Construction of markovian coalescents

Abstract Partition-valued and measure-valued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m , and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x + y at rate к ( x,y ), for some non-negative, symmetric collision rate kernel к ( x,y ). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on к and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel к ( x,y ) = x + y . This process, which arises from the evolution of tree components in a random graph process, has asymptotic properties related to the stable subordinator of index 1/2.