Structure theorem for Gaussian measures

In mathematics, the structure theorem for Gaussian measures shows that the abstract Wiener space construction is essentially the only way to obtain a strictly positive Gaussian measure on a separable Banach space. It was proved in the 1970s by Kallianpur–Sato–Stefan and Dudley–Feldman–le Cam.

There is the earlier result due to H. Satô (1969) ^[1] which proves that "any Gaussian measure on a separable Banach space is an abstract Wiener measure in the sense of L. Gross". The result by Dudley et al. generalizes this result to the setting of Gaussian measures on a general topological vector space.

Statement of the theorem

Let γ be a strictly positive Gaussian measure on a separable Banach space (E, || ||). Then there exists a separable Hilbert space (H, ⟨ , ⟩) and a map i : H → E such that i : H → E is an abstract Wiener space with γ = i_∗(γ^H), where γ^H is the canonical Gaussian cylinder set measure on H.

References

^ H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969.

Dudley, Richard M.; Feldman, Jacob; Le Cam, Lucien (1971). "On seminorms and probabilities, and abstract Wiener spaces". Annals of Mathematics. Second Series. 93 (2): 390–408. doi:10.2307/1970780. ISSN 0003-486X. JSTOR 1970780. MR 0279272.

[1] H. Satô, Gaussian Measure on a Banach Space and Abstract Wiener Measure, 1969.

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