Fractional Hypoellipticity for Degenerate Kinetic Fokker-Planck Equations with Multiplicative Lévy Noise: Critical Degeneracy Thresholds and Hydrodynamic Limits

This paper develops a complete mathematical theory for degenerate kinetic Fokker-Planck equations with multiplicative Lévy noise. The main novelty is the treatment of state-dependent noise intensity $\sigma \left(x,v\right)$ that may vanish on sets of positive measure, a situation arising naturally in applications with heterogeneous media or discontinuous coefficients. We establish three fundamental contributions: 1) a well-posedness theory for solutions in fractional Sobolev spaces under minimal regularity assumptions on $\sigma$ ; 2) a critical degeneracy threshold theorem proving that hypoelliptic regularization persists even when $\sigma \left(x,v\right)$ vanishes, provided the degeneracy measure satisfies $\delta <\alpha {\lambda }_1/\left(2+\alpha \right)$ , where ${\lambda }_1$ is the mixing rate associated with the transport operator; 3) rigorous hydrodynamic limits yielding effective fractional diffusion equations with spatially-dependent coefficients $\bar{\sigma }\left(x\right)$ . The analysis combines weighted energy methods, fractional commutator estimates, and compensated compactness techniques to handle the interplay between degeneracy and nonlocality. Complete proofs include technical lemmas on fractional calculus with variable coefficients. Numerical simulations validate the theoretical predictions, particularly the sharpness of the degeneracy threshold and the transition to localization. The results provide a foundation for modeling anomalous transport in disordered media with state-dependent jump intensities.