Properties of Measure-preserving Shifts on the Wiener Space

Abstract

Let (W, H, μ) be an abstract Wiener space and let T: W → W be a measurable map of the form T = I _w + u, where u is a Wiener functional with values in the Cameron-Martin space H. Assume that μ is invariant under T and that almost surely, the map h → u(w + h) is Fréchet differentiable on H. Then we prove some pointwise properties of T; namely, that T has a universally measurable right inverse that is absolutely continuous, and the set of nondegeneracy of T has full μ-measure. Finally, in the case E[|∧|] ≤ 1 (see the introduction), we show that T, in fact, is almost surely a bijection.