Weierstrass' function and Brownian motion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T12:56:21Z http://mathoverflow.net/feeds/question/59089 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/59089/weierstrass-function-and-brownian-motion Weierstrass' function and Brownian motion Simon Lyons 2011-03-21T18:00:18Z 2011-03-21T19:21:25Z <p>Is there a known connection between <a href="http://en.wikipedia.org/wiki/Weierstrass_function" rel="nofollow">Weierstrass' function</a> </p> <p>$W_\alpha (x) = \sum_{n=0}^\infty b^{- n \alpha} \cos(b^n x)$ </p> <p>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.</p> <p>The expression reminds me a little of the Karhunen-loeve expansion of Brownian motion, but I don't see how the two might relate.</p> <p>Many thanks.</p> http://mathoverflow.net/questions/59089/weierstrass-function-and-brownian-motion/59097#59097 Answer by BSteinhurst for Weierstrass' function and Brownian motion BSteinhurst 2011-03-21T19:20:50Z 2011-03-21T19:20:50Z <p>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. </p> <p>Convergence of the Weierstrass-Mandelbrot process to Fractional Brownian Motion'' Murad Taqqu and Vladas Pipiras. Fractals. 8 (2000) 369-384. </p> http://mathoverflow.net/questions/59089/weierstrass-function-and-brownian-motion/59098#59098 Answer by Shai Covo for Weierstrass' function and Brownian motion Shai Covo 2011-03-21T19:21:25Z 2011-03-21T19:21:25Z <p>You might find <a href="http://portail.mathdoc.fr/PMO/PDF/K_KAHANE-69.pdf" rel="nofollow">this account</a> useful. In particular, see the end of Section 1, page 7, and the end of Section 4.</p>