-product we get a natural construction of two distinct alpha-stable Ornstein–Uhlenbeck processes via the Lamperti transformation for 0 < alpha < 2. Also a new class of mixed linear fractional alpha-stable motions is introduced.