Some measure - preserving point transformations on the Wiener space and their ergodicity
نویسندگان
چکیده
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 rotations which are second quantizations of the unitary operators on the Cameron-Martin space, are characterized. Finally, the ergodicity of the transformation dY t = γ(t)dW t , 0 ≤ t ≤ 1 where W is n-dimensional Wiener and γ is non random is characterized.
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