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

TL;DR

This paper derives the hydrodynamic limit of a kinetic spray model with random forcing, showing that the limiting density obeys a stochastic conservation law influenced by the stationary Markov process.

Contribution

It introduces a novel kinetic model with space-dependent random forcing and rigorously derives its hydrodynamic limit using the perturbed test function method.

Findings

The limiting density satisfies a stochastic conservation equation in Stratonovich form.

The drift and diffusion coefficients are explicitly determined by the stationary Markov process.

The approach provides a framework for analyzing similar kinetic systems with random perturbations.

Abstract

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