Smoluchowski's Coagulation Equation: Uniqueness, Non-Uniqueness and a Hydrodynamic Limit for the Stochastic Coalescent

. Sufficient conditions are given for existence and uniqueness in Smoluchowski 's coagulation equation, for a wide class of coagulation kernels and initial mass distributions. An example of non-uniqueness is constructed. The stochastic coalescent is shown to converge weakly to the solution of Smoluchowski's equation. 1. Introduction Coagulation of particles, in pairs and over time, within a large system of particles, is a phenomenon widely observed and widely postulated in scientific models. Examples arise in the study of aerosols, of phase separation in liquid mixtures, in polymerization and astronomy. Typically, it is argued that the rate at which pairs of particles coagulate depends, for physical reasons, in a given way, on some positive parameter associated to each particle, such as size or mass. In the models we shall consider, it is further argued that the effects of spatial fluctuations in the size or mass density are negligible---for example, by supposing that the particles pe..