Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees

• 成果类型: Article
• 署名作者: Donnelly, P; Evans, SN; Fleischmann, K; Kurtz, TG; Zhou, XW
• 署名单位: University of Oxford; Leibniz Association; Weierstrass Institute for Applied Analysis & Stochastics; University of California System; University of California Berkeley; University of Wisconsin System; University of Wisconsin Madison; University of Wisconsin System; University of Wisconsin Madison; University of British Columbia
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2000
• 页码: 1063-1110
• 关键词: hierarchically interacting diffusions; measure-valued diffusion; capacity; systems; motion

摘要

Analogues of stepping-stone models are considered where the site-space is continuous, the migration process is a general Markov process, and the type-space is infinite. Such processes were defined in previous work of the second author by specifying a Feller transition semigroup in terms of expectations of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interacting particle systems. It is shown that, under a mild condition on the migration process, the continuum-sites stepping-stone process has continuous sample paths. The case when the migration process is Brownian motion on the circle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show that a tree-like random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdorff and packing dimension both almost surely equal to 1/2 and, moreover, this space is capacity equivalent to the middle-1/2 Canter set land hence also to the Brownian zero set).