Coagulation and diffusion: a probabilistic perspective on the Smoluchowski PDE

The Smoluchowski coagulation-diffusion PDE is a system of partial differential equations modelling the evolution in time of mass-bearing Brownian particles which are subject to shortrange pairwise coagulation. This survey presents a fairly detailed exposition of the kinetic limit derivation of the Smoluchowski PDE from a microscopic model of many coagulating Brownian particles that was undertaken in [10]. It presents heuristic explanations of the form of the main theorem before discussing the proof, and presents key estimates in that proof using a novel probabilistic technique. The survey’s principal aim is an exposition of this kinetic limit derivation, but it also contains an overview of several topics which either motivate or are motivated by this derivation. ∗Department of Statistics, University of Oxford. This survey developed from a graduate course given at the University of Geneva in the autumn of 2012. The course was supported by the Swiss Doctoral Program in Mathematics. The author is supported principally by EPSRC grant EP/I004378/1.