Selfsimilar free additive processes and freely selfdecomposable distributions

In the paper by Fan\cite{F06}, he introduced the marginal selfsimilarity of non-commutative stochastic processes and proved the marginal distributions of selfsimilar processes with freely independent increments are freely selfdecomposable. In this paper, we firstly introduce a new definition, stronger than Fan's one in general, of selfsimilarity via linear combinations of non-commutative stochastic processes, although their two definitions are equivalent for non-commutative stochastic processes with freely independent increments. We secondly prove the converse of Fan's result, to complete the relationship between selfsimilar free additive processes and freely selfdecomposable distributions. Furthermore, we construct stochastic integrals with respect to free additive processes for representing the background driving free L{\'e}vy processes of freely selfdecomposable distributions. A relationship between freely selfdecomposable distributions and their background driving free L{\'e}vy processes in terms of their free cumulant transforms is also given, and several examples are discussed.Comment: 31 pages, no figur