Self-decomposability and Lévy processes in free probability. (English) Zbl 1024.60022

Quoting D. Voiculescu [in: Lectures on probability theory and statistics. Lect. Notes Math. 1738, 279-349 (2000; Zbl 1015.46037)], free probability theory = noncommutative probability theory + free independence. A noncommutative probability space is a unital algebra \(\mathcal A\) over \(\mathcal C\) (the set of complex numbers) endowed with a linear functional \(\tau : {\mathcal A} \rightarrow {\mathcal C}\), \(\tau(1)= 1\). Elements \(a \in {\mathcal A}\) are called “random variables”. \(({\mathcal A},\tau)\) is a \(W^*\)-probability space if the pair is isomorphic to a von Neuman algebra (or \(W^*\)-algebra) and some vector state. Let us identify \(\mathcal A\) with an algebra \({\mathcal B}({\mathcal H})\) of operators on a Hilbert space \({\mathcal H}\). Given any selfadjoint operator \(a \in {\mathcal B}({\mathcal H})\), the spectrum \(\text{sp}(a)\) is contained in \(R\) and there exists a unique probability measure \(\mu_a\) on \(R\), concentrated on \(\text{sp}(a)\), satisfying \(\tau(f(a)) = \int_R f(t)\mu_a(dt)\) for all bounded Borel functions \(f\) on \(R\). The measure \(\mu_a\) is called the (spectral) distribution of \(a\) wrt \(\tau\).
Once having the notion of “free independence” in a \(W^*\)-probability space (cf. Voiculescu, loc. cit.) we can introduce a free additive convolution of (spectral) distributions: let \(a\) and \(b\) be selfadjoint operators in \({\mathcal B}({\mathcal H})\). If \(a\) and \(b\) are freely independent, then we write \(\mu_a \boxplus \mu_b\) instead of \(\mu_{a+b}\). The class of \(\boxplus\)-infinitely divisible (spectral) distributions is studied e.g. in papers by D. Voiculescu (1986), H. Maassen [J. Funct. Anal. 106, 409-438 (1992; Zbl 0784.46047)] and H. Bercovici and D. Voiculescu [Indiana Univ. Math. J. 42, 733-773 (1993; Zbl 0806.46070)]. The present paper is devoted to the \(\boxplus\)-selfdecomposability. The authors prove that the \(\Lambda\)-bijection, introduced by Bercovici and Pata, maps the class of the classically self-decomposable probability measures onto the free counterpart. Further, they study Lévy processes in free probability and construct stochastic integrals wrt such processes.


60G51 Processes with independent increments; Lévy processes