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

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 1/2 and, moreover, this space is capacity equivalent to the middle--1/2 Cantor set (and hence also to the Brownian zero set).