Smooth perturbations of a function with a smooth local time

Geman, D.; Horowitz, J.

doi:10.1090/S0002-9947-1981-0626487-X

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.

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