On some properties of polynomials measurable with respect to a Gaussian measure

Abstract

As it is known (see [1]), for every measurable poly� nomial f with respect to a Gaussian measure γ there exists a version with the following property: for almost each x, the function (x + h) is a continuous polynomial in h on the Cameron–Martin space of the measure γ. It is shown in this paper that there exists a version with this property which is a homogeneous algebraic polynomial, moreover, for a homogeneous polynomial, any version that is an algebraic and homogeneous will be suitable. At the same time, in the nonhomogeneous case, not every such version will be suitable, which will be seen in the examples below. We recall some concepts used below. Let X be a locally convex space with the topological dual space X* and the Borel σ�field � (X). A Borel probability measure γ on X is called Radon (see [1, 2]) if for every B and every e > 0 there exists a compact set K ⊂ B such that γ(B\K) < e. We also assume throughout that the mea� sure γ is a centered Gaussian measure (see [1, 3, 4]), i.e., for every functional f ∈ X*, its image γ ° f –1 is a centered Gaussian measure on the real line, in other words, f is a centered Gaussian random variable on (X, � (X), γ). We use the symbol H(γ) to denote the Cameron–Martin space of the measure γ, which con� sists of all vectors with the finite Cameron–Martin norm