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October, 1981 A Global Intrinsic Characterization of Brownian Local Time
Edwin Perkins
Ann. Probab. 9(5): 800-817 (October, 1981). DOI: 10.1214/aop/1176994309


Let $B(t)$ be a Brownian motion with local time $s(t, x)$. Paul Levy showed that for each $x, s(t, x)$ is a.s. equal to the limit as $\delta$ approaches zero of $\delta^{1/2}$ times the number of excursions from $x$, exceeding $\delta$ in length, that are completed by $B$ up to time $t$. The aim of the present paper is to show that the exceptional null sets, which may depend on $x$, can be combined into a single null set off which the above convergence is uniform in $x$. The proof uses nonstandard analysis to construct a simple combinatorial representation for the local time of a Brownian motion constructed by R. M. Anderson.


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Edwin Perkins. "A Global Intrinsic Characterization of Brownian Local Time." Ann. Probab. 9 (5) 800 - 817, October, 1981.


Published: October, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0469.60081
MathSciNet: MR628874
Digital Object Identifier: 10.1214/aop/1176994309

Primary: 02H25
Secondary: 60C05 , 60G45 , 60J55 , 60J65

Keywords: Brownian motion , Local time , nonstandard analysis

Rights: Copyright © 1981 Institute of Mathematical Statistics


Vol.9 • No. 5 • October, 1981
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