Quantum Potential Theory

This volume contains the notes of lectures given at the School “Quantum Potential Theory: Structure and Applications to Physics”. This school was held at the Alfried Krupp Wissenschaftskolleg in Greifswald from February 26 to March 9, 2007. We thank the lecturers for the hard work they - complished in preparing and giving these lectures and in writing these notes. Theirlecturesgiveanintroductiontocurrentresearchintheirdomains,which is essentially self-contained and should be accessible to Ph.D. students. We hope that this volume will help to bring together researchers from the areas of classical and quantum probability, functional analysis and operator al- bras, and theoretical and mathematical physics, and contribute in this way to developing further the subject of quantum potential theory. We are greatly indebted to the Alfried Krupp von Bohlen und Halbach- Stiftung for the ?nancial support, without which the school would not have been possible. We are also very thankful for the support by the University of Greifswald and the University of Franche-Comt´ e.One of the organisers(UF) wassupportedby a MarieCurieOutgoingInternationalFellowshipofthe EU (Contract Q-MALL MOIF-CT-2006-022137). Special thanks go to Melanie Hinz who helped with the preparation and organisation of the school and who took care of all of the logistics. Finally, we would like to thank all the students for coming to Greifswald and helping to make the school a success.

This volume contains the revised and completed notes of lectures given at the school "Quantum Potential Theory: Structure and Applications to Physics," held at the Alfried-Krupp-Wissenschaftskolleg in Greifswald from February 26 to March 10, 2007.

Quantum potential theory studies noncommutative (or quantum) analogs of classical potential theory. These lectures provide an introduction to this theory, concentrating on probabilistic potential theory and it quantum analogs, i.e. quantum Markov processes and semigroups, quantum random walks, Dirichlet forms on C* and von Neumann algebras, and boundary theory. Applications to quantum physics, in particular the filtering problem in quantum optics, are also presented.