The Annals of Probability

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

Peter Donnelly, Steven N. Evans, Klaus Fleischmann, Thomas G. Kurtz, and Xiaowen Zhou

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Analogues of stepping-stone models are considered where the sitespace 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 in .nite 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 Cantor set (and hence also to the Brownian zero set).

Article information

Ann. Probab., Volume 28, Number 3 (2000), 1063-1110.

First available in Project Euclid: 18 April 2002

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60G57: Random measures 60J60: Diffusion processes [See also 58J65]

Coalesce partition right process annihilate dual diffusion exchangeable particle system vector measure tree Hausdorff dimension packing dimension capacity equivalence fractal


Donnelly, Peter; Evans, Steven N.; Fleischmann, Klaus; Kurtz, Thomas G.; Zhou, Xiaowen. Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees. Ann. Probab. 28 (2000), no. 3, 1063--1110. doi:10.1214/aop/1019160326.

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