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Percolation Dimension of Brownian Motion in $R^3$

   
@article{ECP993,
	author = {Chad Fargason},
	title = {Percolation Dimension of Brownian Motion in $R^3$},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {3},
	year = {1998},
	keywords = {Percolation dimension, boundary dimension, intersection exponent},
	abstract = {Let $B(t)$ be a Brownian motion in $R^3$. A subpath of the Brownian path $B[0,1]$ is a continuous curve $\gamma(t)$, where $\gamma[0,1] \subseteq B[0,1]$ , $\gamma(0) = B(0)$, and $\gamma(1) = B(1)$. It is well-known that any subset $S$ of a Brownian path must have Hausdorff dimension $\text{dim} (S) \leq 2.$ This paper proves that with probability one there exist subpaths of $B[0,1]$ with Hausdorff dimension strictly less than 2. Thus the percolation dimension of Brownian motion in $R^3$ is strictly less than 2.},
	pages = {no. 7, 51-63},
	issn = {1083-589X},
	doi = {10.1214/ECP.v3-993},    
        url = {http://ecp.ejpecp.org/article/view/993}}