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Transactions of the American Mathematical Society

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Smooth perturbations of a function with a smooth local time


Authors: D. Geman and J. Horowitz
Journal: Trans. Amer. Math. Soc. 267 (1981), 517-530
MSC: Primary 60J55; Secondary 26A45, 28A15
DOI: https://doi.org/10.1090/S0002-9947-1981-0626487-X
MathSciNet review: 626487
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Abstract: A real Borel function on $ [0,\,1]$ has a local time if its occupation measure up to each time $ t$ (equivalently: its increasing, equimeasurable rearrangement on $ [0,\,t]$) is absolutely continuous; the local time $ {\alpha _t}(x)$ is then the density. An inverse relationship exists between the smoothness of the local time in $ (t,\,x)$ and that of the original function. The sum of a function with a smooth local time and a well-behaved (e.g. absolutely continuous) function is shown to have a local time, which inherits certain significant properties from the old local time, and for which an explicit formula is given. Finally, using a probabilistic approach, examples are given of functions having local times of prescribed smoothness.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1981-0626487-X
Keywords: Occupation measures, local time, functions of a real variable, bounded variation, Gaussian process
Article copyright: © Copyright 1981 American Mathematical Society

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