Sharp hypocoercive convergence estimates for underdamped Langevin dynamics with specular reflection
Abstract
We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain Ω⊂ℝd by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal μx∝ e−U satisfies a Poincaré inequality on Ω with constant m>0, ∇2U⪰−K Id and Ω is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in L2, with an explicit rate that scales like √m, which is optimal when U is convex. Since the normally reflected overdamped dynamics converges exactly at rate m, this establishes a square-root acceleration for constrained sampling in the small-gap regime when m is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan--Li--Lu. The proof adapts the modified L2 hypocoercivity method of Dolbeault--Mouhot--Schmeiser with the gap-shifted corrector of Fan--Li--Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of ∂Ω and is nonnegative for convex Ω. Finally, we extend our results to the setting where the domain Ω is non-convex. We obtain an explicit contraction rate that depends on the domain.
Related
- The Fenchel Game of Underdamped Langevin Dynamics: Insights into Accelerated Convergence
- Reverse Transportation and Harnack Inequalities for Underdamped Langevin in the Non-convex Setting
- Annealed Underdamped Langevin Dynamics
- Underdamped Anchored Langevin Dynamics
- Exponential Contraction for Underdamped Langevin Diffusions with Superlinear Forces
BibTeX
@misc{du2026sharp,
title = {{Sharp hypocoercive convergence estimates for underdamped Langevin dynamics with specular reflection}},
author = {Hengrong Du and Qi Feng and Lingjiong Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17022},
eprint = {2608.17022},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17022},
}
RIS
TY - UNPB
TI - Sharp hypocoercive convergence estimates for underdamped Langevin dynamics with specular reflection
AU - Hengrong Du
AU - Qi Feng
AU - Lingjiong Zhu
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2608.17022
UR - https://arxiv.org/abs/2608.17022
AB - We study the underdamped (kinetic) Langevin dynamics confined to a bounded domain Ω⊂ℝ^d by specular reflection of the velocity at the boundary. This process is the natural momentum-based analogue of the normally reflected overdamped Langevin diffusion, and it is used in practice for constrained sampling. Assuming only that the position marginal μ_x∝ e^−U satisfies a Poincaré inequality on Ω with constant m>0, ∇^2U⪰−K Id and Ω is a convex domain, we prove that the law converges to the Gibbs measure exponentially fast in L^2, with an explicit rate that scales like √m, which is optimal when U is convex. Since the normally reflected overdamped dynamics converges exactly at rate m, this establishes a square-root acceleration for constrained sampling in the small-gap regime when m is small, matching the acceleration known in the unconstrained case. The explicit rate is the same in the unconstrained setting of Fan--Li--Lu. The proof adapts the modified L^2 hypocoercivity method of Dolbeault--Mouhot--Schmeiser with the gap-shifted corrector of Fan--Li--Lu. The specular symmetry makes the transport operator antisymmetric, and that the corrector automatically selects the Neumann realization of the overdamped generator, which is precisely the boundary condition that keeps every auxiliary function inside the specular class. The Bochner identity used in the whole-space argument is replaced by a weighted Reilly formula, whose boundary contribution involves the second fundamental form of ∂Ω and is nonnegative for convex Ω. Finally, we extend our results to the setting where the domain Ω is non-convex. We obtain an explicit contraction rate that depends on the domain.
ER -