Self-similarity of free stochastic processes

In this paper we study self-similarity of free stochastic processes. We establish the non-commutative counterpart of Lamperti's self-similar processes. We develop the characterization of noncommutative self-similar processes through a modification of Voiculescu transform, the free cumulant transform. We study the connection between free self-similarity, strict boxed plus-stability and boxed plus-self-decomposability. In particular, we derive the properties of free self-similar processes and their connection to strict boxed plus-stability and boxed plus-self-decomposability, that turn out to be consistent with their classical analogue.