We deene 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 It^ o's predictable representation theorem, as well as product form and functional form of It^ o's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener s...

Fractional Lévy processes generalize fractional Brownian motion in a natural way. We go a step further and extend the usual fractional Riemann-Liouville kernels to the more general class of regularly varying functions with the corresponding fractional integration parameter. The resulting stochastic processes are called generalized fractional Lévy processes (GFLP). Moreover, we define stochastic...

In this paper, we prove a central limit theorem for a sequence of multiple Skorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted quadratic variation of the fractional Brownian motion are discussed.

The class of second-order difference equations that arise in the diffraction of a plane wave at skew incidence on an impedance wedge is investigated. For a typical equation which was recently solved by the introduction of elliptic integrals, it is shown that the solution can be obtained in a very trivial manner from a convolution-type integral equation and the results are in complete agreement ...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...