Regularity property of Donsker's delta function

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Let L be the space of rapidly decreasing smooth functions on ℝ and L* its dual space. Let (L2)+ and (L2)- be the spaces of test Brownian functionals and generalized Brownian functionals, respectively, on the white noise space L* with standard Gaussian measure. The Donsker delta function δ(B(t)-x) is in (L2)- and admits the series representation {Mathematical expression}, where Hn is the Hermite polynomial of degree n. It is shown that for φ in (L2)+, gt,φ(x)≡〈δ(B(t)-x), φ〉 is in L and the linear map taking φ into gt,φ is continuous from (L2)+ into L. This implies that for f in L* is a generalized Brownian functional and admits the series representation {Mathematical expression}, where ξn,t is the Hermite function of degree n with parameter t. This series representation is used to prove the Ito lemma for f in L*, {Mathematical expression}, where ∂s* is the adjoint of {Mathematical expression}-differentiation operator ∂s. © 1984 Springer-Verlag New York Inc.

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Applied Mathematics & Optimization

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