Gelation and Localization in Multicomponent Coagulation with Multiplicative Kernel Through Branching Processes

The multicomponent coagulation equation is a generalization of the Smoluchowski coagulation equation, where the size of a particle is described by a vector. Similar to the original Smoluchowski equation, the multicomponent coagulation equation exhibits gelation behavior when supplied with a multiplicative kernel. Additionally, a new type of behaviour called localization is observed due to the multivariate nature of the particle size distribution. Here we extend the branching process representation technique, which we introduced to study differential equations in our previous work, and apply it to find a concise probabilistic solution of the multicomponent coagulation equation supplied with monodisperse initial conditions. We also provide short proofs for the gelation time and characterisation the localization phenomenon.