N ov 1 99 8 CONTINUUM – SITES STEPPING – STONE MODELS , COALESCING EXCHANGEABLE 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).