Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

Abstract

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