Stochastic homogenization of diffusions in turbulence driven by non-local symmetric L\'evy operators

TL;DR

This paper studies the homogenization of turbulent diffusion processes driven by non-local symmetric Lévy operators, establishing new regularity estimates for correctors in ergodic random environments.

Contribution

It introduces novel $W_{loc}^{1,q}$ estimates for correctors in the stochastic homogenization of non-local Lévy-driven diffusions with unbounded drift fields.

Findings

Established $W_{loc}^{1,q}$ estimates for correctors.

Extended homogenization theory to unbounded divergence-free drifts.

Analyzed effects of non-local Lévy operators in turbulent diffusion.

Abstract

We investigate the stochastic homogenization of a class of turbulent diffusions generated by non-local symmetric L\'evy operators with divergence-free drift fields in ergodic random environments, where neither the drift fields nor their associated stream functions are assumed to be bounded. A pivotal step in our proof is the establishment of $W_{loc}^{1,q}$ estimates with $q\in (1,2)$ for the corresponding correctors, under mild prior regularity conditions imposed on the L\'evy measure and the stream function.