Selfsimilar Free Additive Processes and Freely Selfdecomposable Distributions

In the paper by Fan (Inf Dim Anal Quant Probab Rel Topics 9:451–469, 2006), he introduced marginal selfsimilarity of non-commutative stochastic processes and proved that distributions selfsimilar with freely independent increments are selfdecomposable. this paper, we firstly introduce a new definition, stronger than Fan’s in general, via linear combinations processes, although two definitions equivalent for increments. We secondly prove converse result, to complete relationship between free additive selfdecomposable distributions. Furthermore, construct integrals respect representing background driving Lévy A their terms cumulant transforms is also given, several examples discussed.