Ornstein-Uhlenbeck theory in finite dimension

Rapport, 2012

The contents of these notes were presented during ten lectures, in November 2011, by Peter Sjögren in Gothenburg. The text was written by Adam Andersson who participated and is improved after the careful reading by Peter Sjögren. Ornstein-Uhlenbeck theory can be described as a model of harmonic analysis in which Lebesgue measure is everywhere replaced by a Gaussian measure. The theory has applications in quantum physics and probability theory. If one passes to infinite dimensions and places the theory in a probabilistic context, one gets the Malliavin calculus. In Chapter 1, the basic theory is developed. This concerns the Hermite polynomials, the Ornstein-Uhlenbeck operator and most importantly its semigroup. The Hermite polynomials form an orthogonal system with respect to the Gaussian measure in Euclidean space. It turns out that they are the eigenfunctions of the Ornstein-Uhlenbeck operator, and since this operator is self-adjoint and positive semidefinite, the semigroup can be defined spectrally. An explicit kernel is derived for the semigroup, known as the Mehler kernel. It will be of central importance in this text. In Chapter 2, boundary convergence for the semigroup is considered, i.e., the limiting behavior of the semigroup as the “time” tends to zero. This is done by introducing a maximal operator for the semigroup and proving that it is of weak type (1,1). This result implies almost everywhere convergence for integrable boundary functions. In Chapter 3, first-order Riesz operators related to the Ornstein- Uhlenbeck operator are treated. Explicit off-diagonal kernels for these operators are found. It is finally proved that the Riesz operators are of weak type (1,1).