Periodic homogenization for singular L\'evy SDEs

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.