Geodesic
On local times for functions and stochastic processes.~I
Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 4, pp. 798-812

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Let $X(t)$, $0\le t\le 1$, be a real-valued measurable function having a local time $\alpha (t,u)$, $0\le t\le 1$, $u\in\mathbf{R}$. If the latter is continuous in $t$ for a.e. $u$, then the distribution. $F(t,x)=\int_\mathbf{R}\mathbb{I}(\alpha(t,u)>x)\,du$ and the monotone rearrangement $\alpha^*(t,u)=\inf\{x:F(t,x)$ of the local time $\alpha(t,u)$ are the local times for $\xi(s)=\alpha(s,X(s))$ and $\xi^*(s)=F(s,X(s))$, $0\le s\le 1$, respectively.
Keywords: local time, distribution and monotone rearrangement of a function, Brownian motion.
Mots-clés : orthogonal decomposition 
@article{TVP_1995_40_4_a6,
     author = {F. S. Nasyrov},
     title = {On local times for functions and stochastic {processes.~I}},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {798--812},
     publisher = {mathdoc},
     volume = {40},
     number = {4},
     year = {1995},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1995_40_4_a6/}
}
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F. S. Nasyrov. On local times for functions and stochastic processes.~I. Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 4, pp. 798-812. http://geodesic.mathdoc.fr/item/TVP_1995_40_4_a6/