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Differentiability of Loeb measures

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The Strength of Nonstandard Analysis

Abstract

We introduce a general definition of S-differentiability of an internal measure and compare different special cases. It will be shown how S-differentiability of an internal measure yields differentiability of the associated Loeb measure. We give some examples.

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References

  1. E. Aigner, Differentiability of Loeb measures and applications, PhD thesis, Universität München, (in prep.)

    Google Scholar 

  2. S. Albeverio, J.E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.

    MATH  Google Scholar 

  3. V.I. Averbukh, O.G. Smolyanov and S.V. Fomin, “Generalized functions and differential equations in linear spaces, I. Differentiable measures”, Trudi of Moscow Math. Soc., 24 (1971) 140–184.

    Google Scholar 

  4. V.I. Bogachev, “Differential properties of measures on infinite dimensional spaces and the Malliavin calculus”, Acta Univ. Carolinae, Math. Phys., 30 (1989) 9–30.

    MATH  MathSciNet  Google Scholar 

  5. V.I. Bogachev, “Smooth Measures, the Malliavin Calculus and Approximations in Infinite Dimensional Spaces”, Acta Univ. Carolinae, Math. Phys., 31 (1990) 9–23.

    MATH  MathSciNet  Google Scholar 

  6. V.I. Bogachev, “Differentiable Measures and the Malliavin Calculus”, Journal of Mathematical Sciences, 87 (1997) 3577–3731.

    Article  MATH  MathSciNet  Google Scholar 

  7. N. Cutland, Loeb Measures in Practice: Recent Advances, Lecture Notes in Mathematics 1751, Springer-Verlag, 2000.

    Google Scholar 

  8. N. Cutland and S.-A. Ng, A nonstandard approach to the Malliavin calculus, in Applications of Nonstandard-Analysis to Analysis, Functional Analysis Propability Theory and Mathematical Physics (eds. S. Albeverio, W.A.J. Luxemburg and M. Wolff), D.Reidel-Kluwer, Dordrecht, 1995.

    Google Scholar 

  9. S.V. Fomin, Differential measures in linear spaces, in Proc. Int. Congr. of Mathematicians. Sec.5, Izd. Mosk. Univ., Moscow, 1966.

    Google Scholar 

  10. A.V. Kirillov, “Infinite-dimensional analysis and quantum theory as semimartingale calculus”, Russian Math. Surveys, No. 3 (1994) 41–95.

    Google Scholar 

  11. D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995.

    Google Scholar 

  12. H. Osswald, Malliavin calculus in abstract Wiener spaces. An introduction, Book manuscript, 2001.

    Google Scholar 

  13. A.V. Skorohod, Integration in Hilbert Spaces, Ergebnisse der Mathematik, Springer-Verlag, Berlin, New-York, 1974.

    Google Scholar 

  14. O.G. Smolyanov and H.V. Weizsäcker, “Formulae with logarithmic derivatives of measures related to the quantization of infinite-dimensional Hamilton systems”, Russian Math. Surveys, 551 (1996) 357–358.

    Google Scholar 

  15. O.G. Smolyanov and H.V. Weizsäcker, “Smooth probability measures and associated differential operators”, Inf. Dim. Analysis, Quantum Prob. and Rel. Topics, 2 (1999) 51–78.

    Article  MATH  Google Scholar 

  16. H.V. Weizsäcker, Differenzierbare Maße und Stochastische Analysts, 6 Vorträge am Graduiertenkolleg’ stochastische Prozesse und probabilistische Analysis’. Januar/Februar, Berlin, 1998.

    Google Scholar 

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© 2007 Springer-Verlag Wien

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Aigner, E. (2007). Differentiability of Loeb measures. In: van den Berg, I., Neves, V. (eds) The Strength of Nonstandard Analysis. Springer, Vienna. https://doi.org/10.1007/978-3-211-49905-4_17

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