Well-posedness of kinetic McKean-Vlasov equations

Abstract

We consider the McKean-Vlasov equation $d{X}_t=b(t,X_t,[X_t])dt+\sigma (t,X_t,[X_t])d{W}_t$ 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 $\sigma$ 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.