Quantum Extensions of the Exotic Laplacians and Associated Quantum Heat Semigroups

The hierarchy of exotic Laplacians is a natural extension of the usual Levy Laplacian. In the present paper it is shown that the heat semigroup associated to each Laplacian in the above mentioned hierarchy admits a natural and explicit quantum extension in which the Weyl operators are eigen–operators of the given Laplacian, just as trigonometric exponentials are eigenvectors of the usual Laplacian. Starting with this remark we construct a multiplicity of quantum Markov generators with the property that their restrictions on the position algebra or the momentum algebra or both coincide with the exotic Laplacian of order α ∈ R+. A crucial tool in this construction is that, for any α ∈ R+, the exotic Laplacian of order α coincides on a dense domain with the usual Laplacian associated to the white noise with values in the Ces`aro space of order α. In fact we conjecture that this domain is a core for the given exotic Laplacian.