The Frontier of a Brownian Path Is Multifractal

The Frontier of a Brownian

We consider the multifractal spectrum of harmonic measure of a Brownian motion path in two or three dimensions. We show that the multifractal spectrum is nontrivial and relate the spectrum to the intersection exponent. As a corollary we show that harmonic measure on a three dimension Brownian motion path is carried on a set of Hausdorr dimension strictly less than two.

Multifractal Nature of Two Dimensional Simple Random Walk Paths

The multifractal spectrum of discrete harmonic measure of a two dimensional simple random walk path is considered. It is shown that the spectrum is the same as for Brownian motion, is nontrivial, and can be given in terms of a quantity known as the intersection exponent.

A simple path to asymptotics for the frontier of a branching Brownian motion

Fine regularity of Lévy processes and linear (multi)fractional stable motion

In this work, we investigate the fine regularity of Lévy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Lévy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alp...

Geometric and Fractal Properties of Brownian Motion and Random Walk Paths in Two and Three Dimensions

There is a close relationship between critical exponents for proa-bilities of events and fractal properties of paths of Brownian motion and random walk in two and three dimensions. Cone points, cut points, frontier points, and pioneer points for Brownian motion are examples of sets whose Hausdorr dimension can be given in terms of corresponding exponents. In the latter three cases, the exponent...