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Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space
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  • Published: 06 January 2007

Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space

  • Ali Süleyman Üstünel1 &
  • Moshe Zakai2 

Probability Theory and Related Fields volume 139, pages 207–234 (2007)Cite this article

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  • 9 Citations

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Abstract

Let (W, H, μ) be the classical Wiener space. Assume that U = I W  + u is an adapted perturbation of identity, i.e., u : W → H is adapted to the canonical filtration of W. We give some sufficient analytic conditions on u which imply the invertibility of the map U. In particular it is shown that if \({u\in {\rm ID}_{p,1}(H)}\) is adapted and if \({\exp(\frac{1}{2}\|\nabla u\|_2^2-\delta u)\in L^q(\mu)}\) , where p −1 + q −1 = 1, then I W  + u is almost surely invertible. With the help of this result it is shown that if \({\nabla u\in L^\infty(\mu,H\otimes H)}\) , then the Girsanov exponential of u times the Wiener measure satisfies the logarithmic Sobolev inequality and this implies the invertibility of U = I W  +  u . As a consequence, if, there exists an integer k ≥  1 such that \({\|\nabla^k u\|_{H^{\otimes(k+1)}}\in L^\infty(\mu)}\) , then I W  +  u is again almost surely invertible under the almost sure continuity hypothesis of \({t\to\nabla^i \dot{u}_t}\) for i ≤  k − 1.

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Authors and Affiliations

  1. Department of Infres, ENST, Paris 46, rue Barrault, 75013, Paris, France

    Ali Süleyman Üstünel

  2. Department of Electrical Engineering, Technion, Haifa, 32000, Haifa, Israel

    Moshe Zakai

Authors
  1. Ali Süleyman Üstünel
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  2. Moshe Zakai
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Correspondence to Ali Süleyman Üstünel.

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Üstünel, A.S., Zakai, M. Sufficient conditions for the invertibility of adapted perturbations of identity on the Wiener space. Probab. Theory Relat. Fields 139, 207–234 (2007). https://doi.org/10.1007/s00440-006-0044-z

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  • Received: 26 April 2005

  • Revised: 27 October 2006

  • Published: 06 January 2007

  • Issue Date: September 2007

  • DOI: https://doi.org/10.1007/s00440-006-0044-z

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Keywords

  • Malliavin calculus
  • H-C1 maps
  • Multiplicity
  • Change of variables formula
  • Wiener space
  • Wiener measure
  • Adapted
  • Perturbation of identity
  • Carleman’s inequality
  • Logarithmic sobolev inequality

Mathematics Subject Classifications (2000)

  • 60H07
  • 60H05
  • 60H25
  • 60G15
  • 60G30
  • 60G35
  • 46G12
  • 47H05
  • 47H1
  • 35J60
  • 35B65
  • 35A30
  • 46N10
  • 49Q20
  • 58E12
  • 26A16
  • 28C20
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