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.
Citation
E. Csaki. M. Csorgo. A. Foldes. P. Revesz. "How Big are the Increments of the Local Time of a Wiener Process?." Ann. Probab. 11 (3) 593 - 608, August, 1983. https://doi.org/10.1214/aop/1176993504
Information