Effective dimension of points visited by Brownian motion

Abstract

We consider the individual points on a Martin-Löf 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-Löf random relative to $t$ and so the effective dimension of $(t,f(t))$ is 2.

The conference version was published as: The law of the iterated logarithm for algorithmically random paths of Brownian motion, Logical Foundations of Computer Science, Lecture Notes in Computer Science 4514 (2007), 310--317

Cited by in corpus (5)

• Martin-Löf randomness and Galton-Watson processes
• Kolmogorov complexity and strong approximation of Brownian motion
• Fourier spectra of measures associated with algorithmically random Brownian motion
• Algorithmic randomness for Doob's martingale convergence theorem in continuous time
• Diophantine properties of Brownian motion: recursive aspects