In this paper we determine the distributional behavior of sums of free (in the sense of Voiculescu) identically distributed, infinitesimal random variables. The theory is shown to parallel the classical theory of independent random variables, though the limit laws are usually quite different. Our work subsumes all previously known instances of weak convergence of sums of free, identically distributed random variables. In particular, we determine the domains of attraction of stable distributions in the free theory. These freely stable distributions are studied in detail in the appendix, where their unimodality and duality properties are demonstrated.

We de®ne stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Itoà 's predictable representation theorem, as well as product form and functional form of Itoà 's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.Mathematics Subject Classi®cation (1991): 60H05, 46L50, 81S25

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We give a formula for the density of the free convolution of an arbitrary probability measure with a semicircular distribution. We use this formula to establish a certain number of regularity properties of the measures obtained in this way.Introduction. In recent years, a new type of harmonic analysis of measures on the real line has been developped, centered around the definition of free convolution of measures, introduced by D. Voiculescu. Many classical results in the theory of addition of independent random variables have their counterpart in this new theory, such as the law of large numbers, the central limit theorem, the Lévy-Khintchine formula, etcetera. We refer to [VDN] for an introduction to these topics. One of the main problems in dealing with free convolution is that its definition is rather indirect, and one does not get easily formulas for computing densities of free convolution of measures. In this note, we propose such a formula when one of the measures is a semi-circular distribution, and we use it to prove certain regularizing properties of the operation of taking the free convolution with a semi-circular distribution. The semi-circular distribution is the analogue in free probability theory of the gaussian distribution, so that what we are investigating is an explicit formula for the solution to the free analogue of the heat equation. In particular we prove that the measures obtained by free convolution with a semi-circular distribution of variance t have always a continuous density whose cube is Lipschitz, with Lipschitz constant less than 3(4π 3 t 2 ) −1 , and we also give optimal bounds on the density of such measures, when the starting measure has a bounded density. We also give a formula for the Hilbert ransform of the density.This topic has several connections with other areas of mathematics. The free convolution has an interpretation in terms of spectra of large random matrices, see [V2], [S]. The convolution with a semi-circular distribution plays an important role in the theory of free entropy developped by D. Voiculescu see [V3]