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Euler operator and homogeneous hida distributions

Authors:	Liu Kai Yan Jia-an

Institution:	(1) Department of Mathematics, Huazhong University of Science and Technology, 43007 Wuhan, China;(2) Institute of Appl. Math., Academia Sinica, 100080 Beijing, China

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 $- \left( {\frac{d}{{dt}}} \right)^2 + 1 + t^2$ . In this paper the Euler operator Δ_E is defined as the sum $\sum\nolimits_i {\left\langle { \cdot ,e_i } \right\rangle \partial _i }$ , 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 _tϕ)(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=λ_ϕ. 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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