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

Analogues of stepping-stone models are considered where the site-space is c ontinuous, the migration process is a general Markov process, and the type- space is infinite. Such processes were defined in previous work of the seco nd author by specifying a Feller transition semigroup in terms of expectati ons of suitable functionals for systems of coalescing Markov processes. An alternative representation is obtained here in terms of a limit of interact ing particle systems. It is shown that, under a mild condition on the migra tion process, the continuum-sites stepping-stone process has continuous sam ple paths. The case when the migration process is Brownian motion on the ci rcle is examined in detail using a duality relation between coalescing and annihilating Brownian motion. This duality relation is also used to show th at a tree-like random compact metric space that is naturally associated to an infinite family of coalescing Brownian motions on the circle has Hausdor ff and packing dimension both almost surely equal to 1/2 and, moreover, thi s space is capacity equivalent to the middle-1/2 Canter set land hence also to the Brownian zero set).