Smooth perturbations of a function with a smooth local time
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- by D. Geman and J. Horowitz PDF
- Trans. Amer. Math. Soc. 267 (1981), 517-530 Request permission
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
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Additional Information
- © Copyright 1981 American Mathematical Society
- 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