Let $L$ be a local time. It is well known that there exist a law of the iterated logarithm and a modulus of continuity for $L$. Motivated by the case of real Brownian motion, we study the existence of fast points and slow points of $L$. We prove the existence of such points by considering the right-continuous inverse of $L$, which is a subordinator.
"Slow Points and Fast Points of Local Times." Ann. Probab. 27 (1) 150 - 165, January 1999. https://doi.org/10.1214/aop/1022677257