Restrictions of Brownian motion

TL;DR

This paper investigates the restrictions on the size of subsets of the domain of Brownian motion where the process can be Hölder continuous or have finite variation, establishing almost sure bounds on such sets' dimensions.

Contribution

It proves almost sure bounds on the Hausdorff dimension of sets where Brownian motion exhibits Hölder continuity or finite variation, using Kaufman's dimension doubling theorem.

Findings

No set with Hausdorff dimension greater than 1/2 allows Hölder continuity of Brownian motion.

No set with Hausdorff dimension greater than β/2 admits finite β-variation of Brownian motion.

Zero set and deterministic constructions demonstrate the optimality of these bounds.

Abstract

Let $\{ B(t) \colon 0\leq t\leq 1\}$ be a linear Brownian motion and let $\dim$ denote the Hausdorff dimension. Let $\alpha>\frac12$ and $1\leq \beta \leq 2$. We prove that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac12$ and $B\colon A\to\mathbb{R}$ is $\alpha$-H\"older continuous. The proof is an application of Kaufman's dimension doubling theorem. As a corollary of the above theorem, we show that, almost surely, there exists no set $A\subset[0,1]$ such that $\dim A>\frac{\beta}{2}$ and $B\colon A\to\mathbb{R}$ has finite $\beta$-variation. The zero set of $B$ and a deterministic construction witness that the above theorems give the optimal dimensions.