Snakes and spiders: Brownian motion on R-trees

• 成果类型: Article
• 署名作者: Evans, SN
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s004400050010
• 发表日期: 2000
• 页码: 361-386
• 关键词: markov-processes; random-walks; graphs

摘要

We consider diffusion processes on a class of R-trees. The processes are defined in a manner similar to that of Le Gall's Brownian snake. Each point in the tree has a real-valued height or generation, and the height of the diffusion process evolves as a Brownian motion. When the height process decreases the diffusion retreats back along a lineage, whereas when the height process increases the diffusion chooses among branching lineages according to relative weights given by a possibly infinite measure on the family of lineages. The class of R-trees we consider can have branch points with countably infinite branching and lineages along which the branch points have points of accumulation. We give a rigorous construction of the diffusion process, identify its Dirichlet form, and obtain a necessary and sufficient condition for it to be transient. We show that the tail sigma-field of the diffusion is always trivial and draw the usual conclusion that bounded space-time harmonic functions are constant. In the transient case, we identify the Martin compactification and obtain the corresponding integral representations of excessive and harmonic functions. Using Ray-Knight methods, we show that the only entrance laws for the diffusion are the trivial ones that arise from starting the process inside the state-space. Finally, we use the Dirichlet form stochastic calculus to obtain a semimartingale description of the diffusion that involves local time additive functionals associated with each branch point of the tree.