Selfsimilar free additive processes and freely selfdecomposable distributions

TL;DR

This paper explores the relationship between selfsimilar free additive processes and freely selfdecomposable distributions, introducing a stronger selfsimilarity definition and establishing new theoretical connections and representations.

Contribution

It introduces a new, stronger definition of selfsimilarity for non-commutative processes and proves the converse of Fan's result, linking selfsimilar free additive processes to freely selfdecomposable distributions.

Findings

Established the equivalence of two selfsimilarity definitions for processes with freely independent increments.

Proved the converse of Fan's result relating selfsimilar free additive processes to freely selfdecomposable distributions.

Constructed stochastic integrals to represent background driving free Lévy processes.

Abstract

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.