Abstract
We prove the following results about the images and multiple points of an $N$-parameter, $d$-dimensional Brownian sheet $B =\{B(t)\}_{t \in R_+^N}$: (1) If $\text{dim}_H F \leq d/2$, then $B(F)$ is almost surely a Salem set. (2) If $N \leq d/2$, then with probability one $\text{dim}_H B(F) = 2 \text{dim} F$ for all Borel sets of $R_+^N$, where $\text{dim}_H$ could be everywhere replaced by the ``Hausdorff,'' ``packing,'' ``upper Minkowski,'' or ``lower Minkowski dimension.'' (3) Let $M_k$ be the set of $k$-multiple points of $B$. If $N \leq d/2$ and $ Nk > (k-1)d/2$, then $\text{dim}_H M_k = \text{dim}_p M_k = 2 Nk - (k-1)d$, a.s. The Hausdorff dimension aspect of (2) was proved earlier; see Mountford (1989) and Lin (1999). The latter references use two different methods; ours of (2) are more elementary, and reminiscent of the earlier arguments of Monrad and Pitt (1987) that were designed for studying fractional Brownian motion. If $N>d/2$ then (2) fails to hold. In that case, we establish uniform-dimensional properties for the $(N,1)$-Brownian sheet that extend the results of Kaufman (1989) for 1-dimensional Brownian motion. Our innovation is in our use of the sectorial local nondeterminism of the Brownian sheet (Khoshnevisan and Xiao, 2004).
Highlights
Let B = {B(t)}t∈RN+ denote the (N, d)-Brownian sheet
It has been long known that fractional Brownian motion is locally nondeterministic (LND, see Pitt, 1978) whereas the Brownian sheet B is not
It has recently been shown that the Brownian sheet satisfies the following “sectorial” local nondeterminism (Khoshnevisan and Xiao, 2004): Lemma 1.1 (Sectorial LND) Let B0 be an (N, 1)-Brownian sheet
Summary
Khoshnevisan and Xiao (2004) have applied the sectorial LND of the Brownian sheet to study the distributional properties of the level set. First we consider the Fourier dimension of the image B(F ) for a general (N, d)-Brownian sheet, where F ⊂ (0, ∞)N is a fixed Borel set. Consider F to be the zero set B−1(0) The following establishes this uniform dimension result in the non-trivial case that N ≤ d/2.
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