Smooth Perturbations of a Function with a Smooth Local Time
D. Geman, J. Horowitz · Transactions of the American Mathematical Society · 1981
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.