Geometric properties of 2-dimensional Brownian paths

Geometric Properties of 2-dimensional Brownian Paths

Let A be the set of all points of the plane C, visited by 2-dimensional Brownian motion before time 1. With probability 1, all points of A are “twist points” except a set of harmonic measure zero. “Twist points” may be continuously approached in C \ A only along a special spiral. Although negligible in the sense of harmonic measure, various classes of “cone points” are dense in A, with probabil...

Intersection properties of Brownian paths

This review presents a modern approach to intersections of Brownian paths. It exploits the fundamental link between intersection properties and percolation processes on trees. More precisely, a Brownians path is intersect-equivalent to certain fractal percolation. It means that the intersection probabilities of Brownian paths can be estimated up to constant factors by survival probabilities of ...

Branching Properties of Brownian Paths and Trees

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...

Geometric Properties of Fractional Brownian Sheets

Let B = B(t), t ∈ R+ be an (N, d)-fractional Brownian sheet with Hurst index H = (H1, . . . , HN ) ∈ (0, 1) . Our objective of the present paper is to characterize the anisotropic nature of B in terms of H. We prove the following results: (1) B is sectorially locally nondeterministic. (2) By introducing a notion of “dimension” for Borel measures and sets, which is suitable for describing the an...