On Fourier transform of generalized Brownian functionals

AbstractLet (L2)Ḃ− and (L2)ḃ− be the spaces of generalized Brownian functionals of the white noises Ḃ and ḃ, respectively. A Fourier transform from (L2)Ḃ− into (L2)ḃ− is defined by ϕ̂(ḃ) = ∫S∗: exp[−i ∫Rḃ(t) Ḃ(t) dt]: ḃϕ(Ḃ) dμ(Ḃ), where : :ḃ denotes the renormalization with respect to ḃ and μ is the standard Gaussian measure on the space S∗ of tempered distributions. It is proved that the Fourier transform carries Ḃ(t)-differentiation into multiplication by iḃ(t). The integral representation and the action ofϕ̂ as a generalized Brownian functional are obtained. Some examples of Fourier transform are given