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
E. Aigner, Differentiability of Loeb measures and applications, PhD thesis, Universität München, (in prep.)
S. Albeverio, J.E. Fenstad, R. Høegh-Krohn and T. Lindstrøm, Nonstandard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York, 1986.
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.
V.I. Bogachev, “Differential properties of measures on infinite dimensional spaces and the Malliavin calculus”, Acta Univ. Carolinae, Math. Phys., 30 (1989) 9–30.
V.I. Bogachev, “Smooth Measures, the Malliavin Calculus and Approximations in Infinite Dimensional Spaces”, Acta Univ. Carolinae, Math. Phys., 31 (1990) 9–23.
V.I. Bogachev, “Differentiable Measures and the Malliavin Calculus”, Journal of Mathematical Sciences, 87 (1997) 3577–3731.
N. Cutland, Loeb Measures in Practice: Recent Advances, Lecture Notes in Mathematics 1751, Springer-Verlag, 2000.
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.
S.V. Fomin, Differential measures in linear spaces, in Proc. Int. Congr. of Mathematicians. Sec.5, Izd. Mosk. Univ., Moscow, 1966.
A.V. Kirillov, “Infinite-dimensional analysis and quantum theory as semimartingale calculus”, Russian Math. Surveys, No. 3 (1994) 41–95.
D. Nualart, The Malliavin Calculus and Related Topics, Probability and its Applications, Springer-Verlag, 1995.
H. Osswald, Malliavin calculus in abstract Wiener spaces. An introduction, Book manuscript, 2001.
A.V. Skorohod, Integration in Hilbert Spaces, Ergebnisse der Mathematik, Springer-Verlag, Berlin, New-York, 1974.
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.
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.
H.V. Weizsäcker, Differenzierbare Maße und Stochastische Analysts, 6 Vorträge am Graduiertenkolleg’ stochastische Prozesse und probabilistische Analysis’. Januar/Februar, Berlin, 1998.
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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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DOI: https://doi.org/10.1007/978-3-211-49905-4_17
Publisher Name: Springer, Vienna
Print ISBN: 978-3-211-49904-7
Online ISBN: 978-3-211-49905-4
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