Periodic homogenization for singular L\'evy SDEs

TL;DR

This paper extends periodic homogenization theory to multidimensional SDEs with singular Besov drifts and stable Lévy noise, establishing invariant measures, spectral gaps, and convergence results for both Brownian and stable processes.

Contribution

It introduces a novel homogenization framework for SDEs with singular drifts beyond the Young regime, using paracontrolled distributions and maximum principles.

Findings

Existence and uniqueness of invariant measures with positive density.

Spectral gap established for the associated semigroup.

Diffusion converges to Brownian motion or stable process depending on noise type.

Abstract

We generalize the theory of periodic homogenization for multidimensional SDEs with additive Brownian and stable L\'evy noise for $\alpha\in (1,2)$ to the setting of singular periodic Besov drifts of regularity $\beta\in ((2-2\alpha)/3,0)$ beyond the Young regime. For the martingale solution from Kremp, Perkowski '22 projected onto the torus, we prove existence and uniqueness of an invariant probability measure with strictly positive Lebesgue density exploiting the theory of paracontrolled distributions and a strict maximum principle for the singular Fokker-Planck equation. Furthermore, we prove a spectral gap on the semigroup of the diffusion and solve the Poisson equation with singular right-hand side equal to the drift itself. In the CLT scaling, we prove that the diffusion converges in law to a Brownian motion with constant diffusion matrix. In the pure stable noise case, we rescale in the scaling that the stable process respects and show convergence to the stable process itself. We conclude on the periodic homogenization result for the singular parabolic PDE.