A new look to branching Brownian motion from a particle-based reaction–diffusion dynamics point of view

Aim of this note is to analyze branching Brownian motion within the class of models introduced in some recent papers and called chemical diffusion master equations (CDMEs). These models provide a description for the probabilistic evolution of chemical reaction kinetics associated with spatial diffusion of individual particles. We derive an infinite system of Fokker-Planck equations that rules the probabilistic evolution of the single particles generated by the branching mechanism and analyze its properties using Malliavin Calculus techniques. Another key ingredient of our approach is the McKean representation for the solution of the Fisher-Kolmogorov-Petrovskii-Piskunov equation and a stochastic counterpart of that equation. We also derive the reaction-diffusion partial differential equation solved by the average concentration field of the branching Brownian system of particles.