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.