Continuum--sites stepping--stone models, coalescing exchangeable partitions, and random trees
- Donnelly, Peter
- Evans, Steven N.
- Fleischmann, Klaus
- Kurtz, Thomas G.
- Zhou, Xiaowen
2010 Mathematics Subject Classification
- 60K35 60G57 60J60
- coalesce, partition, right process, annihilate, dual, diffusion, exchangeable, vector measure, tree, Hausdorff dimension, packing dimension, capacity equivalence, fractal
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 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 ½ and, moreover, this space is capacity equivalent to the middle -½ Cantor set (and hence also to the Brownian zero set).
- Ann. Probab., 28(2000), pp. 1063-1110