Images of Gaussian random fields: Salem sets and interior points
Volume 176 / 2006
Studia Mathematica 176 (2006), 37-60
MSC: Primary 60G15, 60G17; Secondary 60G60, 42B10, 43A46, 28A80.
DOI: 10.4064/sm176-1-3
Abstract
Let $X = \{X(t),\, t \in \mathbb R^N\}$ be a Gaussian random field in $\mathbb R^d$ with stationary increments. For any Borel set $E \subset \mathbb R^N$, we provide sufficient conditions for the image $X(E)$ to be a Salem set or to have interior points by studying the asymptotic properties of the Fourier transform of the occupation measure of $X$ and the continuity of the local times of $X$ on $E$, respectively. Our results extend and improve the previous theorems of Pitt \cite{Pitt78} and Kahane \cite{Kahane85a, Kahane85b} for fractional Brownian motion.
