  Let $W(t), t\ge 0$ be standard Brownian motion. We study the size of the time
intervals $I$ which are admissible for the long range of slow increase, namely
given a real $z>0$, $$ \sup_{t\in I}{|W(t)|\over \sqrt t} \le z, $$ and we
estimate their number of occurences. We obtain optimal results in terms of
class test functions and, by means of the quantitative Borel-Cantelli lemma, a
fine frequency result concerning their occurences. Using Sakhanenko's
invariance principe to transfer the results to the Kubilius model, we derive
applications to the prime number divisor function. We obtain refinements of
some results recently proved by Ford and Tenenbaum.
