Stochastic homogenization of stable-like process with divergence free drift
Xin Chen, Kun Yin
TL;DR
This paper addresses the stochastic homogenization of a stable-like jump process with a divergence-free drift in a stationary ergodic environment, allowing unbounded coefficients. It formulates a non-local, perturbed Dirichlet form and analyzes the resolvent equation for scaled forms, proving that $u_{\lambda,f}^{\varepsilon,\omega}$ converges in $L^2$ to a deterministic limit $\bar{u}_{\lambda,f}$ solving $\lambda \bar{u}-\bar{L}\bar{u}=f$, where $\bar{L}$ has the kernel $\mathds E[\mu]^2/|z|^{d+\alpha}$. The approach combines variational regularization, fractional Sobolev compactness, and ergodic averaging to handle unbounded drift and nonlocal interactions, extending prior results on stochastic turbulence and non-local homogenization to this broader setting. The results provide a rigorous effective equation for homogenized jump processes in random media, with explicit effective jump intensity given by $\mathds E[\mu]^2$.
Abstract
In this paper we will study homogenization of for stable-like process with divergence-free drift in ergodic environments. In particular, neither the drift nor the stream function are required to be bounded.