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      Entropy, invertibility and variational calculus of adapted shifts on Wiener space

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      Author
      Üstünel, A.S.
      Date
      2009
      Source Title
      Journal of Functional Analysis
      Print ISSN
      0022-1236
      Electronic ISSN
      1096-0783
      Volume
      257
      Issue
      11
      Pages
      3655 - 3689
      Language
      English
      Type
      Article
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      Abstract
      In this work we study the necessary and sufficient conditions for a positive random variable whose expectation under the Wiener measure is one, to be represented as the Radon-Nikodym derivative of the image of the Wiener measure under an adapted perturbation of identity with the help of the associated innovation process. We prove that the innovation conjecture holds if and only if the original process is almost surely invertible. We also give variational characterizations of the invertibility of the perturbations of identity and the representability of a positive random variable whose total mass is equal to unity. We prove in particular that an adapted perturbation of identity U = IW + u satisfying the Girsanov theorem, is invertible if and only if the kinetic energy of u is equal to the entropy of the measure induced with the action of U on the Wiener measure μ, in other words U is invertible ifffrac(1, 2) under(∫, W) | u |H 2 d μ = under(∫, W) frac(d U μ, d μ) log frac(d U μ, d μ) d μ . The relations with the Monge-Kantorovitch measure transportation are also studied. An application of these results to a variational problem related to large deviations is also given. © 2009 Elsevier Inc. All rights reserved.
      Keywords
      Calculus of variations
      Entropy
      Invertibility
      Large deviations
      Malliavin calculus
      Monge transportation
      Permalink
      http://hdl.handle.net/11693/22519
      Published Version (Please cite this version)
      http://dx.doi.org/10.1016/j.jfa.2009.03.015
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