The classical Weyl pseudodifferential calculus is a particular choice of "quantisation" - a way to take functions of the position and momentum operators on R^n. This pseudodifferential calculus allows study of complicated operators to be (mostly) encapsulated by studying their symbols.
Ornstein-Uhlenbeck (OU) operators are analogs of the Laplacian adapted to spaces with Gaussian measure, and arise in many areas including stochastic analysis, quantum field theory and harmonic analysis. They are particularly nasty if tackled analytically and directly, with very rigid structures (for example, the standard OU operator has only H^\infty calculus on L^p, the proof of which takes over 100 pages in full detail!). From one of these origins, the OU operator arises naturally as a "function" of position- and momentum-like operators, which suggests that the ideas of Weyl calculi may be applicable.
This seminar will give an introduction to OU operators, the H^\infty functional calculus, and the Weyl calculus. I will then explain my current work in adapting the Weyl pseudodifferential calculus to the OU setting. This is of a very different flavour to the standard Weyl pseudodifferential calculus, with interesting links to complex analysis and Banach algebras. At present, it seems that using the Weyl calculus splits the problem of studying OU operators into an algebraic part and an analytic part, the analytic part being almost trivial when compared to the analysis used for studying OU directly.