Quantum white noise analysis and quantum stochastic equations

Abstract

We review the basic concepts of quantum stochastic analysis using the universal Itô B*-algebra approach. The thermal quantum white noise as a noncommutative analog of the Lévy-Itô process is characterized in terms of the modular B*-Itô algebra. The main notions and results of classical and quantum stochastics are reformulated in this unifying approach. The Belavkin master equation for a quasi-Markov quantum stochastic dynamics on the predual space of a W*-algebra is presented as a noncommutative version of the Zakai equation driven by the modular quantum white noise. This is done by the noncommutative analog of the Girsanov transformation which we introduce in full QS generality.