Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients

Abstract

We establish quantitative stability and well-posedness for degenerate Fokker--Planck equations with Sobolev-regular and possibly fully degenerate diffusion coefficient. For two classes of rough drifts, stability estimates imply uniqueness, and existence is proved under suitable conditions. First, for anisotropic drifts with partial fractional Sobolev regularity, we prove both existence (as probability densities, essential for the probabilistic interpretation via superposition) and quantitative stability without imposing any growth conditions, covering the full range of fractional exponents and extending previous Lagrangian-flow results. Second, for drifts whose distributional gradients are given by singular integrals (e.g., Biot--Savart velocity fields from $L^1$ vorticity), we obtain stability estimates that allow the diffusion matrix to be time-space dependent and possibly degenerate, removing the prior restrictions of identity diffusion or bounded coefficients. Our results also recover the pure transport regime as a limiting case.