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Images of the Brownian sheet. (English) Zbl 1124.60037

Authors’ abstract: An \(N\)-parameter Brownian sheet in \(\mathbb R^{d}\) maps a non-random compact set\(F\) in \(\mathbb R_{+}^{N}\) to the randomcompact set \(B(F)\) in \(\mathbb R^{d}\). 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 [Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 309–315 (1977; Zbl 0385.60043)], J.-P. Kahane [“Some random series of functions”, Cambridge: Cambridge University Press. (1993; Zbl 0805.60007) and in: Recent progress in Fourier analysis. North-Holland Math. Stud. 111, 65–121 (1985; Zbl 0596.60075)], and D. Khoshnevisan [Trans. Am. Math. Soc. 351, No. 7, 2607–2622 (1999; Zbl 0930.60055)].
(2) If \(\dim _{\varkappa }F>\frac{d}{2}\), then with probability one, we can find a finite number of points \(\zeta _{1},\dots,\zeta _{m}\in \mathbb R^{d}\) such that for any rotation matrix \(\theta \) that leaves \(F\) in \(\mathbb R_{+}^{N}\), 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 [Bull. Lond. Math. Soc. 21, No. 2, 179–185 (1989; Zbl 0668.60044)].
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 T. S. Mountford [Stochastics 23, No. 4, 449–464 (1988; Zbl 0645.60086)].

MSC:

60G15 Gaussian processes
60G17 Sample path properties
28A80 Fractals
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References:

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