Lévy Laplacian for Square Roots of Measures

Abstract: L.Accardi shows that the Banach space of singed measures is homeomorphic to the Hilbert space formed by so-called root measures. In this paper, we redefine root measures in view of the theory of measures on infinite dimensional spaces, and introduce a notion of differentiation, Fourier transform, and convolution product for root measures, and examine those relations. We also study about L\'{e}vy Laplacian on Wiener space as application. It is shown that the symbol of L\'{e}vy Laplacian is equal to the quadratic variation of paths.