The fourier transform in white noise calculus

Hui-Hsiung Kuo

Journal of Multivariate Analysis, 1989, vol. 31, issue 2, 311-327

Abstract: Let 0* be the space of termpered distributions with standard Gaussian measure [mu]. Let (0) [subset of] L2([mu]) [subset of] (0)* be a Gel'fand triple over the white noise space (0*, [mu]). The S-transform (S[phi])([zeta]) = [integral operator]7 [phi](x + [zeta]) d[mu](x), [zeta] [set membership, variant] 0, on L2([mu]) extends to a U-functional U[[phi]]([zeta]) = «exp(·, [zeta]), [phi] å exp(-[short parallel][zeta][short parallel]2/2), [zeta] [set membership, variant] 0, on (0)*. Let 5 consist of [phi] in (0)* such that U[[phi]](-i[zeta]1T) exp[-2-1 [integral operator]T[zeta](t)2 dt], [zeta] [set membership, variant] 0, is a U-functional. The Fourier transform of [phi] in 5 is defined as the generalized Brownian functional [phi] in (0)* such that U[[phi]]([zeta]) = U[[phi]](-i[zeta]1T) exp[-2-1 [integral operator]T[zeta](t)2 dt], [zeta] [set membership, variant] 0. Relations between the Fourier transform and the white noise differentiation [not partial differential]t and its adjoint [not partial differential]t* are proved. Results concerning the Fourier transform and the Gross Laplacian [Delta]G, the number operator N, and the Volterra Laplacian [Delta]V are obtained. In particular, ([Delta]G*[phi])? = -[Delta]G*[phi] and [([Delta]V + N)[phi]]? = -([Delta]V + N)[phi]. Many examples of the Fourier transform are given.

Keywords: B; (t)-derivative; Gaussian; measure; generalized; Brownian; functional; Gross; Laplacian; Lévy; Laplacian; number; operator; renormalization; U-functional; Volterra; Laplacian (search for similar items in EconPapers)
Date: 1989
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