THE EXOTIC (HIGHER ORDER LÉVY) LAPLACIANS GENERATE THE MARKOV PROCESSES GIVEN BY DISTRIBUTION DERIVATIVES OF WHITE NOISE

Abstract

We introduce, for each a ∈ ℝ+, the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov5 extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat semi-group associated to any exotic Laplacian of order ≥ 1/2 and to give its explicit expression in terms of infinite dimensional Fourier transform.