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

Abstract

Let $W(t)$ be a standard Wiener process with local time $L(x, t)$. Put $L(t) = L(0, t)$ and $L^\ast(t) = \sup_{-\infty < x < \infty} L(x, t)$. We study the almost sure behaviour of small increments of $L(t)$ and also, the joint behaviour of $L(t)$ and the last excursion, $U(t)$. The increment problem of $L(x, t)$ are also studied uniformly in $x$. This implies a $\lim \inf$-type law of the iterated logarithm for $L^\ast(t)$ due to Kesten (1965), in which case the exact constant, not known before, is also determined.