Images of the Brownian Sheet

An N-parameter Brownian sheet in Rd maps a non-random compact set F in R+ to the random compact set B(F ) in R d. We prove two results on the image-set B(F ): (1) It has positive d-dimensional Lebesgue measure if and only if F has positive d 2 -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985), and Khoshnevisan (1999). (2) If dimH F > d 2 , then with probability one, we can find a finite number of points ζ1, . . . , ζm ∈ Rd such that for any rotation matrix θ that leaves F in R+ , one of the ζi’s is interior to B(θF ). In particular, B(F ) has interior-points a.s. This verifies a conjecture of T. S. Mountford (1989). This paper contains two novel ideas: To prove (1), we introduce and analyze a family of bridged sheets. Item (2) is proved by developing a notion of “sectorial local-non-determinism (LND).” Both ideas may be of independent interest. We showcase sectorial LND further by exhibiting some arithmetic properties of standard Brownian motion; this completes the work initiated by Mountford (1988).