Cameron–Martin theorem

In mathematics, the Cameron–Martin theorem or Cameron–Martin formula (named after Robert Horton Cameron and W. T. Martin) is a theorem of measure theory that describes how abstract Wiener measure changes under translation by certain elements of the Cameron–Martin Hilbert space.

The standard Gaussian measure γⁿ on n-dimensional Euclidean space Rⁿ is not translation-invariant. (In fact, there is a unique translation invariant Radon measure up to scale by Haar's theorem: the n-dimensional Lebesgue measure, denoted here dx.) Instead, a measurable subset A has Gaussian measure

Here ${\displaystyle \langle x,x\rangle _{\mathbf {R} ^{n}}}$ refers to the standard Euclidean dot product in Rⁿ. The Gaussian measure of the translation of A by a vector h ∈ Rⁿ is

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