Well-posedness of kinetic McKean-Vlasov equations

TL;DR

This paper proves the well-posedness of a class of degenerate kinetic McKean-Vlasov equations with law-dependent coefficients under H"older continuity and a weak H"ormander condition, using a novel, PDE-free approach.

Contribution

It establishes the existence and uniqueness of solutions for a previously unaddressed class of kinetic McKean-Vlasov equations with law-dependent diffusion.

Findings

Proved well-posedness under H"older and weak H"ormander conditions.

Introduced a PDE-free proof technique based on sub-Riemannian geometry.

Filled a gap in the theory of kinetic McKean-Vlasov equations with law-dependent coefficients.

Abstract

We consider the McKean-Vlasov equation $dX_t = b(t, X_t, [X_t])dt + \sigma(t, X_t, [X_t])dW_t$ 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 H\"older continuous coefficients and a weak H\"ormander 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 $\sigma$ 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.