Asymptotic behavior of a class of multiple time scales stochastic kinetic equations

TL;DR

This paper analyzes the asymptotic behavior of stochastic kinetic equations with two small parameters, showing convergence to a diffusion PDE through combined diffusion approximation and averaging techniques.

Contribution

It introduces a novel analysis of stochastic kinetic equations with dual time scale parameters, extending existing methods to the regime where e9 and d6 are not necessarily equal.

Findings

Density converges to a linear diffusion PDE as e9,d6 to 0.

The approach combines diffusion approximation and averaging in a unified framework.

The proof uses stopping times and perturbed test functions adapted to the general regime.

Abstract

We consider a class of stochastic kinetic equations, depending on two time scale separation parameters $\epsilon$ and $\delta$: the evolution equation contains singular terms with respect to $\epsilon$, and is driven by a fast ergodic process which evolves at the time scale $t/\delta^2$. We prove that when $(\epsilon,\delta)\to (0,0)$ the density converges to the solution of a linear diffusion PDE. This is a mixture of diffusion approximation in the PDE sense (with respect to the parameter $\epsilon$) and of averaging in the probabilistic sense (with respect to the parameter $\delta$). The proof employs stopping times arguments and a suitable perturbed test functions approach which is adapted to consider the general regime $\epsilon\neq \delta$.