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July 2000 Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees
Peter Donnelly, Steven N. Evans, Klaus Fleischmann, Thomas G. Kurtz, Xiaowen Zhou
Ann. Probab. 28(3): 1063-1110 (July 2000). DOI: 10.1214/aop/1019160326

Abstract

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).

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Peter Donnelly. Steven N. Evans. Klaus Fleischmann. Thomas G. Kurtz. Xiaowen Zhou. "Continuum-sites stepping-stone models, coalescing exchangeable partitions and random trees." Ann. Probab. 28 (3) 1063 - 1110, July 2000. https://doi.org/10.1214/aop/1019160326

Information

Published: July 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1023.60082
MathSciNet: MR1797304
Digital Object Identifier: 10.1214/aop/1019160326

Subjects:
Primary: 60K35
Secondary: 60G57 , 60J60

Keywords: annihilate , capacity equivalence , Coalesce , diffusion , dual , Exchangeable , Fractal , Hausdorff dimension , Packing dimension , Particle system , Partition , Right process , tree , vector measure

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 3 • July 2000
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