Weyl Pseudodifferential Operators in Ornstein-Uhlenbeck Settings

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 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 original 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.

After briefly explaining these and other relevant concepts, I will explain my current work in adapting the Weyl calculus to the OU setting. This is of a very different flavour to the standard Weyl calculus. Recently, I have re-proven the optimal functional calculus result for the standard OU operator using the Weyl calculus, with a proof that shows the Weyl calculus splits the analytic and algebraic difficulties of such problems, and vastly simplifies the analytic difficulties. I will explain some ideas which could potentially help with the algebraic difficulties in more general cases.