Probabilistic approach of some discrete and continuous coagulation equations with diffusion

The diffusive coagulation equation models the evolution of the local concentration n(t,x,z) of particles having position and size z at time t, for a system in which a coagulation phenomenon occurs. The aim of this paper is to introduce a probabilistic approach and a numerical scheme for this equation. We first delocalise the interaction, by considering a "mollified" model. This mollified model is naturally related to a -valued nonlinear stochastic differential equation, in a certain sense. We get rid of the nonlinearity of this S.D.E. by approximating it with an interacting stochastic particle system, which is (exactly) simulable. By using propagation of chaos techniques, we show that the empirical measure of the system converges to the mollified diffusive equation. Then we use the smoothing properties of the heat kernel to obtain the convergence of the mollified solution to the true one. Numerical results are presented at the end of the paper.