Ornstein-Uhlenbeck theory in finite dimension

The contents of these notes were presented during ten lectures, in November 2011, by PeterSj\uf6gren in Gothenburg. The text was written by Adam Andersson who participated and isimproved after the careful reading by Peter Sj\uf6gren. Ornstein-Uhlenbeck theory can bedescribed as a model of harmonic analysis in which Lebesgue measure is everywhere replacedby a Gaussian measure. The theory has applications in quantum physics and probabilitytheory. 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 theHermite polynomials, the Ornstein-Uhlenbeck operator and most importantly its semigroup.The Hermite polynomials form an orthogonal system with respect to the Gaussian measure inEuclidean space. It turns out that they are the eigenfunctions of the Ornstein-Uhlenbeckoperator, and since this operator is self-adjoint and positive semidefinite, the semigroup canbe defined spectrally. An explicit kernel is derived for the semigroup, known as the Mehlerkernel. It will be of central importance in this text. In Chapter 2, boundary convergence forthe semigroup is considered, i.e., the limiting behavior of the semigroup as the “time” tends tozero. This is done by introducing a maximal operator for the semigroup and proving that it isof weak type (1,1). This result implies almost everywhere convergence for integrableboundary 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. Itis finally proved that the Riesz operators are of weak type (1,1)