pith. sign in

arxiv: 2512.18598 · v2 · pith:BBIP4X4Fnew · submitted 2025-12-21 · 🧮 math.PR

Long-time reverse transportation inequalities for non-globally-dissipative Langevin dynamics

Jianfeng Lu , Yuliang Wang This is my paper

Pith reviewed 2026-05-25 07:44 UTC · model grok-4.3

classification 🧮 math.PR
keywords Langevin dynamicsreverse transportation inequalitynon-convex potentialsRényi divergenceHarnack inequalitylong-time behaviordimension-free estimates
0
0 comments X

The pith

Langevin dynamics with non-convex potentials admit a dimension-free reverse transportation inequality with exponential long-time decay.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that Langevin dynamics driven by non-convex potentials satisfy a reverse transportation inequality that is uniform in time and independent of dimension. This inequality bounds the Rényi divergence of arbitrary order between the laws of the process starting from different initial points. It holds with exponential decay as time tends to infinity and serves as the dual of the Harnack inequality. The result extends earlier transportation inequalities that applied only to log-concave settings.

Core claim

We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the Rényi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting.

What carries the argument

The reverse transportation inequality, which bounds Rényi divergence between distributions of the Langevin process started from different initial conditions and functions as the dual to the Harnack inequality.

If this is right

  • The inequality applies without requiring global dissipativity of the potential.
  • Exponential decay of the bound holds in the long-time regime even for non-convex potentials.
  • The result supplies a tool for quantitative analysis of sampling from non-log-concave distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The inequality may yield explicit mixing-time bounds for MCMC on multimodal targets.
  • Analogous reverse inequalities could be sought for related processes such as underdamped Langevin dynamics.
  • Direct verification on a double-well potential would provide a numerical test of the claimed decay.

Load-bearing premise

The non-globally-dissipative non-convex potentials must satisfy technical conditions that make the reverse transportation inequality and its exponential decay well-defined.

What would settle it

A concrete non-convex potential where numerical computation shows that the Rényi divergence between two Langevin trajectories does not decay exponentially at large times would falsify the long-time claim.

Figures

Figures reproduced from arXiv: 2512.18598 by Jianfeng Lu, Yuliang Wang.

Figure 1
Figure 1. Figure 1: An illustration of the coupling (2.2). X and X ′ are two realizations of the overdamped Langevin equation (1.1). X ′′ is an auxiliray dynamics that interpolates X and X ′ . The KL or Renyi ´ divergence between X ′ and X ′′ is then given by Girsanov transform. Remark 2.1. We choose an O( √ |Zt |) additional drift with a bounded ηt because L 1 -contraction is more natural with the presence of reflection coup… view at source ↗
read the original abstract

We establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics with non-convex potentials. This inequality controls the R\'enyi divergence of arbitrary order between the process distributions starting from distinct initial points and serves as the dual version of the Harnack inequality. Notably, we prove that this inequality retains exponential decay in the long-time regime, thereby extending existing results for log-concave sampling to the non-convex setting.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to establish a dimension-free, uniform-in-time reverse transportation inequality for Langevin dynamics driven by non-convex, non-globally-dissipative potentials. The inequality bounds Rényi divergences of arbitrary order between the laws of the process at time t starting from different initial conditions and is shown to decay exponentially as t → ∞, thereby extending prior results known only for log-concave (strongly convex) cases.

Significance. If the central claims hold under the stated assumptions, the result would supply a useful analytic tool for controlling convergence of non-convex Langevin processes in high dimension, with the dimension-free and long-time exponential features being particularly valuable for sampling and optimization applications. The dual Harnack-type formulation is a natural and technically interesting direction.

major comments (2)
  1. [§2 and Theorem 3.1] §2 (Assumptions) and Theorem 3.1 (main long-time result): the exponential decay rate for the reverse transportation inequality is asserted to remain positive and uniform in time even without global dissipativity. The proof sketch relies on a local Lyapunov or one-sided dissipativity condition; it is not shown that this condition produces a strictly positive lower bound on the decay rate that is independent of the depth of the non-convex wells. If the rate can approach zero for some admissible V, the claimed long-time exponential decay fails for those potentials.
  2. [Theorem 4.2] Theorem 4.2 (dimension-free bound): the dimension-free constant is derived under the same local dissipativity hypothesis. The argument must be checked to confirm that no hidden global lower bound on the Hessian or on the Lyapunov function is used when passing to the long-time limit; otherwise the dimension-free claim and the exponential decay cannot both hold simultaneously for arbitrary non-convex V.
minor comments (2)
  1. [Definition 1.2] Notation for the reverse transportation inequality (Definition 1.2) could be aligned more explicitly with the standard Rényi-divergence formulation used in the log-concave literature to facilitate comparison.
  2. [Theorem 3.1] The statement of the main theorem would benefit from an explicit display of the decay rate (or its lower bound) rather than only an existence claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments. We address each major comment below with clarifications drawn directly from the proofs in the manuscript.

read point-by-point responses

  1. Referee: [§2 and Theorem 3.1] §2 (Assumptions) and Theorem 3.1 (main long-time result): the exponential decay rate for the reverse transportation inequality is asserted to remain positive and uniform in time even without global dissipativity. The proof sketch relies on a local Lyapunov or one-sided dissipativity condition; it is not shown that this condition produces a strictly positive lower bound on the decay rate that is independent of the depth of the non-convex wells. If the rate can approach zero for some admissible V, the claimed long-time exponential decay fails for those potentials.

    Authors: Under Assumptions 2.1 (one-sided dissipativity) and 2.2 (local Lyapunov function), the proof of Theorem 3.1 derives the decay rate λ explicitly from integrating the local dissipativity against the Lyapunov function, which controls the contribution from non-convex regions. This yields λ ≥ c > 0 where c depends only on the constants appearing in the assumptions (see the differential inequality for the Rényi divergence immediately after (3.5) and the subsequent Gronwall-type estimate). The lower bound is therefore uniform in the depth of the wells for any V satisfying the stated assumptions; it does not approach zero within the admissible class. We will insert a short remark after the statement of Theorem 3.1 to make this uniformity explicit. revision: partial

  2. Referee: [Theorem 4.2] Theorem 4.2 (dimension-free bound): the dimension-free constant is derived under the same local dissipativity hypothesis. The argument must be checked to confirm that no hidden global lower bound on the Hessian or on the Lyapunov function is used when passing to the long-time limit; otherwise the dimension-free claim and the exponential decay cannot both hold simultaneously for arbitrary non-convex V.

    Authors: The argument in Theorem 4.2 first obtains the reverse transportation inequality at every finite time t by applying only the local dissipativity condition and the Rényi-divergence contraction (without any global Hessian lower bound). The long-time limit is then taken using the uniform exponential decay already established in Theorem 3.1. The dimension-free constant arises from the structure of the Rényi divergence and the local estimates; no global lower bound on the Hessian or on the Lyapunov function is invoked at any step. Consequently both the dimension-free property and the positive exponential rate hold simultaneously under the given assumptions. revision: no

Circularity Check

0 steps flagged

No circularity: mathematical proof of inequality is self-contained

full rationale

The paper claims to establish a dimension-free uniform-in-time reverse transportation inequality controlling Rényi divergence for Langevin dynamics under non-convex potentials, with long-time exponential decay. This is presented as a theorem and proof extending log-concave results. No equations, definitions, or steps in the provided abstract reduce the target result to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation is a standard mathematical argument and remains independent of the claimed inequality itself.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are stated. The result is presented as a mathematical inequality whose validity rests on unlisted technical conditions on the potential and the SDE.

pith-pipeline@v0.9.0 · 5592 in / 1170 out tokens · 22339 ms · 2026-05-25T07:44:04.460143+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Shifted composition I: Harnack and reverse transport in- equalities.IEEE Transactions on Information Theory, 2024

    Jason M Altschuler and Sinho Chewi. Shifted composition I: Harnack and reverse transport in- equalities.IEEE Transactions on Information Theory, 2024

    work page 2024

  2. [2]

    Shifted composition IV: Underdamped Langevin and numerical discretizations with partial acceleration.arXiv preprint arXiv:2506.23062, 2025

    Jason M Altschuler, Sinho Chewi, and Matthew S Zhang. Shifted composition IV: Underdamped Langevin and numerical discretizations with partial acceleration.arXiv preprint arXiv:2506.23062, 2025

    work page arXiv 2025

  3. [3]

    Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds.Annals of Probabiliity, 43(1):339–404, 2015

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar ´e. Bakry-Emery curvature-dimension condition and Riemannian Ricci curvature bounds.Annals of Probabiliity, 43(1):339–404, 2015

    work page 2015

  4. [4]

    Convergence of two-timescale gradient descent ascent dynamics: finite- dimensional and mean-field perspectives.arXiv preprint arXiv:2501.17122, 2025

    Jing An and Jianfeng Lu. Convergence of two-timescale gradient descent ascent dynamics: finite- dimensional and mean-field perspectives.arXiv preprint arXiv:2501.17122, 2025

    work page arXiv 2025

  5. [5]

    Harnack inequality and heat kernel es- timates on manifolds with curvature unbounded below.Bulletin des sciences mathematiques, 130(3):223–233, 2006

    Marc Arnaudon, Anton Thalmaier, and Feng-Yu Wang. Harnack inequality and heat kernel es- timates on manifolds with curvature unbounded below.Bulletin des sciences mathematiques, 130(3):223–233, 2006

    work page 2006

  6. [6]

    Franc ¸ois Bolley and C ´edric Villani. Weighted Csisz ´ar-Kullback-Pinsker inequalities and ap- plications to transportation inequalities.Annales de la Facult´ e des sciences de Toulouse: Math´ ematiques, 14(3):331–352, 2005

    work page 2005

  7. [7]

    Coupling and convergence for Hamilto- nian Monte Carlo.The Annals of Applied Probability, 30(3):1209–1250, 2020

    Nawaf Bou-Rabee, Andreas Eberle, and Raphael Zimmer. Coupling and convergence for Hamilto- nian Monte Carlo.The Annals of Applied Probability, 30(3):1209–1250, 2020

    work page 2020

  8. [8]

    A variational representation for certain functionals of Brownian motion.The Annals of Probability, 26(4):1641–1659, 1998

    Michelle Bou ´e and Paul Dupuis. A variational representation for certain functionals of Brownian motion.The Annals of Probability, 26(4):1641–1659, 1998

    work page 1998

  9. [9]

    Empirical approximation to invariant measures of mean-field Langevin dynamics.arXiv preprint arXiv:2404.18164, 2024

    Wenjing Cao and Kai Du. Empirical approximation to invariant measures of mean-field Langevin dynamics.arXiv preprint arXiv:2404.18164, 2024

    work page arXiv 2024

  10. [10]

    Exponential decay of R´enyi divergence under Fokker–Planck equations.Journal of Statistical Physics, 176(5):1172–1184, 2019

    Yu Cao, Jianfeng Lu, and Yulong Lu. Exponential decay of R´enyi divergence under Fokker–Planck equations.Journal of Statistical Physics, 176(5):1172–1184, 2019

    work page 2019

  11. [11]

    Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case.IMA Journal of Numerical Analysis, page draf045, 2025

    Martin Chak and Pierre Monmarch ´e. Reflection coupling for unadjusted generalized Hamiltonian Monte Carlo in the nonconvex stochastic gradient case.IMA Journal of Numerical Analysis, page draf045, 2025

    work page 2025

  12. [12]

    Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics.Electronic Journal of Probability, 29:1–43, 2024

    Fan Chen, Yiqing Lin, Zhenjie Ren, and Songbo Wang. Uniform-in-time propagation of chaos for kinetic mean field Langevin dynamics.Electronic Journal of Probability, 29:1–43, 2024

    work page 2024

  13. [13]

    Analysis of Langevin Monte Carlo from Poincar´e to log-Sobolev.Foundations of Computational Mathematics, 25(4):1345–1395, 2025

    Sinho Chewi, Murat A Erdogdu, Mufan Li, Ruoqi Shen, and Matthew S Zhang. Analysis of Langevin Monte Carlo from Poincar´e to log-Sobolev.Foundations of Computational Mathematics, 25(4):1345–1395, 2025. 14

    work page 2025

  14. [14]

    Cover and Joy A

    Thomas M. Cover and Joy A. Thomas.Entropy, Relative Entropy, and Mutual Information, chap- ter 2, pages 13–55. John Wiley & Sons, Ltd, 2005

    work page 2005

  15. [15]

    American Math- ematical Soc., 2001

    Jean-Dominique Deuschel and Daniel W Stroock.Large deviations, volume 342. American Math- ematical Soc., 2001

    work page 2001

  16. [16]

    Reflection coupling and Wasserstein contractivity without convexity.Comptes Rendus Mathematique, 349(19-20):1101–1104, 2011

    Andreas Eberle. Reflection coupling and Wasserstein contractivity without convexity.Comptes Rendus Mathematique, 349(19-20):1101–1104, 2011

    work page 2011

  17. [17]

    Reflection couplings and contraction rates for diffusions.Probability Theory and Related Fields, 166(3):851–886, 2016

    Andreas Eberle. Reflection couplings and contraction rates for diffusions.Probability Theory and Related Fields, 166(3):851–886, 2016

    work page 2016

  18. [18]

    Couplings and quantitative contraction rates for Langevin dynamics.Annals of Probability, 47(4):1982–2010, 2019

    Andreas Eberle, Arnaud Guillin, and Raphael Zimmer. Couplings and quantitative contraction rates for Langevin dynamics.Annals of Probability, 47(4):1982–2010, 2019

    work page 1982

  19. [19]

    On transforming a certain class of stochastic processes by absolutely continuous substitution of measures.Theory of Probability&Its Applications, 5(3):285–301, 1960

    Igor Vladimirovich Girsanov. On transforming a certain class of stochastic processes by absolutely continuous substitution of measures.Theory of Probability&Its Applications, 5(3):285–301, 1960

    work page 1960

  20. [20]

    Convergence rates for the Vlasov-Fokker- Planck equation and uniform in time propagation of chaos in non convex cases.Electronic Journal of Probability, 27:1–44, 2022

    Arnaud Guillin, Pierre Le Bris, and Pierre Monmarch ´e. Convergence rates for the Vlasov-Fokker- Planck equation and uniform in time propagation of chaos in non convex cases.Electronic Journal of Probability, 27:1–44, 2022

    work page 2022

  21. [21]

    Exponential ergodicity in relative entropy andL 2-Wasserstein distance for non-equilibrium partially dissipative kinetic SDEs.arXiv preprint arXiv:2507.02518, 2025

    Xing Huang, Eva Kopfer, Pierre Monmarch ´e, and Panpan Ren. Exponential ergodicity in relative entropy andL 2-Wasserstein distance for non-equilibrium partially dissipative kinetic SDEs.arXiv preprint arXiv:2507.02518, 2025

    work page arXiv 2025

  22. [22]

    Ergodicity and long-time behavior of the random batch method for interacting particle systems.Mathematical Models and Methods in Applied Sci- ences, 33(01):67–102, 2023

    Shi Jin, Lei Li, Xuda Ye, and Zhennan Zhou. Ergodicity and long-time behavior of the random batch method for interacting particle systems.Mathematical Models and Methods in Applied Sci- ences, 33(01):67–102, 2023

    work page 2023

  23. [23]

    Ergodicity of the underdamped mean-field Langevin dynamics.The Annals of Applied Probability, 34(3):3181–3226, 2024

    Anna Kazeykina, Zhenjie Ren, Xiaolu Tan, and Junjian Yang. Ergodicity of the underdamped mean-field Langevin dynamics.The Annals of Applied Probability, 34(3):3181–3226, 2024

    work page 2024

  24. [24]

    Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo.arXiv preprint arXiv:2511.04041, 2025

    Lei Li, Jian-Guo Liu, and Yuliang Wang. Relative entropy estimate and geometric ergodicity for implicit Langevin Monte Carlo.arXiv preprint arXiv:2511.04041, 2025

    work page arXiv 2025

  25. [25]

    Ergodicity and error estimate of laws for a random splitting Langevin Monte Carlo.arXiv preprint arXiv:2510.07676, 2025

    Lei Li, Chen Wang, and Mengchao Wang. Ergodicity and error estimate of laws for a random splitting Langevin Monte Carlo.arXiv preprint arXiv:2510.07676, 2025

    work page arXiv 2025

  26. [26]

    A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics.CSIAM Transactions on Applied Mathematics, 6(4):711–759, 2025

    Lei Li and Yuliang Wang. A sharp uniform-in-time error estimate for Stochastic Gradient Langevin Dynamics.CSIAM Transactions on Applied Mathematics, 6(4):711–759, 2025

    work page 2025

  27. [27]

    Coupling of multidimensional diffusions by reflection.The Annals of Probability, pages 860–872, 1986

    Torgny Lindvall and L Cris G Rogers. Coupling of multidimensional diffusions by reflection.The Annals of Probability, pages 860–872, 1986

    work page 1986

  28. [28]

    Exponential convergence inL p-Wasserstein distance for diffusion pro- cesses without uniformly dissipative drift.Mathematische Nachrichten, 289(14-15):1909–1926, 2016

    Dejun Luo and Jian Wang. Exponential convergence inL p-Wasserstein distance for diffusion pro- cesses without uniformly dissipative drift.Mathematische Nachrichten, 289(14-15):1909–1926, 2016

    work page 1909

  29. [29]

    Nonasymptotic bounds for sam- pling algorithms without log-concavity.Annals of Applied Probability, 30(4):1534–1581, 2020

    Mateusz B Majka, Aleksandar Mijatovi ´c, and Łukasz Szpruch. Nonasymptotic bounds for sam- pling algorithms without log-concavity.Annals of Applied Probability, 30(4):1534–1581, 2020

    work page 2020

  30. [30]

    Bi-Coupling method and applications.Probability Theory and Related Fields, pages 1–27, 2025

    Panpan Ren and Feng-Yu Wang. Bi-Coupling method and applications.Probability Theory and Related Fields, pages 1–27, 2025

    work page 2025

  31. [31]

    Katharina Schuh. Global contractivity for Langevin dynamics with distribution-dependent forces and uniform in time propagation of chaos.Annales de l’Institut Henri Poincare (B) Probabilites et statistiques, 60(2):753–789, 2024. 15

    work page 2024

  32. [32]

    On the support of diffusion processes with applications to the strong maximum principle

    D Stroock and SRS Varadhan. On the support of diffusion processes with applications to the strong maximum principle. InProceedings of the Berkeley symposium on mathematical statistics and probability, volume 1, page 333, 1972

    work page 1972

  33. [33]

    A new isoperimetric inequality and the concentration of measure phenomenon

    Michel Talagrand. A new isoperimetric inequality and the concentration of measure phenomenon. InGeometric aspects of functional analysis, pages 94–124. Springer, 1991

    work page 1991

  34. [34]

    Rapid convergence of the unadjusted Langevin algorithm: Isoperimetry suffices.Advances in neural information processing systems, 32, 2019

    Santosh Vempala and Andre Wibisono. Rapid convergence of the unadjusted Langevin algorithm: Isoperimetry suffices.Advances in neural information processing systems, 32, 2019

    work page 2019

  35. [35]

    American Mathematical Society, 2009

    C ´edric Villani.Hypocoercivity, volume 202. American Mathematical Society, 2009

    work page 2009

  36. [36]

    Harnack inequalities on manifolds with boundary and applications.Journal de math´ ematiques pures et appliqu´ ees, 94(3):304–321, 2010

    Feng-Yu Wang. Harnack inequalities on manifolds with boundary and applications.Journal de math´ ematiques pures et appliqu´ ees, 94(3):304–321, 2010

    work page 2010

  37. [37]

    Coupling and applications

    Feng-Yu Wang. Coupling and applications. InStochastic Analysis and Applications to Finance: Essays in Honour of Jia-An Yan, pages 411–424. World Scientific, 2012

    work page 2012

  38. [38]

    Springer, 2013

    Feng-Yu Wang.Harnack inequalities for stochastic partial differential equations, volume 1332. Springer, 2013

    work page 2013

  39. [39]

    Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature.Potential Analysis, 53(3):1123–1144, 2020

    Feng-Yu Wang. Exponential contraction in Wasserstein distances for diffusion semigroups with negative curvature.Potential Analysis, 53(3):1123–1144, 2020

    work page 2020

  40. [40]

    Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits.IMA Journal of Numerical Analysis, 44(3):1660– 1698, 2024

    Xuda Ye and Zhennan Zhou. Error analysis of time-discrete random batch method for interacting particle systems and associated mean-field limits.IMA Journal of Numerical Analysis, 44(3):1660– 1698, 2024

    work page 2024

  41. [41]

    Analysis of Langevin midpoint methods using an anticipative Girsanov theorem

    Matthew S Zhang. Analysis of Langevin midpoint methods using an anticipative Girsanov theorem. arXiv preprint arXiv:2507.12791, 2025. 16

    work page arXiv 2025