Diffusion-approximation for a kinetic equation with perturbed velocity redistribution process

TL;DR

This paper derives a diffusion approximation for a kinetic equation with stochastic velocity perturbations, resulting in a stochastic PDE for the density, under certain mixing conditions.

Contribution

It introduces a novel diffusion-approximation framework for kinetic equations with stochastic velocity perturbations, linking microscopic randomness to macroscopic stochastic PDEs.

Findings

Established diffusion limit under mixing hypotheses

Derived a stochastic PDE for the macroscopic density

Validated the approximation through rigorous analysis

Abstract

We derive the hydrodynamic limit of a kinetic equation with a stochastic, short range perturbation of the velocity operator. Under some mixing hypotheses on the stochastic perturbation, we establish a diffusion-approximation result: the limit we obtain is a parabolic stochastic partial differential equation on the macroscopic parameter, the density here.