Exotic laplacians and associated stochastic processes

In this paper, we give a decomposition of the space of tempered distributions by the Cesaro norm, and for any a > 1/2 we construct directly from the exotic trace an infinite dimensional separable Hilbert space H-c,H-2a-1 on which the exotic trace plays the role as the usual trace. This implies that the Exotic Laplacian coincides with the Volterra-Gross Laplacian in the Boson Fock space Gamma(H-c,H-2a-1) over the Hilbert space H-c,H-2a-1. Finally we construct the Brownian motion naturally associated to the Exotic Laplacian of order 2a - 1 and we find an explicit expression for the associated heat semigroup.