Sectorial local non-determinism and the geometry of the brownian sheet

Abstract

We prove the following results about the images and multiple points of an N-parameter, d-dimensional Brownian sheet B = (B(t))t∈ℝN+ : (1) If dimH F ≤ d/2, then B(F) is almost surely a Salem set. (2) If N ≤ d/2, then with probability one dimH B(F) = 2dimH F for all Borel sets F ⊂ ℝN+, where “dimH” could be everywhere replaced by the “Hausdorff,” “packing,” “upper Minkowski,” or “lower Minkowski dimension.” (3) Let Mk be the set of k-multiple points of B. If N ≤ d/2 and Nk > (k - 1)d/2, then dimHMk = dimPMk = 2Nk - (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). © 2006 Applied Probability Trust.