Measuring the range of an additive Levy process

• 成果类型: Article
• 署名作者: Khoshnevisan, D; Xiao, YM; Zhong, YQ
• 署名单位: Utah System of Higher Education; University of Utah; Michigan State University; Chinese Academy of Sciences
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 1097-1141
• 关键词: stationary independent increments; potential-theory; multiparameter processes; Hausdorff Dimension; markov properties; stable processes; sets; indexes

摘要

The primary goal of this paper is to study the range of the random field X (t) = Sigma(j=1)(N) X-j (t(j)), where X-1,...,X-N are independent Levy processes in R-d. To cite a typical result of this paper, let us suppose that psi(i) denotes the Levy exponent of X-i 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 R-d There Exists xi bar right arrow Pi(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 Levy processes initiated in a companion by paper Khoshnevisan, Xiao and Zhong.