Images of the Brownian sheet

Abstract:

An $N$-parameter Brownian sheet in $\mathbf {R}^d$ maps a non-random compact set $F$ in $\mathbf {R}^N_+$ to the random compact set $B(F)$ in $\mathbf {R}^d$. 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 $\frac d 2$-dimensional capacity. This generalizes greatly the earlier works of J. Hawkes (1977), J.-P. Kahane (1985), and Khoshnevisan (1999). (2) If $\dim _{_\mathcal {H}}F > \frac d 2$, then with probability one, we can find a finite number of points $\zeta _1,\ldots ,\zeta _m\in \mathbf {R}^d$ such that for any rotation matrix $\theta$ that leaves $F$ in $\mathbf {R}^N_+$, one of the $\zeta _i$’s is interior to $B(\theta F)$. 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).