Hausdorff dimension of graphs of one-dimensional Brownian motion First question here, my apologies if it is a duplicate or inappropriate.
There is a page on Wikipedia listing fractals by Hausdorff dimension and it includes the graph of a "regular Brownian function" as having Hausdorff dimension 1.5. Does this mean that a Brownian motion has Hausdorff dimension almost always, or is it the expected value, or something else?
I've tried to look for references (besides the Wikipedia one that I can't get) and googling for more information to little avail.
If it's not true that the graph of a Brownian motion has Hausdorff dimension 1.5 almost always, is it true that the Hausdorff dimension >1 almost always?

Yes, the Hausdorff dimension of the graph is 3/2 almost surely. 
A good explanation can be found in Chapter 4 of Peres and Morters's 
book on Brownian motion. See Theorem 4.29 on page 110.
