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 L¹ 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.

Energy stability of explicit--implicit--null time discretizations for nonlinear degenerate parabolic equations with state-dependent coefficients
Yue Wu, Chi-Wang Shu
arXiv25 Oct 2026PreprintarXiv licenceFull textNumerical analysis & scientific computingarXiv
Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
Jie Xiong, Wen Xu
arXiv6 Sep 2026PreprintCC BY-SA 4.0Full textProbabilityarXiv
On well-posedness for second-order degenerate parabolic equations with unbounded lower-order terms
Khalid Baadi
Advances in Calculus of Variation29 Sep 2026Peer-reviewedCC BY 4.0Full textAnalysis & PDEsHAL
WEAKLY VISCOUS NONLOCAL WAVE EQUATIONS WITH VARIABLE COEFFICIENTS: WELL-POSEDNESS, RUNGE APPROXIMATION, AND INVERSE PROBLEMS
Philipp Zimmermann
HAL26 Oct 2026PreprintCC BY 4.0Full textAnalysis & PDEsHAL
Well-posedness of Filtering Equations in Weighted Sobolev Spaces with Unbounded System Coefficients
Zeju Sun, Songlin Zhou, Stephen S. -T. Yau
arXiv3 Sep 2026PreprintarXiv licenceFull textOptimisation & controlarXiv

Cite

BibTeX RIS

BibTeX

@misc{li2026wellposedness,
  title = {{Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients}},
  author = {Huaiqian Li and Liying Mu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01894},
  eprint = {2611.01894},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01894},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Well-posedness and Quantitative Stability for Degenerate Fokker--Planck Equations with Rough Coefficients
AU  - Huaiqian Li
AU  - Liying Mu
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01894
UR  - https://arxiv.org/abs/2611.01894
AB  - 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.
ER  -

Abstract

Related

Cite