Exponential Contraction for Underdamped Langevin Diffusions with Superlinear Forces

Abstract

Underdamped Langevin diffusions model kinetic sampling and thermally driven inertial dynamics, but quantitative convergence becomes difficult when degenerate noise is coupled with a nonconvex superlinear force. We establish unit-prefactor exponential contraction in an explicit weighted Kantorovich cost for potentials satisfying radial confinement, a hypocoercive Lyapunov condition, and \(|\nabla U(x)-\nabla U(y)|\leq L_1(1+|x|^\ell+|y|^\ell)|x-y|\), \(0\leq\ell\leq2\). No smallness condition is imposed on \(L_1\) when \(\ell<2\); the endpoint \(\ell=2\) is covered under an explicit threshold. The proof combines a regularized reflection--synchronous coupling with an exponential Lyapunov weight. Fixed coefficients encode the kinetic geometry, while the radius-dependent local Lipschitz constant enters only the concavity profile. This separation reduces the localization loss to \(\exp\{C(R^\ell+R)\}\) and permits Lyapunov absorption. A smooth covariance correction preserves both marginals, while tightness and occupation-time estimates remove the singularity at a vanishing coupling direction. We obtain uniqueness of the Gibbs measure and exponential convergence for every initial law with finite transportation cost. In particular, polynomial potentials of degree \(p\in(2,4)\) are covered for arbitrary local Lipschitz coefficient.