## 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
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