Multiple points of trajectories of multiparameter fractional Brownian motion
成果类型：
Article
署名作者：
Talagrand, M
署名单位：
Sorbonne Universite; Centre National de la Recherche Scientifique (CNRS); University System of Ohio; Ohio State University
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s004400050200
发表日期：
1998
页码：
545-563
关键词：
摘要：
Consider 0 < alpha < 1 and the Gaussian process Y(t) on R-N with covariance E(Y(s)Y(t)) = \t\(2 alpha) + \s\(2 alpha) - \t - s\(2 alpha), where \t\ is the Euclidean norm of t. Consider independent copies X-1,...,X-d of Y and the process X(t) = (X-1(t),...,X-d(t)) valued in R-d. When kN less than or equal to (k - 1)alpha d, we show that the trajectories of X do not have k-multiple points. If N < alpha d and kN > (k - 1)alpha d, the set of k-multiple points of the trajectories X is a countable union of sets of finite Hausdorff measure associated with the function phi(epsilon) = epsilon(kN/alpha-(k-1)d) (log log(1/epsilon))(k). If N = alpha d, we show that the set of k-multiple points of the trajectories of X is a countable union of sets of finite Hausdorff measure associated with the function phi(epsilon) = epsilon(d)(log(1/epsilon) log log log 1/epsilon)(k). (This includes the case k = 1.).
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