A Global Intrinsic Characterization of Brownian Local Time

Abstract

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.