Levy Processes, Pseudo-differential Operators and Dirichlet Forms in the Heisenberg group.

• davis applebaum
• serge cohen

Code(s) de Classification MSC:

Résumé: N. Jacob and his colleagues have recently made many interesting investigations of Markov processes in Euclidean space where the infinitesimal generator of the associated semigroup is a pseudo-differential operator in the Kohn-Nirenberg sense. We wish to extend this programme to the Heisenberg group where we can utilise the Weyl calculus to build pseudo-differential operators and we begin by considering Levy processes. We obtain the general form of symbol for infinitesimal generators. We then investigate a natural sub-class of group-valued processes whose components are a levy process in phase space and the associated Levy area process on the real line. These are applied to give a new probabilistic proof for Mehler's formula. In addition, we describe some properties of the generator in its Schrödinger representation and of the associated Dirichlet form.

Date: 2001-11-23

Prépublication numéro: LSP-2001-10