Diffusion-approximation in stochastically forced kinetic equations

TL;DR

This paper derives a stochastic hydrodynamic limit for kinetic equations with stochastic forces, revealing enhanced diffusion phenomena compared to deterministic models.

Contribution

It introduces a novel derivation of the hydrodynamic limit for kinetic equations with stochastic forcing, highlighting the emergence of a stochastic PDE and enhanced diffusion effects.

Findings

Hydrodynamic limit is a scalar second-order stochastic PDE.

Stochastic forcing leads to enhanced diffusion.

The model extends deterministic kinetic equations to include stochastic effects.

Abstract

We derive the hydrodynamic limit of a kinetic equation where the interactions in velocity are modelled by a linear operator (Fokker-Planck or Linear Boltzmann) and the force in the Vlasov term is a stochastic process with high amplitude and short-range correlation. In the scales and the regime we consider, the hydrodynamic equation is a scalar second-order stochastic partial differential equation. Compared to the deterministic case, we also observe a phenomenon of enhanced diffusion.