Domain of Exotic Laplacian Constructed by Wiener Integrals of Exponential White Noise Distributions (original) (raw)
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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 Cesaro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesaro space of order 2a). In particular, for a = 1=2 one nds the usual Levy 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 Smolyanov 5 extending the well known Cesaro 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 in nite dimensional Fourier transform.
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Let φ be a function defined on L 2 (0, 1). In 1922 P. Lévy introduced the Laplacian ∆ L φ of φ for harmonic analysis on the space L 2 (0, 1) by ∆φ(x) = lim N →∞ 1 N N X n=1 φ (x)e n , e n , where {en; n ≥ 1} is an orthonormal basis for L 2 (0, 1). This Laplacian, called the Lévy Laplacian, has no finite dimensional analogue. It is intrinsically infinite dimensional and possesses peculiar properties. In 1975 T. Hida introduced white noise theory which can be used to study Lévy's functional analysis. In this paper we study recent progress on some properties of the Lévy Laplacian from the white noise viewpoint, e.g., its relationships to the Lévy group, the spherical mean, Wiener processes, Gross Laplacian, and the Fourier transform. More significantly, we discuss K. Saitô's novel idea on the diagonalization of the Lévy Laplacian operator to construct a domain of this operator and to find semigroups and stochastic processes generated by the Lévy Laplacian. We present some of recent results due to Saitô and his collaborators.
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