CORDIS

The proposed project aims to develop a Hida-Malliavin-type calculus for quantum stochastic processes. Quantum stochastic processes are models for the evolution of quantum systems subject to noise. They also have applications in the theory of operator algebras, for example for the construction of dilations, and in classical probability, where they provide examples of processes with interesting and sometimes surprising properties. We take Wigner functions as densities of quantum stochastic processes and us e techniques on non-commutative harmonic analysis and infinite-dimensional analysis, in particular white noise calculus, to obtain sufficient conditions for quantum stochastic processes obtained as solutions of quantum stochastic differential equations to have smooth Wigner functions. The tools which we intend to develop can also be used to study other properties of quantum stochastic processes such as their asymptotic behaviour. Furthermore, we plan to apply them to realistic physical models from quantum optics.