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Let $W(t), t\ge 0$ be standard Brownian motion. We study the size of the timeintervals $I$ which are admissible for the long range of slow increase, namelygiven a real $z>0$, $$ \sup_{t\in I}{|W(t)|\over \sqrt t} \le z, $$ and weestimate their number of occurences. We obtain optimal results in terms ofclass test functions and, by means of the quantitative Borel-Cantelli lemma, afine frequency result concerning their occurences. Using Sakhanenko'sinvariance principe to transfer the results to the Kubilius model, we deriveapplications to the prime number divisor function. We obtain refinements ofsome results recently proved by Ford and Tenenbaum.

Instants of small amplitude of Brownian motion and application to the Kubilius model