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Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces

Elena Issoglio, Stefano Pagliarani, Francesco Russo, Davide Trevisani

TL;DR

The paper addresses the weak well-posedness of a degenerate kinetic McKean–Vlasov SDE with a distributional drift in anisotropic Besov spaces of negative regularity, under a weak Hörmander condition. The authors develop a comprehensive analytic framework, including anisotropic Besov/Schauder theory and regularized (mollified) problems for both nonlinear Fokker–Planck and backward Kolmogorov PDEs, to establish existence and stability of solutions in law via a nonlinear martingale problem MKMP. They first analyze the linear kinetic SDE through MP(ℬ,μ_0) and then tackle the nonlinear, law-dependent drift by constructing a fixed point in suitable Besov/Hölder spaces, proving existence and uniqueness of MKMP(F,b,u^0). The work also proves continuity of the density u_t and convergence of regularized problems, providing a rigorous bridge between singular PDE theory and McKean–Vlasov kinetic dynamics with degenerate noise. This offers a rigorous foundation for singular mean-field kinetic models and advances the analytical toolbox for hypoelliptic, distributional drift SDEs with density feedback.

Abstract

The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak H{ö}rmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.

Degenerate McKean-Vlasov equations with drift in anisotropic negative Besov spaces

TL;DR

The paper addresses the weak well-posedness of a degenerate kinetic McKean–Vlasov SDE with a distributional drift in anisotropic Besov spaces of negative regularity, under a weak Hörmander condition. The authors develop a comprehensive analytic framework, including anisotropic Besov/Schauder theory and regularized (mollified) problems for both nonlinear Fokker–Planck and backward Kolmogorov PDEs, to establish existence and stability of solutions in law via a nonlinear martingale problem MKMP. They first analyze the linear kinetic SDE through MP(ℬ,μ_0) and then tackle the nonlinear, law-dependent drift by constructing a fixed point in suitable Besov/Hölder spaces, proving existence and uniqueness of MKMP(F,b,u^0). The work also proves continuity of the density u_t and convergence of regularized problems, providing a rigorous bridge between singular PDE theory and McKean–Vlasov kinetic dynamics with degenerate noise. This offers a rigorous foundation for singular mean-field kinetic models and advances the analytical toolbox for hypoelliptic, distributional drift SDEs with density feedback.

Abstract

The paper is concerned with a McKean-Vlasov type SDE with drift in anisotropic Besov spaces with negative regularity and with degenerate diffusion matrix under the weak H{ö}rmander condition. The main result is of existence and uniqueness of a solution in law for the McKean-Vlasov equation, which is formulated as a suitable martingale problem. All analytical tools needed are derived in the paper, such as the well-posedness of the Fokker-Planck and Kolmogorov PDEs with distributional drift, as well as continuity dependence on the coefficients. The solutions to these PDEs naturally live in anisotropic Besov spaces, for which we developed suitable analytical inequalities, such as Schauder estimates.
Paper Structure (20 sections, 37 theorems, 256 equations)

This paper contains 20 sections, 37 theorems, 256 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Statement of the problem
  3. Literature review
  4. Setting and main result
  5. Analytical tools
  6. Anisotropic Besov Spaces
  7. Kinetic-type semigroups and Schauder's estimates
  8. Anisotropic and intrinsic Hölder spaces
  9. Regularizations
  10. Non-linear Fokker-Planck singular PDE
  11. Well-posedeness of the PDE
  12. The regularized equation
  13. Backward Kolmogorov PDE with singular drift
  14. Well-posedness of the PDE
  15. Continuity properties
  16. ...and 5 more sections

Key Result

Theorem 1.6

Let Assumptions ass:kolmogorov_op and ass:beta and b hold. Let also Assumption ass:phi hold with $\Phi=F,\widetilde{F}$. Then there exists a unique solution to MKMP$( F, b,u^0)$.

Theorems & Definitions (104)

  • Remark 1.2
  • Example 1.3
  • Theorem 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: Bernstein's type inequality
  • Corollary 2.5
  • proof
  • Definition 2.6
  • ...and 94 more