Resumen de Measuring the range of an additive Lévy process

Davar Khoshnevisan, Yimin Xiao, Yuquan Zhong

• The primary goal of this paper is to study the range of the random field X(t)=∑Nj=1Xj(tj), where X1,…,XN\vspace*{-1pt} are independent Lévy processes in \Rd.

  To cite a typical result of this paper, let us suppose that Ψi denotes the Lévy exponent of Xi for each i=1,…,N. Then, under certain mild conditions, we show that a necessary and sufficient condition for X(\RN+) to have positive d-dimensional Lebesgue measure is the integrability of the function \Rd∋ξ↦∏Nj=1R{1+Ψj(ξ)}−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.