Euler operator and homogeneous hida distributions |
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Authors: | Liu Kai Yan Jia-an |
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Affiliation: | (1) Department of Mathematics, Huazhong University of Science and Technology, 43007 Wuhan, China;(2) Institute of Appl. Math., Academia Sinica, 100080 Beijing, China |
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Abstract: | Let (S)⊄L 2(S′(∔),μ)⊄(S)* be the Gel'fand triple over the white noise space (S′(∔),μ). Let (e n ,n>-0) be the ONB ofL 2(∔) consisting of the eigenfunctions of the s.a. operator . In this paper the Euler operator Δ E is defined as the sum , where ∂ i stands for the differential operatorD e i. It is shown that Δ E is the infinitesimal generator of the semigroup (T t ), where (T t ϕ)(x)=ϕ(e t x) for ϕ∈(S). Similarly to the finite dimensional case, the λ-order homogeneous test functionals are characterized by the Euler equation: Δ Eϕ =λϕ. Via this characterization the λ-order homogeneous Hida distributions are defined and their properties are worked out. Supported by the National Natural Science Foundation of China. |
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