THE ORNSTEIN-UHLENBECK PROCESS ASSOCIATED WITH THE LÉVY
LAPLACIAN AND ITS DIRICHLET FORM
Luigi Accardi
Vladimir I. Bogachev
Abstract: We prove the existence of an Ornstein-Uhlenbeck type process associated with the
Lévy Laplacian. Like the classical case, the law of the Lévy Brownian motion at time 1 is an
invariant probability of this process. The corresponding semigroup is explicitly described and
the related Dirichlet form is constructed. There exist other parallels with the classical
situation such as the hypercontractivity of the semigroup, an analogue of the Cameron-Martin
space, etc. However, unlike the classical case in our setting the cylindrical functions do not
form a core of the Dirichlet form, in fact the form is identically zero on them. In this
sense the Lévy Ornstein-Uhlenbeck process provides an example of a new type of a
gradient-type (or classical) Dirichlet form which is essentially infinite dimensional.
2000 AMS Mathematics Subject Classification: Primary: -; Secondary: -;
Key words and phrases: -