Exponential Contraction for Underdamped Langevin Diffusions with Superlinear Forces

Abstract

We establish exponential contraction for underdamped Langevin diffusions with nonconvex confining potentials and possibly superlinear forces. The force satisfies a polynomial local Lipschitz bound with exponent $\ell \ge 0$, and the potential has a lower bound of order $m\ge 2$, with $3\ell <2m$. Under radial confinement and a potential--dissipation inequality, we construct an explicit weighted Kantorovich transportation cost in which the Markov semigroup contracts probability laws exponentially. The estimate holds for all nonnegative times, with prefactor one and an explicit positive rate, without a smallness condition on the local Lipschitz coefficient. The proof combines localized reflection--synchronous coupling with an exponential Lyapunov weight. A radius-dependent shift of the position--velocity difference balances the local force contribution to the distance drift, while distinct localization scales in position and velocity exploit the growth of the potential. Together, these choices yield a dissipative estimate for the weighted cost with a vanishing regularization error. The regularized couplings preserve the Langevin marginals, so passage to weak limits yields contraction for all initial laws with finite transportation cost. Consequently, the dynamics converge exponentially to the Gibbs measure within the corresponding finite-cost class. The assumptions include quartic double-well potentials of arbitrary amplitude. Numerical experiments for asymmetric quartic wells examine transportation between two initial laws and quantify the conservatism of the explicit rate.