Q-MALL · Malliavin Calculus for Quantum Stochastic Processes
6РП — Действия „Мария Кюри“
- Период
- 2006-11-06 → 2009-11-05
- Финансиране от ЕС
- 243 977 €
- Участници
- 2
- Схема
- OIF
Линиите свързват координатора с партньорите. За проекти отпреди 2014 г. CORDIS не винаги дава точни координати. Тези точки са на ниво град или държава.
Накратко на български
Квантовият стохастичен анализ изследва математическите модели на случайни процеси, като например тези в квантовата оптика. Тези инструменти помагат за по-доброто разбиране на квантовите диференциални уравнения и стохастичните модели в биологичните системи.
Кратко обяснение, генерирано от езиков модел по текста на CORDIS. Оригиналът е по-долу.
Резултати накратко
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.
Текст от CORDIS, на английски · Данни: CORDIS, © Европейски съюз
Цел на проекта
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.
Оригинален текст от CORDIS (на английски).
Участници
Връзки
Данни: CORDIS, © Европейски съюз