×

Sectorial local non-determinism and the geometry of the Brownian sheet. (English) Zbl 1111.60020

Summary: We prove the following results about the images and multiple points of an \(N\)-parameter, \(d\)-dimensional Brownian sheet \(B=\{B(t)\}_{t\in \mathbb{R}_+^N}\):
(1) If \(\dim_HF\leq d/2\), then \(B(F)\) is almost surely a Salem set.
(2) If \(N\leq d/2\), then with probability one \[ \dim_HB(F) =2\dim_HF\quad\text{for all Borel sets }F\subset\mathbb{R}_+^N, \] where “\(\dim_H\)” could be everywhere replaced by the “Hausdorff”, “packing”, “upper Minkowski”, or “lower Minkowski dimension”.

MSC:

60G15 Gaussian processes
28A80 Fractals
60G17 Sample path properties
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI EuDML