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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?

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up vote 6 down vote accepted

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.

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Thanks! I actually found that book, but just searching for "Hausdorff dimension" didn't bring up the theorem. I should be more thorough in my skimming. – minimalrho Jul 21 '11 at 3:07
@minimalrho Glad to be of help! – Byron Schmuland Jul 21 '11 at 3:14

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