Measuring the range of an additive Lévy process

Abstract

The primary goal of this paper is to study the range of the random field $X(t) = \sum_{j=1}^N X_j(t_j)$, where $X_1,\ldots, X_N$\vspace*{-1pt} are independent Lévy processes in $\R^d$. To cite a typical result of this paper, let us suppose that $\Psi_i$ denotes the Lévy exponent of $X_i$ for each $i=1,\ldots,N$. Then, under certain mild conditions, we show that a necessary and sufficient condition for $X(\R^N_+)$ to have positive $d$-dimensional Lebesgue measure is the integrability of the function $\R^d \ni \xi \mapsto \prod_{j=1}^N \Re \{ 1+ \Psi_j(\xi)\}^{-1}$. This extends a celebrated result of Kesten and of Bretagnolle in the one-parameter setting. Furthermore, we show that the existence of square integrable local times is yet another equivalent condition for the mentioned integrability criterion. This extends a theorem of Hawkes to the present random fields setting and completes the analysis of local times for additive Lévy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.