Noncommutative Analysis and Probability.

The project concerns quantum stochastic analysis and quantum Markov processes.
Algebraic structure in quantum stochastic analysis springs from the Itô-type relations enjoyed by quantum stochastic integrals. Deeper structure is revealed through use of Guichardet's symmetric measure space model of Boson Fock space, with its union and partition operations. It is now clear that the algebraic structure of multiple quantum Wiener integrals may be expressed via a purely combinatorial set convolution operation. The binary operation is on integrands which map finite subsets of the time interval into the full Fock space over the multiplicity space of the noise, augmented by one dimension for time. The project will explore this set convolution and investigate its application to quantum Markov processes on algebraic structures such as quantum groups.