Bilinear Coagulation Equations

Abstract

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form $\pi (y)\cdot A\pi (x)$ for a vector of conserved quantities $\pi$, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time $t_{\mathrm{g}}\in (0,\infty )$, which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.

Generalises the previous version to focus on general coagulation processes of bilinear type, without restricting to the single example of the previous version. The previous results are mentioned as motivation, and all results of the previous version can be obtained from this more general version