Stochastic Calculus in Fock Space

Abstract

In this chapter, we reach the main topic of these notes, non-commutative stochastic calculus for adapted families of operators on Fock space, with respect to the basic operator martingales. This calculus is a direct generalization of the classical Ito integration of adapted stochastic processes w.r.t. Brownian motion, or other martingales. Its physical motivation is quantum mechanical evolution in the presence of a “quantum noise”. Stochastic calculus has also been developed for fermions in a series of papers by Barnett, Streater and Wilde [BSW], and in abstract versions by Accardi, Fagnola, Quaegebeur.