MAPS THAT TAKE GAUSSIAN MEASURES TO GAUSSIAN MEASURES

摘要

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 Φ(x_1+x_2/2~(1/2))=Φ(x_1)+Φ(x_2)/2~(1/2) for W~2-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.