Geodesic
On local times for functions and stochastic processes
Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 2, pp. 284-299

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Let $X(t)$, $0\l t\l 1$, be a real-valued measurable function having a local time $\alpha(t,u)$, $0\l t\l 1$, $u\in\mathbb R$. If the latter is continuous in $t$ for a.a. $u$, then the distribution $F(t,x)=\int_\mathbb R\mathbb{I}\{\alpha(t,u) > x\}\,du$ and the monotone rearrangement $\alpha^*(t,u)=\inf\{x\: F(t,x) u\}$ of the local time $\alpha(t,u)$ are the local times for $\xi(s)=\alpha(s,X(s))$ and $\xi^*(s)=F(s,\xi(s))$, $0\l s\l 1$, respectively.
Keywords: local time, distribution and monotonere arrangement of a function, Brownian motion.
Mots-clés : orthogonal decomposition 
@article{TVP_1996_41_2_a3,
     author = {F. S. Nasyrov},
     title = {On local times for functions and stochastic processes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {284--299},
     publisher = {mathdoc},
     volume = {41},
     number = {2},
     year = {1996},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1996_41_2_a3/}
}
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F. S. Nasyrov. On local times for functions and stochastic processes. Teoriâ veroâtnostej i ee primeneniâ, Tome 41 (1996) no. 2, pp. 284-299. http://geodesic.mathdoc.fr/item/TVP_1996_41_2_a3/