New: The lecture notes have appeared

• D. Applebaum, B.V.R. Bhat, J. Kustermans, J.M. Lindsay, Quantum Independent Increment Processes I: From Classical Probability to Quantum Stochastic Calculus, (U. Franz and M. Schürmann, eds.), Springer Lecture Notes in Mathematics, Vol. 1865, 2005.
• O.E. Barndorff-Nielsen, U. Franz, R. Gohm, B. Kümmerer, S. Thorbjørnsen, Quantum Independent Increment Processes II: Quantum Lévy Processes, Classical Probability, and Applications, (U. Franz and M. Schürmann, eds.), Springer Lecture Notes in Mathematics, Vol. 1866, 2006.

Photos

Program

First week (March 10-15):

Classical Lévy Processes

Lecturer: David Applebaum
Syllabus/abstract including references (as PostScript file)
Syllabus:

1. Lecture: Infinite divisibility and Lévy processes with examples. Stable laws and processes. Subordination.
2. Lecture: Semigroups, generators and resolvents. Hunt's formula and pseudo-differential operator representation for the generator. Subordination of semigroups.
3. Lecture: The Lévy-Itô decomposition and interlacing structure.
4. Lecture: Stochastic integration and SDEs.
5. Lecture: Lévy Processes on Lie groups. Hunt's formula and interlacing. Unitary representations of processes. Lévy processes in the Heisenberg group.
6. Lecture: Representations of current groups. The Lévy-Itô decomposition in Fock space and the road to quantum stochastic calculus.

The first four lectures are in Euclidean space.
Notes:

• Lectures on Classical Lévy Processes in Euclidean Spaces and Groups
  Available as PostScipt file

Quantum Stochastic Calculus

Lecturer: Martin Lindsay
Syllabus/abstract including references (as PostScript file)
Syllabus:

1. Lecture. Spaces: Fock, operator and matrix.
2. Lecture. Processes: martingales and Markovian cocycles.
3. Lecture. Integrals: quantum Wiener, noncausal stochastic and (multiple) quantum Itô.
4. Lecture. Stochastic differential equations.
5. Lecture. Cocycle types: contractive, positive, homomorphic.
6. Lecture. Perturbation and dilation of cocycles.

The action will take place on operator spaces.
Notes:

• Quantum Stochastic Calculus Lecture Notes
  Available as PostScipt file

Quantum Groups

Lecturer: Johan Kustermans
Syllabus/abstract including references (as PostScript file)
Syllabus:

• The definition of a locally compact quantum group: motivational examples and special cases. The classical case, compact and discrete quantum groups.
• The necessary background material on one-parameter groups of automorphisms and Tomita-Takesaki theory for weights.
• The general definition of a locally compact quantum group and its basic consequences. The antipode and its polar decomposition, the multiplicative unitary and the dual quantum group.
• An overview of examples of non-compact quantum groups together with a detailed discussion of the main peculiarities of their construction.
• Locally compact quantum groups co-acting on von Neumann algebras and quantum subgroups. Bicrossed products. Unitary implementation of a coaction. Extensions.

Notes:

• Locally compact quantum groups
  Available as PostScipt file

Second week (March 17-22)

On the Roles of Classical and Free Lévy Processes in Theory and Applications

Lecturers: Ole Barndorff-Nielsen and Steen Thorbjørnsen
Syllabus/abstract including references (as PostScript file)
Syllabus:

• Background material, in summary form, on classical infinite divisibility and independent increment processes, and independently scattered random measures. (With some particular emphasis on selfdecomposability.)
• Recent applications of classical infinite divisibility and Lévy processes in physics and finance
• Background material on quantum physics.
• Background material on operator algebras.
• Free independence: Infinite divisibility; Bercovici-Pata bijection; Lévy processes; Lévy-Itô representation; Sato processes; OU processes. Multivariate aspects.

Notes:

• Title page (available as PostScript file)
• O.E. Barndorff-Nielsen and Neil Shepard, A Chapter and Two Appendices from "Financial Volatility: semimartingales, Lévy processes, and stochastic volatility"
• O.E. Barndorff-Nielsen and S. Thorbjørnsen, Selfdecomposability and Levy processes in free probability, Bernoulli 8(3) (2002), 323-366.
  available as PostScript file
• O.E. Barndorff-Nielsen and S. Thorbjørnsen, Levy Laws in Free Probability, Proceedings of the National Academy of Science, vol. 99(26) (Dec. 2002), 16568-16575.
  available as PDF file
• O.E. Barndorff-Nielsen and S. Thorbjørnsen, Levy Processes in Free Probability, Proceedings of the National Academy of Science, vol. 99(26) (Dec. 2002), 16576-16580.
  available as PDF file
• Slides of the lectures by Steen Thorbjørnsen
  available as PDF file

Quantum Markov Processes and Applications in Physics

Lecturer: Burkhard Kümmerer
Syllabus/abstract (including references as PostScript file)
Syllabus:

1. Quantum Markov Processes
2. Scattering Theory for Quantum Markov Processes
3. Quantum Markov Processes in Physics
4. Ergodic Theory of Repeated Mesurement

Notes:

• Quantum Markov Processes and Applications to Physics

Dilations, Cocycles, and Product Systems

Lecturer: B.V.R. Bhat
Syllabus/abstract including references (as PostScript file)
Syllabus:

• Dilation theory basics: Sz. Nagy dilation, Stinespring representation for completely positive maps, Quantum Dynamical Semigroups and Weak Markov flows
• E_0-semigroups, cocycle conjugacy and classification of E_0-semigroups using product systems. Units, index and type classification of product systems of Hilbert spaces.
• Behaviour of domination under dilation, Quantum Stochastic Calculus in E_0-semigroup theory. Minimality of HP flows.
• Current state of affairs in product system classification and product systems of Hilbert C*-modules.

Notes:

• Dilations, Cocycles, and Product Systems
  Available as PostScipt file

Lévy Processes on Quantum Groups and Dual Groups

Lecturers: Uwe Franz and Michael Schürmann
Syllabus/abstract including references (as PostScript file)
Syllabus:

1. Lecture: Lévy processes on involutive bialgebras. Generators, triples, and the representation theorem.
2. Lecture: Examples. Lévy processes on the non-commutative analogue of U(n). Relation to dilations.
3. Lecture: The five independences: tensor, free, boolean, monotone, and anti-monotone. Dual groups/H-algebras/cogroups.
4. Lecture: Tensor, free, boolean, monotone and anti-monotone Lévy processes on dual groups/H-algebras/cogroups. Reduction to Lévy processes on involutive bialgebras

Notes:

• Lévy Processes on Quantum Groups and Dual Groups
  Available as PostScipt file