Ornstein-Uhlenbeck semigroup: a Weyl calculus point of view

The Ornstein-Uhlenbeck semigroup is a variant of the heat semigroup that plays a fundamental role in quite distinct areas of mathematics.

In the first half of this talk, I will attempt to present the perspectives of at least three of these areas: analysis of PDE, stochastic calculus, and mathematical physics. In the first, the Ornstein-Uhlenbeck semigroup is the heat semigroup of $\mathbb{R}^n$ endowed with the Gaussian measure. In the second, it describes expected values of functions of a simple mean returning process. In the third, it is an observable in quantum mechanics (acting on a Fock space) counting the number of particles.

In the second half of the talk, I will present results that I have recently obtained with Jan van Neerven (Delft, Netherlands). We look at the Ornstein-Uhlenbeck semigroup as arising from the (non-commutative) Weyl functional calculus of a pair of position and momentum operators. This makes sense both from the point of view of mathematical physics and the point of view of stochastic calculus. In doing so, we set up a Gaussian analogue of pseudo-differential calculus, which looks promising from the point of view of analysis of PDE. To demonstrate its potential, we recover, and in some cases extend, all (arguably) classical and recent results regarding the $L^p-L^q$ behaviour of this semigroup, using nothing but H\"older's inequalities in our norm estimates! The idea is that, with an appropriate point of view that separates the geometric and the algebraic difficulties, the analysis becomes almost as simple as its counterpart for the usual heat semigroup.