Cameron–Martin theorem

E59985

mathematical theorem theorem in functional analysis theorem in probability theory

The Cameron–Martin theorem is a fundamental result in probability theory and functional analysis that characterizes how Gaussian measures on infinite-dimensional spaces change under shifts by elements of a special Hilbert subspace (the Cameron–Martin space).

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Observed surface forms (1)

Surface form	Occurrences
Cameron–Martin space	1

Statements (47)

Predicate	Object
instanceOf	mathematical theorem ⓘ theorem in functional analysis ⓘ theorem in probability theory ⓘ
appliesTo	Gaussian measures on infinite-dimensional spaces ⓘ
characterizes	quasi-invariance of Gaussian measures under translations ⓘ
concernsProperty	absolute continuity of shifted Gaussian measures ⓘ mutual singularity of measures ⓘ quasi-invariance under translations ⓘ
describes	change of Gaussian measures under shifts ⓘ
field	functional analysis ⓘ measure theory ⓘ probability theory ⓘ
gives	explicit formula for Radon–Nikodym derivative of shifted Gaussian measure ⓘ
hasComponent	definition of Cameron–Martin space as reproducing kernel Hilbert space of Gaussian measure ⓘ
hasConsequence	characterization of admissible shifts of Gaussian paths ⓘ description of support of Gaussian measures on Banach spaces ⓘ
historicalPeriod	20th century mathematics ⓘ
holdsFor	centered Gaussian measures ⓘ non-degenerate Gaussian measures ⓘ
involves	Cameron–Martin theorem self-linksurface differs ⓘ surface form: Cameron–Martin space Gaussian measure ⓘ Hilbert spaces ⓘ Radon–Nikodym derivative ⓘ separable Banach spaces ⓘ translation operator ⓘ
isAbout	Cameron–Martin Hilbert subspace ⓘ shift invariance properties of Gaussian measures ⓘ structure of Gaussian measures on Banach spaces ⓘ
mathematicalDomain	Gaussian measure theory ⓘ infinite-dimensional analysis ⓘ
namedAfter	Richard H. Cameron ⓘ W. T. Martin ⓘ
relatedTo	Gaussian process ⓘ Girsanov theorem ⓘ Hilbert space embedding of Cameron–Martin space ⓘ Wiener measure ⓘ abstract Wiener space ⓘ
statesThat	translation by a vector outside the Cameron–Martin space makes the Gaussian measure mutually singular with the original ⓘ translation by an element of the Cameron–Martin space preserves equivalence class of a Gaussian measure ⓘ
usedIn	Brownian motion analysis ⓘ Gaussian processes theory ⓘ Malliavin calculus ⓘ infinite-dimensional integration ⓘ large deviations theory ⓘ path space measures ⓘ stochastic analysis ⓘ theory of abstract Wiener spaces ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Cameron–Martin theorem → involves → Cameron–Martin theorem self-linksurface differs ⓘ

this entity surface form: Cameron–Martin space

Girsanov theorem → relatedTo → Cameron–Martin theorem ⓘ