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Malliavin Calculus for Quantum Stochastic Processes

Final Activity Report Summary - Q-MALL (Malliavin Calculus for Quantum Stochastic Processes)

In the framework of his Marie-Curie fellowship on quantum Malliavin calculus, Uwe Franz established the expected connection between his own prior work in collaboration with Remi Leandre, Rene Schott and Nicolas Privault, as well as the prior work by Nobuaki Obata, the Japanese scientist in charge of this project and Un Cig Ji. He showed that the derivatives introduced by Obata and Ji in the context of quantum white noise analysis extended the derivation operator defined by Franz, Leandre and Schott. This results were very fruitful, because they allowed to remove unnecessary restrictions and to transfer Obata and Jis results, e.g. on integral representations of quantum semi-martingales or on Bogoljubov transformations to the setting of quantum Malliavin calculus. This led to new powerful tools for studying quantum stochastic differential equations which arise in models, such as in quantum optics.

The next step would be to generalise the calculus to other algebraic structures, e.g. Lie algebras and quantum groups, and to other notions of independence. For this purpose, Franz studied monotone convolutions and stochastic processes with monotonically independent increments and made important progress by extending the monotone convolution to non-compactly supported measures and by studying the bijection between monotone increment processes and Loewner chains. In collaboration with Skalski and Tomatsu he classified idempotent states and roots of the Haar state on quantum groups.

Furthermore, in collaboration with former colleagues from the Institut for biomathematics in Neuherberg, Munich, in particular with Stefan Zeiser, he worked on the study of stochastic models for biological systems. The goal of this work was to get a better understanding of the processes involved in gene expression. They showed that the class of piece-wise deterministic Markov processes introduced by Davies was very powerful for both the simulation of such systems as well as for the rigorous mathematical study of their asymptotic behaviour.