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An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions

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Convexity and Concentration

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 161))

Abstract

One way to interpret smoothness of a measure in infinite dimensions is quasi-invariance of the measure under a class of transformations. Usually such settings lack a reference measure such as the Lebesgue or Haar measure, and therefore we cannot use smoothness of a density with respect to such a measure. We describe how a functional inequality can be used to prove quasi-invariance results in several settings. In particular, this gives a different proof of the classical Cameron-Martin (Girsanov) theorem for an abstract Wiener space. In addition, we revisit several more geometric examples, even though the main abstract result concerns quasi-invariance of a measure under a group action on a measure space.

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Acknowledgements

The author is grateful to Sasha Teplyaev and Tom Laetsch for useful discussions and helpful comments.

This research was supported in part by NSF Grant DMS-1007496.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Maria Gordina

Authors
  1. Maria Gordina
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Correspondence to Maria Gordina .

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Editors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, New Jersey, USA

    Eric Carlen

  2. Department of Mathematical Sciences, University of Delaware, Newark, Delaware, USA

    Mokshay Madiman

  3. Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, USA

    Elisabeth M. Werner

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Gordina, M. (2017). An Application of a Functional Inequality to Quasi-Invariance in Infinite Dimensions. In: Carlen, E., Madiman, M., Werner, E. (eds) Convexity and Concentration. The IMA Volumes in Mathematics and its Applications, vol 161. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-7005-6_8

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