A new class of random multiplicative and statistically self-similar measures is defned on IR. It is the limit of measure-valued martingales constructed by multiplying random functions attached to the points of a statistically self-similar Poisson point process in a strip of the plane. Several...

We show that DLA follows a surprising new scaling rule. It expresses that the screened region, in which the harmonic measure is tiny, increases more than proportionately as the cluster grows. This scaling rule also gives indirect evidence that the harmonic measure of lattice DLA follows a...

We have constructed diffusion limited aggregates off-lattice, that is, using Brownian motion instead of random walk, and the equipotential and the current lines have also been computed in the continuous plane. This construction (describes as being off off-lattice) is illustrated for the...

A sharp distinction is drawn between general multiplicative multifractals, as originally introduced by the author, and the more familiar but more restricted class defined by Frisch and Parisi and by Halsey et al. All the general multiplicative multifractals are exactly renormalizable, by design....

Using two new methods of geometric analysis, this paper establishes that DLA clusters are definitely not self-similar. Compared to small clusters, the morphology of large clusters (of sizes up to 30 million particles) can be characterized, both visually and quantitatively, as being far more...

In diverse sciences that lack Hamiltonians, the analysis of complex systems is helped by the powerful tools provided by renormalization, fixed points and scaling. As one example, an intrinsic form of exact renormalizability was long used by the author in economics and related fields, most...

The right-hand side of the ƒ(α) curve of the harmonic measure on DLA is undefined. This does not necessarily imply that the harmonic measure and the DLA geometry are not self-similar. We show for off-lattice DLA that the right-hand tail satisfies a different rescaling rule. This Cauchy...

This short paper advances and defends a strong statement concerning financial modeling. It argues that, even when the present fractal models become superseded, fractal tools are bound to remain central to finance. The reasons are that the main feature of price records is roughness and that the...