The Ornstein–Uhlenbeck Operator and the Ornstein–Uhlenbeck Semigroup

In this chapter we are going to define and study the Ornstein–Uhlenbeck operator and the Ornstein–Uhlenbeck semigroup. They are analogous, in the Gaussian harmonic analysis, to the Laplacian and the heat semigroup in the classical case. Then, we study an important property of the Ornstein–Uhlenbeck semigroup, the hypercontractivity property, and some of its applications.