Title

On admissible shifts of generalized white noises

Document Type

Article

Publication Date

1-1-1982

Abstract

Let S be the Schwartz space of rapidly decreasing real functions. The dual space S* consists of the tempered distributions and the relation S ⊂ L2(R) ⊂ S* holds. Let γ be the Gaussian white noise on S* with the characteristic functional γ(ξ) = exp{-∥ξ∥2/2}, ξ ∈ S, where ∥·∥ is the L2(R)-norm. Let ν be the Poisson white noise on S* with the characteristic functional ν(ξ) = expΥ{hooked}R∫R {[exp(iξ(t)u)] - 1 - (1 + u2)-1(iξ(t)u)} dη(u)dt), ξ ∈ S, where the Lévy measure is assumed to satisfy the condition ∫Ru2dη(u) < ∞. It is proved that γ*ν has the same dichotomy property for shifts as the Gaussian white noise, i.e., for any ω ∈ S*, the shift (γ*ν)ω of γ*ν by ω and γ*ν are either equivalent or orthogonal. They are equivalent if and only if when ω ∈ L2(R) and the Radon-Nikodym derivative is derived. It is also proved that for the Poisson white noice νω is orthogonal to ν for any non-zero ω in S*. © 1982.

Publication Source (Journal or Book title)

Journal of Multivariate Analysis

First Page

80

Last Page

88

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