  A well-known result of Arratia shows that one can make rigorous the notion of
starting an independent Brownian motion at every point of an arbitrary closed
subset of the real line and then building a set-valued process by requiring
particles to coalesce when they collide. Arratia noted that the value of this
process will be almost surely a locally finite set at all positive times, and a
finite set almost surely if the initial value is compact: the key to both of
these facts is the observation that, because of the topology of the real line
and the continuity of Brownian sample paths, at the time when two particles
collide one or the other of them must have already collided with each particle
that was initially between them. We investigate whether such instantaneous
coalescence still occurs for coalescing systems of particles where either the
state space of the individual particles is not locally homeomorphic to an
interval or the sample paths of the individual particles are discontinuous. We
give a quite general criterion for a coalescing system of particles on a
compact state space to coalesce to a finite set at all positive times almost
surely and show that there is almost sure instantaneous coalescence to a
locally finite set for systems of Brownian motions on the Sierpinski gasket and
stable processes on the real line with stable index greater than one.
