Noncommutative Probability

Martin Lindsay, Alex Belton, Adam Skalski and PhD students Batakrishna Das and Oliver Margetts are working in diverse aspects of noncommutative probability. Recent graduates include Nicholas Weatherall and Orawan Sanhan

Three areas under intense current development are described below.

Quantum stochastic analysis. The stochastic calculus of quantum probability was initiated by Hudson and Parthasarathy in the mid-1980's. Since then it has undergone considerable development and extension (see 81S25 ), and has become the prototype for other noncommutative stochastic calculi. Recent advances have crucially used operator space theory - in particular a new form of tensor product well-suited to marrying the categories of topology (from a state space) and measure (from the randomness). Current research is focusing on quantum stochastic processes on (locally) compact quantum groups in the sense of (Kustermans, Vaes and) Woronowicz.

Discrete approximation of quantum processes. Quantum-probabilistic random walks were introduced by Accardi, Bach, Lindsay, Parthasarathy and others in the late eighties. They have attracted much recent attention (by Attal, Bouten and others). Belton has developed a general framework for the convergence of quantum random walks and, with Franz, Skalski has established a natural approximation scheme for quantum Lévy processes.

Noncommutative Dirichlet forms. The generators of symmetric Markov semigroups were characterised by Beurling and Deny in the fifties, and in the fully noncommutative case by Goldstein and Lindsay in the late nineties. Through gaining a proper understanding of the adjoint of a map between von Neumann algebras with a state or weight (examples include time reversal for Markov processes and conditional expectation) the first step in extension to nonsymmetric semigroups has recently been achieved. A long-term goal of this work is the canonical association of a quantum Markov process to each noncommutative Dirichlet form. Research in collaboration with Rajarama Bhat indicates that this will require significant refinement of existing theory.

• D. Applebaum, B.V.R. Bhat, J. Kustermans and J.M. Lindsay, Quantum Independent Increment Processes I, Lecture Notes in Mathematics Springer (2005) Springer
• S. Attal and A.C.R. Belton, The chaotic-representation property for a class of normal martingales, Probability Theory and Related Fields 139 (2007), nos. 3-4, 543-562. PTRF
• A.C.R. Belton, On the path structure of a semimartingale arising from monotone probability theory, Annales de l'Institut Henri Poincaré (Probabability and Statistics) 44 (2008), 258-279. AIHP arXiv
• A.C.R. Belton, Approximation via toy Fock space - the vacuum-adapted viewpoint, Preprint, arXiv
• U. Franz and A.G. Skalski, Approximation of quantum Levy processes by quantum random walks, Proceedings of the Indian Academy of Sciences. Mathematical Sciences 118 (2008), no. 2, 281-288. PIAS arXiv
• J.M. Lindsay and A.G. Skalski, Quantum stochastic convolution cocycles II, Communications in Mathematical Physics 280 (2008), no. 3, 575-610. CMP arXiv
• J.M. Lindsay and S.J. Wills, Quantum stochastic operator cocycles via associated semigroups, Mathematical Proceedings of the Cambridge Philosophical Society 142 (2007), no. 3, 535-556. MPCPS arXiv
• A.G. Skalski, Completely positive quantum stochastic convolution cocycles and their dilations, Mathematical Proceedings of the Cambridge Philosophical Society, 143 (2007), 201-219. MPCPS arXiv