Structure theorem for Gaussian measures
Mathematical theorem
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∗(γH), where γH 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.
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Analysis in topological vector spaces
- Abstract Wiener space
- Classical Wiener space
- Bochner space
- Convex series
- Cylinder set measure
- Infinite-dimensional vector function
- Matrix calculus
- Vector calculus
- Cameron–Martin theorem
- Inverse function theorem
- Feldman–Hájek theorem
- No infinite-dimensional Lebesgue measure
- Sazonov's theorem
- Structure theorem for Gaussian measures
