Some measure-preserving point transformations on the Wiener space and their ergodicity
Probability
2007-05-23 v2
Abstract
Suppose that T is a map of the Wiener space into itself, of the following type: T=I+u where u takes its values in the Cameron-Martin space H. Assume also that u is a finite sum of H-valued multiple Ito-Wiener integrals. In this work we prove that if T preserves the Wiener measure, then necessarily u is in the first Wiener chaos and the transformation corresponding to it is a rotation in the sense of [9]. Afterwards the ergodicity and mixing of such transformations, which are second quantizations of the unitary operators on the Cameron-Martin space, are characterized. Finally, the ergocity of the transformation dY_t=gamma(t)dW_t, 0 \le t \le 1 where W is n-dimensional Wiener and gamma is non random is characterized
Keywords
Cite
@article{arxiv.math/0002198,
title = {Some measure-preserving point transformations on the Wiener space and their ergodicity},
author = {A. S. Ustunel and M. Zakai},
journal= {arXiv preprint arXiv:math/0002198},
year = {2007}
}