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Well-posedness of kinetic McKean-Vlasov equations

Andrea Pascucci, Alessio Rondelli

TL;DR

The paper addresses the well-posedness of degenerate kinetic McKean-Vlasov equations with diffusion coefficients that depend on the law, formulated as $dV_t = b(t,Z_t,[Z_t])dt + \sigma(t,Z_t,[Z_t])dW_t$, $dX_t = V_t dt$. It advances the theory by employing a duality framework and a sub-Riemannian, anisotropic Hölder setting to handle the measure-dependent diffusion without PDEs in the measure argument. The main contributions are the existence and uniqueness of a weak (and under extra conditions strong) solution, the strong Markov and Feller properties, and two-sided Gaussian bounds on the transition density $p$, with extensions to strong well-posedness when the associated linear SDE is solvable. This work lays groundwork for propagation of chaos in second-order kinetic systems and informs mean-field kinetic modeling with distribution-dependent diffusion.

Abstract

We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + σ(t, X_t, [X_t])dW_t$ where $[X_t]$ is the law of $X_t$. We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion $W$ is strictly smaller than that of the solution $X$, as commonly required in classical models of collisional kinetic theory. Assuming Hölder continuous coefficients and a weak Hörmander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously unexplored case where the diffusion coefficient $σ$ depends on the law $[X_t]$. Notably, our proof employs a simplified and direct argument eliminating the need for PDEs involving derivatives with respect to the measure argument. A critical ingredient is the sub-Riemannian metric structure induced by the corresponding Fokker-Planck operator.

Well-posedness of kinetic McKean-Vlasov equations

TL;DR

The paper addresses the well-posedness of degenerate kinetic McKean-Vlasov equations with diffusion coefficients that depend on the law, formulated as , . It advances the theory by employing a duality framework and a sub-Riemannian, anisotropic Hölder setting to handle the measure-dependent diffusion without PDEs in the measure argument. The main contributions are the existence and uniqueness of a weak (and under extra conditions strong) solution, the strong Markov and Feller properties, and two-sided Gaussian bounds on the transition density , with extensions to strong well-posedness when the associated linear SDE is solvable. This work lays groundwork for propagation of chaos in second-order kinetic systems and informs mean-field kinetic modeling with distribution-dependent diffusion.

Abstract

We consider the McKean-Vlasov equation where is the law of . We specifically consider the kinetic case, where the equation is degenerate because the dimension of the Brownian motion is strictly smaller than that of the solution , as commonly required in classical models of collisional kinetic theory. Assuming Hölder continuous coefficients and a weak Hörmander condition, we prove the well-posedness of the equation. This result advances the existing literature by filling a crucial gap: it addresses the previously unexplored case where the diffusion coefficient depends on the law . Notably, our proof employs a simplified and direct argument eliminating the need for PDEs involving derivatives with respect to the measure argument. A critical ingredient is the sub-Riemannian metric structure induced by the corresponding Fokker-Planck operator.
Paper Structure (6 sections, 4 theorems, 49 equations)

This paper contains 6 sections, 4 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Motivations and related literature
  4. Anisotropic Hölder spaces
  5. A duality result
  6. Proof of Theorem \ref{['t1']}

Key Result

Theorem 1.3

Under Assumptions H1,H2 and H3 there exists a unique weak solution of e1 with initial distribution $[X_{0}]=\bar{{\mu}}_{0}$. The solution is a strong Markov and Feller process, and admits a transition density $p$ satisfying the two-sided Gaussian estimate for some positive constants $C_{\pm},\lambda_{\pm}$, where ${\Gamma}^{\lambda}$ is the Gaussian kernel in Gau.

Theorems & Definitions (12)

  • Theorem 1.3: Weak well-posedness
  • Remark 1.4
  • Definition 2.1: Anisotropic norm
  • Definition 2.2: Anisotropic spaces
  • Remark 2.4
  • Example 2.5: Zhao, Example 1
  • Example 2.6
  • Definition 3.1: Strong Lie solution
  • Theorem 3.2
  • Lemma 3.4: Duality
  • ...and 2 more