#
Ornstein-Uhlenbeck theory in finite dimension

Report, 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).