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

TL;DR

This paper investigates the durations during which Brownian motion remains within a scaled amplitude, estimates their frequency, and applies these findings to the Kubilius model and prime number divisor functions, refining existing results.

Contribution

It introduces optimal estimates for the occurrence of small amplitude intervals in Brownian motion and applies these to the Kubilius model, enhancing understanding of prime divisor functions.

Findings

Optimal frequency estimates for small amplitude intervals in Brownian motion

Application of results to the Kubilius model for prime number analysis

Refinements of recent results by Ford and Tenenbaum

Abstract

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.