Effective dimension of points visited by Brownian motion

TL;DR

This paper investigates the effective dimension of points on Brownian motion paths, showing that Khintchine's law applies at almost all points and identifying points with effective dimension less than 1.

Contribution

It demonstrates that Khintchine's law holds at almost all points on Brownian paths and finds points with effective dimension below 1, beyond the trivial origin case.

Findings

Khintchine's law of the iterated logarithm holds at almost all points

Existence of points with effective dimension less than 1

For almost all times, the path is Martin-Löf random relative to time

Abstract

We consider the individual points on a Martin-L\"of random path of Brownian motion. We show (1) that Khintchine's law of the iterated logarithm holds at almost all points; and (2) there exist points (besides the trivial example of the origin) having effective dimension $<1$. The proof of (1) shows that for almost all times $t$, the path $f$ is Martin-L\"of random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.