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Random rotations of the Wiener path
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  • Published: September 1995

Random rotations of the Wiener path

  • A. S. Üstünel1,2 &
  • M. Zakai1,2 

Probability Theory and Related Fields volume 103, pages 409–429 (1995)Cite this article

  • 126 Accesses

  • 20 Citations

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Summary

Let (W, H, μ) be an abstract Wiener space and letR(w) be a strongly measurable random variable with values in the set of isometries onH. Suppose that ∇Rh is smooth in the Sobolev sense and that it is a quasi-nilpotent operator onH for everyh∈H. It is shown that δ(R(w)h) is again a Gaussian (0, |h| 2 H )-random variable. Consequently, if (e i ,i∈ℕ)⊂W * is a complete, orthonormal basis ofH, then\(\tilde w = \sum\nolimits_i {(\delta R(w)e_i )e_i } \) defines a measure preserving transformation, a “rotation”, onW. It is also shown that if for some strongly measurable, operator valued (onH) random variableR, δ(R(w+k)h) is (0, |h| 2 H )-Gaussian for allk, h∈H, thenR is an isometry and ∇Rh is quasi-nilpotent for allH∈H. The relation between the stochastic calculi for these Wiener pathsw and\(\tilde w\), as well as the conditions of the inverbibility of the map\(w \to \tilde w\) are discussed and the problem of the absolute continuity of the image of the Wiener measure μ under Euclidean motion on the Wiener space (i.e.\(w \to \tilde w\) composed with a shift) is studied.

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Author information

Authors and Affiliations

  1. Département Réseaux, ENST, 46, rue Barrault, F-75013, Paris, France

    A. S. Üstünel & M. Zakai

  2. Department of Electrical Engineering, Technion-Israel Institute of Technology, 32000, Haifa, Israel

    A. S. Üstünel & M. Zakai

Authors
  1. A. S. Üstünel
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  2. M. Zakai
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Additional information

Dedicated to the memory of Albert Badrikian

The research of the second author was supported by the Fund for the Promotion of Research at the Technion

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Üstünel, A.S., Zakai, M. Random rotations of the Wiener path. Probab. Th. Rel. Fields 103, 409–429 (1995). https://doi.org/10.1007/BF01195481

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  • Received: 19 December 1994

  • Revised: 28 April 1995

  • Issue Date: September 1995

  • DOI: https://doi.org/10.1007/BF01195481

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Mathematics Subject Classification

  • 60G30
  • 60H07
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