Given a pair of separable, real Banach spaces E and F and a centered Gaussian measure μ on E , one can ask what sort of Borel measurable maps Φ?: E → F map μ to a centered Gaussian measure on F . Obviously, a sufficient condition is that Φ be linear. On the other hand, linearity is far more than is really needed. Indeed, it suffices to know that Φ has the property that for -almost every ( x 1, x 2) ∈ E 2. In this article, I will first prove a structure theorem which shows that any map Φ which satisfies this property arises from a linear map on the Cameron–Martin space associated with μ on E . I will then investigate which linear maps on the Cameron–Martin space determine a Φ, and finally I will discuss some of the properties of Φ which reflect properties of the linear map from which it is determined.
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