## 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.

### MSC:

 60G51 Processes with independent increments; Lévy processes

### Citations:

Zbl 1015.46037; Zbl 0784.46047; Zbl 0806.46070