How fast are the particles of super-Brownian motion?

• 成果类型: Article
• 署名作者: Mörters, P
• 署名单位: University of Kaiserslautern
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/PL00008801
• 发表日期: 2001
• 页码: 171-197
• 关键词: support; dimension; points; trees; path

摘要

In this paper we investigate fast particles in the range and support of super-Brownian motion in the historical setting. In this setting each particle of super-Brownian motion alive at time t is represented by a path w : [0, t] --> R-d and the state of historical super-Brownian motion is a measure on the set of paths. Typical particles have Brownian paths, however in the uncountable collection of particles in the range of a super-Brownian motion there are some which at exceptional times move faster than Brownian motion. We, determine the maximal speed of all particles during a given time period E, which turns out to be a function of the packing dimension of E. A path w in the support of historical super-Brownian motion at time t is called a-fast if lim sup(h down arrow0) \w(t) - w(t - h)\/root hlog(1/h) greater than or equal to a. We calculate the Hausdorff dimension of the set of a-fast paths in the support and the range of historical super-Brownian motion. A valuable tool in the proofs is a uniform dimension formula for the Brownian snake, which reduces dimension problems in the space of stopped paths to dimension problems on the Line.