Images of the Brownian sheet

Abstract

An N N -parameter Brownian sheet in R d \mathbf {R}^d maps a non-random compact set F F in R + N \mathbf {R}^N_+ to the random compact set B ( F ) B(F) in R d \mathbf {R}^d . We prove two results on the image-set B ( F ) B(F) : (1) It has positive d d -dimensional Lebesgue measure if and only if F F has positive d 2 \frac d 2 -dimensional capacity. This generalizes greatly the earlier works of J. Hawkes  (1977), J.-P. Kahane  (1985), and Khoshnevisan (1999). (2) If dim H ⁡ F > d 2 \dim _{_\mathcal {H}}F > \frac d 2 , then with probability one, we can find a finite number of points ζ 1 , … , ζ m ∈ R d \zeta _1,\ldots ,\zeta _m\in \mathbf {R}^d such that for any rotation matrix θ \theta that leaves F F in R + N \mathbf {R}^N_+ , one of the ζ i \zeta _i ’s is interior to B ( θ F ) B(\theta F) . In particular, B ( F ) 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).