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.