Convergence of stochastic particle systems undergoing advection and coagulation

The convergence of stochastic particle systems representing physical advection, inflow, outflow and coagulation is considered. The problem is studied on a bounded spatial domain such that there is a general upper bound on the residence time of a particle. The laws on the appropriate Skorohod path space of the empirical measures of the particle systems are shown to be relatively compact. The paths charged by the limits are characterised as solutions of a weak equation restricted to functions taking the value zero on the outflow boundary. The limit points of the empirical measures are shown to have densities with respect to Lebesgue measure when projected on to physical position space. In the case of a discrete particle type space a strong form of the Smoluchowski coagulation equation with a delocalised coagulation interaction and an inflow boundary condition is derived. As the spatial discretisation is refined in the limit equations, the delocalised coagulation term reduces to the standard local Smoluchowski interaction. The original Smoluchowski coagulation equation [22] gives a deterministic description of coagulation of an infinite, well-mixed population of particles. Smoluchowski arrived at the equation by considering the volume swept out by a diffusing particle and therefore in some sense from an underlying stochastic model. Heuristic derivations [15, 5], which are more explicitly probabilistic and assume only a general stochastic coagulation process with specified rate, lead via a Kolmogorov forward equation to the same Smoluchowski coagulation equation. An important, and explicit, step in these works is to neglect the correlations between particles. While this assumption was motivated by the need to simplify the problem, it also leads to Markov jump process dynamics that are well suited to simulation [6]. These processes can be used as numerical methods for the Smoluchowski coagulation equation. Rigorous convergence results (existence of a limit point satisfying the Smoluchowski equation) for these stochastic particle methods took some time to develop [9, 16, 2]. Extensive generalisations are now available including general n-particle interactions [3] and [12], the former including particle inflow while the latter provides a CLT result. Some convergence results are also available that go beyond the assumption of a spatially well mixed population. Guiaş [8] considered coagulation in the presence of diffusion on a spatial lattice and showed convergence of the jump processes to a unique limit point satisfying the Smoluchowski equation 1 with diffusion. Particles were not able to leave the domains studied. Analogous results for continuous diffusions, that is not on a spatial lattice, were given first by Lang and Xanh [13] and for more general, but still non-degenerate diffusions by Wells [23] and Yaghouti et al. [25]. 1[24] gives a similar result to that in [8], but for a biologically motivated coagulation model that does not lead to the Smoluchowski equation.