On admissible shifts of generalized white noises

AbstractLet 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) = expϒR∫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∗