## Weierstrass' function and Brownian motion

Is there a known connection between Weierstrass' function

$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$

and Brownian motion? Specifically, when $\alpha = 1/2$, the Weierstrass function has same Holder continuity peoperties that Brownain sample paths do. Some crude Matlab experiments seem to suggest this function has linear quadratic variation for low values of $b$, too.

The expression reminds me a little of the Karhunen-loeve expansion of Brownian motion, but I don't see how the two might relate.

Many thanks.

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I doubt there is a direct connection other than both functions sharing similar properties. There are many functions that have Hölder exponent 1/2 everywhere. I note that it is a well-known open problem to compute the Hausdorff dimension of the graph of the Weierstrass function, while for Brownian images this is well-known to be $3/2$ (this is the conjectured value for the Weierstrass graph when $\alpha=1/2$). This tells us that we don't really understand the shape of the Weierstrass function, while for Brownian motion even very fine geometric information is known. – Pablo Shmerkin Mar 21 2011 at 19:13
@Pablo - It is worth noting that all those fine path properties that are known for Brownian motion come with the 'almost surely' caveat. So it is not surprising that for a specific path candidate less might be known. – BSteinhurst Mar 21 2011 at 20:09
Hi, I thought Brownian Motion had paths with Hölder exponent strictly inferior to 1/2 almost surely ? – The Bridge Mar 21 2011 at 20:29
do random fourier series also converge to brownian motion? – John Mangual Mar 22 2011 at 2:34

Some quick Googling brought me to this paper. The idea is to take the coefficients in the summation to be suitable independent random variables according to a suggestion of Mandelbrot. I can't actually access the paper right now so I can't say if the authors were able to include the case of standard Brownian motion in their results.

Convergence of the Weierstrass-Mandelbrot process to Fractional Brownian Motion'' Murad Taqqu and Vladas Pipiras. Fractals. 8 (2000) 369-384.

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You might find this account useful. In particular, see the end of Section 1, page 7, and the end of Section 4.

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