How Big are the Increments of the Local Time of a Wiener Process?

Abstract

Let $W(t)$ be a standard Wiener process with local time (occupation density) $L(x, t)$. Kesten showed that $L(0, t)$ and $\sup_x L(x, t)$ have the same LIL law as $W(t)$. Exploiting a famous theorem of P. Levy, namely that $\{\sup_{0 \leq s \leq t} W(s), t \geq 0\} {\underline{\underline{\mathscr{D}}}} \{L(0, t), t \geq 0\}$, we study the almost sure behaviour of big increments of $L(0, t)$ in $t$. The very same increment problems in $t$ of $L(x, t)$ are also studied uniformly in $x$. The results in the latter case are slightly different from those concerning $L(0, t)$, and they coincide only for Kesten's above mentioned LIL.