Diffusion-approximation for a kinetic spray-like system with random forcing

We study a kinetic toy model for spray of particles immersed in an ambient fluid, subject to some additional random forcing given by mixing, space-dependent Markov process. Using the perturbed test function method, we derive hydrodynamic limit system. The law limiting density satisfies stochastic conservation equation Stratonovich form, whose drift and diffusion coefficients are completely determined stationary process associated with Markovian perturbation.