We outline the construction of Brownian motion on certain self-similar fractals and introduce the notion of walk dimension. We then show how the probabilistic approach relates to the theory of function spaces on fractals.

The work developed in the paper concerns the multivariate fractional Brownian motion (mfBm) viewed through the lens of the wavelet transform. After recalling some basic properties on the mfBm, we calculate the correlation structure of its wavelet transform. We particularly study the asymptotic behaviour of the correlation, showing that if the analyzing wavelet has a sufficient number of null first order moments, the decomposition eliminates any possible long-range (inter)dependence. The cross-spectral...

Let $E$ be a Banach (or quasi-Banach) space which is shift and scaling invariant (typically a homogeneous Besov or Sobolev space). We introduce a general definition of pointwise regularity associated with $E$, and denoted by $C_E^{\alpha }\left(x_0\right)$. We show how properties of $E$ are transferred into properties of $C_E^{\alpha }\left(x_0\right)$. Applications are given in multifractal analysis.

We show that there are Hilbert spaces constructed from the Hausdorff measures Hs on the real line R with 0 < s < 1 which admit multiresolution wavelets. For the case of the middle-third Cantor set C ⊂ [0,1], the Hilbert space is a separable subspace of L2(R, (dx)s) where s = log3(2). While we develop the general theory of multiresolutions in fractal Hilbert spaces, the emphasis is on the case of scale 3 which covers the traditional Cantor set C.

Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in $C^1[0,1]$ is one-to-one on $F$? If ${\underline{dim}}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B}F=1/2$ for which the answer is no. If $C_{\alpha }$ is the middle-$\alpha$ Cantor set we prove that the answer is yes if and only if $dim\left(C_{\alpha }\right)\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.