Coagulation dynamics under environmental noise: scaling limit to SPDE

Abstract

Motivated by the meteorological question whether a turbulent velocity field accelerates rain droplets merging and favors rain, we prove that for a system of locally interacting diffusions carrying discrete masses, subject to an environmental noise and undergoing mass coagulation, its empirical measures corresponding to different masses converge to a system of Stochastic Partial Differential Equations (SPDEs) with enhanced diffusion coefficient and Smoluchowskitype nonlinearity. Existence, uniqueness and regularity of the SPDEs are proved, which can be of independent interest.

Scaling limit for a second-order particle system with local annihilation

“…In the last section, we prove the uniqueness of the limit PDE. The techniques used here, which are very influenced by Hammond-Rezakhanlou [13], share elements with the author's previous works [10,11], hence certain arguments are made more succinct.…”

“…It should be seen as a form of local equilibrium. Since the function is very close to singular (for small), we need to apply the Itô-Tanaka trick to (11), where the additional localization by would be useful.…”

“…Proof. We only sketch the proof as it is similar to arguments detailed in [13,10,11]. In the sequel, we use Err( , ) to denote a stochastic error that can change from line to line and which vanishes in the following limit…”

“…By Proposition 5 (we can remove the function without changing the value, as explained near (11)) and (0) = 0, we have that…”

“…We apply the Itô-Tanaka trick to (11). The trick itself is a classical one in stochastic analysis, see e.g.…”