Quantitative positivity of transition densities for random perturbations of Hamiltonian systems

David P. Herzog; Kyle L. Liss; Shimaa Elesaely

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Quantitative positivity of transition densities for random perturbations of Hamiltonian systems

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Pith's one-line read For a wide class of noise-perturbed Hamiltonian systems, transition densities are shown to have a positive, ε-independent lower bound at the slow equilibration time t0/ε, with no knowledge of the invariant measure required.

desk verdict A genuinely new small-noise minorization theorem for hypoelliptic Hamiltonian systems; the main risk is the weakly checked time-reversed boundary regularity in Assumption 2.3. read the letter →

arxiv 2509.02448 v1 pith:CHZLP2K5 submitted 2025-09-02 math.PR math.AP

Shimaa Elesaely , David P. Herzog , Kyle L. Liss This is my paper · ORCID

classification math.PRmath.AP MSC 60J6060H1035H1060J35

keywords quantitativeminorizationtransitiondensitypositivitysmall-noisediffusionshypoellipticdynamicsLangevinLorenz-96modelGalerkinNavier-StokesDeGiorgi-Moseriteration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves a quantitative minorization theorem for diffusions obtained by adding small noise ε to a conservative Hamiltonian flow. The main result (Theorem 2.1) says that for every large energy level R there are t0 > 0 and λ > 0, both independent of ε, such that the transition density satisfies inf_{x,y ∈ H_R} p^ε_{t0/ε}(x,y) ≥ λ on the slow, equilibrium time scale 1/ε. The point is that the lower bound is explicit and uniform in the noise strength, and the proof never uses the invariant measure—it does not even require one to exist. That makes the theorem applicable to Langevin dynamics with anisotropic heat baths, oscillator chains with ends at different temperatures, and finite-dimensional fluid models such as Lorenz-96 and Galerkin truncations of Navier-Stokes, where an explicit stationary density is unknown. With a mild extra Lyapunov assumption (Assumption 2.4), the minorization upgrades to the sharp exponential convergence rate e^{-cεt} in weighted total variation, recovering the quantitative strength of functional-analytic hypocoercivity in settings where those methods cannot run.

What carries the argument

The load-bearing object is the time-averaged transition density h^ε(x,y) = α∫_{t0}^∞ e^{-α(t-t0)}q^ε_t(x,y)dt, the one-step density of a kernel that samples the process at an exponentially distributed time. Unlike the raw density q^ε_t, h^ε has uniform-in-ε fractional Sobolev regularity in time and space, which the quantitative parabolic Hörmander estimates and the quantitative Moser iteration (Theorem 7.1) exploit. Two further mechanisms are essential: De Giorgi-type truncations w_k = φ_ε(1 - (4/θ)^k h^ε/δ_R,S), whose L²-smallness implies h^ε is pointwise large; and a quantitative Steinhaus lemma built on an additive-combinatorics sumset theorem, guaranteeing that the set of good times cont

What would settle it

Concrete test: compute inf_{z,w∈H_R} p^γ_{t0/γ}(z,w) numerically for underdamped Langevin dynamics with a double-well potential, for γ = 10^{-2}, 10^{-3}, 10^{-4}. The theorem predicts a positive limit λ independent of γ; an infimum decaying to zero would refute it. Separately, run the same check on the paper's own non-regular example (Remark 2.4), the hypoelliptic process dX_t = Y_t²dt, dY_t = dB_t on the square (-1,1)², where boundary regularity fails: a uniform lower bound there would show Assumption 2.3 is a proof artifact rather than a necessary condition.

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Extended reading notes

Core claim

The central discovery is that quantitative positivity of the time-rescaled density q^ε_t = p^ε_{t/ε} can be reached through its exponentially time-averaged version, h^ε(x,y) = α∫_{t0}^∞ e^{-α(t-t0)}q^ε_t(x,y)dt. The raw density moves on the fast Hamiltonian scale and lacks uniform-in-ε time regularity; the averaged density has it. Quantitative Hörmander-type smoothing and a quantitative Moser iteration furnish uniform upper bounds on h^ε; a De Giorgi-type iteration on smoothed truncations (1 - Ch^ε)_+ converts those upper bounds into a uniform lower bound; and a quantitative Steinhaus lemma from additive combinatorics promotes the averaged lower bound to a fixed-time lower bound at t0/ε, wit

Load-bearing premise

Assumption 2.3: for every energy level R there must be a bounded, continuous-boundary domain containing the sublevel set H_R from which both the diffusion and its time-reversed partner exit instantly from every boundary point. If no such domain exists, the classical solvability step inside the Moser iteration (Theorem 7.1) collapses and the whole chain breaks. The paper checks it for Langevin by citing a 'nearly identical' reversed-drift argument, and for fluid models by rand

Editorial extensions

If this is right

For Langevin dynamics with fairly general confining potentials (Assumption 1.1), the theorem yields inf_{z,w ∈ H_R} p^γ_{t0/γ}(z,w) ≥ λ with λ, t0 independent of the friction γ, on the optimal γ^{-1} equilibration timescale.
Adding the Lyapunov condition (Assumption 2.4) upgrades strict positivity to the exponential rate e^{-cεt} in weighted total variation, with c,C independent of ε — the first such quantitative rate for anisotropically heated chains of oscillators and degenerate fluid models.
Because the invariant measure is never used, the same theorem covers systems where no closed-form stationary density exists (e.g., the non-equilibrium oscillator chain, Remark 3.1 and Example 3.2) and where existence of a stationary distribution is not even proved (Lorenz-96, Galerkin Navier-Stokes, Examples 3.3-3.4).
The paper's stated limitation (Remark 1.1): the proof gives no information on timescales shorter than γ^{-1}; positivity on level sets, i.e. metastable relaxation, is left open.
The method positions probabilistic minorization as quantitatively competitive with L²/hypocoercivity approaches, closing part of the gap between the two convergence-theory styles.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction of t0 in Section 5 is explicit but very conservative (t0 ≤ 5m(1+7nm) with n ≈ 20L/η); in concrete models a direct coupling or spectral argument could plausibly yield far smaller t0, and λ is the quantity worth comparing across models.
The time-averaging device is essentially a template: whenever a degenerate-noise diffusion's density lacks uniform time regularity but its time-average has it, the upper-bounds-to-lower-bounds mantra should transfer — suggesting extensions to infinite-dimensional settings (SPDEs) beyond the finite-dimensional fluid truncations treated here.
The two distinct mechanisms used to verify Assumption 2.3 — hyperbolicity of level sets for Langevin, random convex hulls for fluid models — hint that boundary regularity is a mild geometric condition; a testable conjecture is that generic small perturbations of the drift render almost every level set boundary-regular, removing the assumption's case-by-case character.
If the theorem is right, quantitative minorization follows from uniform hypoellipticity plus weak dissipation plus boundary regularity, without any reversibility or invariant-measure structure; that suggests a systematic program for non-reversible and far-from-equilibrium systems where both L²-hypocoercivity and explicit Gibbs measures are unavailable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core result is new and worth engaging with. Replacing the stationary-density route of Bedrossian–Liss with a time-averaged transition density is a real idea, and it buys something useful: quantitative minorization on the 1/epsilon time scale without knowing or even needing an invariant measure. The uniform-in-epsilon bounds and the quantitative Steinhaus/small-set argument look like solid work. The examples are also meaningful, especially multi-temperature oscillator chains and degenerate fluid models where no invariant measure is known.

The proof is long but mostly careful. The main soft spot is exactly where the stress-test note lands. Assumption 2.3 is load-bearing for the Moser/Dirichlet step in Section 7, and for the headline Langevin example it is verified in Lemma 3.1 by changing the sign of the drift in [22, Theorem 8.4] and saying the proof follows 'nearly identically.' That is not a routine symmetry: the reversed process has drift +epsilon|v|^2, and boundary regularity is sensitive to whether the process exits immediately at degenerate boundary points. The assumption may well be true, but it needs a real proof, not a one-line analogy. For the fluid examples the random-convex-hull construction is more convincing, since non-tangent Brownian noise forces immediate exit regardless of drift. Still, the Langevin case is the paper's advertised application, so this is not a minor footnote.

A second, softer concern is that the parameter-dependent Hormander estimates and parts of the Moser machinery are imported from [8], which has author overlap. That is not disqualifying — I would just want the authors to state clearly which theorems are assumed from prior work and make the paper readable without cross-checking [8] line by line. The citation pattern is otherwise fine.

Who is this for? People working on quantitative ergodicity, hypoelliptic diffusions, and small-noise limits. It deserves a serious referee. My recommendation: send it out, but ask for a complete verification of Lemma 3.1 for the time-reversed Langevin process, or a revised assumption that avoids delegating the reversed boundary regularity check.

Referee Report

2 major / 4 minor

Summary. The paper proves a quantitative small-noise minorization bound for a class of diffusion processes of the form (2.1) with a Hamiltonian-type conservation structure. Under Assumptions 2.1–2.3 (regularity of sublevel sets H_R, a Lyapunov/subinvariance structure, and boundary regularity of an enclosing domain for both the forward and time-reversed process), Theorem 2.1 gives t0, λ > 0, independent of ε, such that inf_{x,y ∈ H_R} p^ε_{t0/ε}(x,y) ≥ λ. Applications include Langevin dynamics, oscillator chains, Lorenz '96, and Galerkin fluid models. The proof uses time-averaged transition densities, quantitative Hörmander estimates, Moser iteration for upper bounds, a De Giorgi-type argument for lower bounds, and a quantitative petite-set argument. A further Lyapunov assumption yields exponential convergence in weighted total variation.

Significance. If the proof is correct, the paper gives a substantial advance: quantitative minorization at the ε^{-1} time scale for a general class of hypoelliptic small-noise systems without requiring knowledge of an invariant measure. The idea of time-averaging the transition density to gain uniform-in-ε time regularity, then converting quantitative upper bounds into quantitative lower bounds, is original and likely to be influential. The paper carefully tracks constants, gives explicit roles for each assumption, and includes a genuinely quantitative Steinhaus-type lemma. The main weakness is the verification of Assumption 2.3 for the time-reversed Langevin process, which is load-bearing and is currently delegated to a one-line analogy with a preprint.

major comments (2)

[Section 3, Lemma 3.1] The verification of Assumption 2.3 for the time-reversed Langevin process is the least supported load-bearing step. Lemma 3.1 dismisses the reversed case with 'the rest of the proof follows in a nearly identical way' after changing the sign of the drift. This is not a routine symmetry: the reversed generator is L^-_ε = -Z0 + εZ + εΣZ_j^2 (Eq. (7.11)), and at boundary points where the noise directions are tangent (for Langevin, v = 0), the drift sign controls whether the process enters or exits the domain. Assumption 2.3 is used in Theorem 7.1 for both M_ε = L_ε and M_ε = L^-_ε (Eq. (7.13) and the choice of O_R), and Lemma 7.1 and Theorem 8.1 invoke Theorem 7.1 with the reversed operator. A failure of boundary regularity for the reversed process would break the Moser upper bound and hence the minorization conclusion. The authors should either provide a complete proof, not a citation to [2
[Sections 6–7, Theorems 6.3/6.4 and Theorem 7.1] The proof of Theorem 7.1 relies on the quantitative Hörmander estimates of Theorems 6.3 and 6.4, but those are quoted from [8] without proof, and the correspondence between Definition 2.1 and the stratified condition S_{s,0}(O)=S(O) used in Theorem 6.4 is only asserted. In particular, Lemma 7.2 states that the product vector-field list satisfies the uniform parabolic Hörmander condition 'because derivatives in x and y commute,' which needs a detailed verification under the parameter-dependent coefficients. These estimates control the H^s regularity in (4.3) and feed directly into Theorems 4.1 and 4.3. The manuscript should either reproduce the relevant arguments or state a precise dictionary from the assumptions to the hypotheses of Theorems 6.3 and 6.4.

minor comments (4)

[Theorem 4.3 statement] The theorem states sup_{x,y ∈ H_R} p_{t0}(x,y) ≥ λ, but the proof establishes the stronger inf_{x,y ∈ H_R} p_{t0}(x,y) ≥ λ, which is also what is used to obtain Theorem 2.1. Please correct the displayed statement.
[Section 8, proof of Lemma 8.1] In the sentence 'Choosing S = (2C_R)^{-1}', the displayed formula is inconsistent with the requirement S > R and with the preceding bound C_R/(1+S) ≤ 1/2. It should be S = 2C_R (or any sufficiently large S).
[Example 3.3, proof of Theorem 3.3] In the estimate verifying Assumption 2.2 (V4), the term '2λ_2|x_3|^2' should read '2λ_3|x_3|^2'.
[Abstract] Minor typo: 'heat bathes' should be 'heat baths'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main minorization theorem is derived from the stated assumptions via quantitative Hörmander, Moser/De Giorgi and small-set arguments; the self-citations to [8] and [22] are independent technical inputs, not disguised conclusions.

full rationale

The paper's central result, Theorem 2.1, is not an input to Assumptions 2.1–2.3 nor a renaming of them. The proof proceeds through genuinely independent intermediate results: Theorem 4.1 gives quantitative upper bounds, Theorem 4.2 gives a time-averaged lower bound, and Theorem 4.3 converts these into a fixed-time minorization using a quantitative Steinhaus/small-set argument. The Moser iteration (Theorem 7.1), the De Giorgi lower-bound argument, and the time-averaged density hε are all proven in the paper from the assumptions, with no fitted parameters or normalization that forces the conclusion. The cited prior works with author overlap ([8], [22], and [13]) are used for parameter-free technical estimates—quantitative Hörmander smoothing and boundary regularity—whose stated assumptions do not include the target minorization bound; under the review rules these count as independent support and do not raise the circularity score. One limitation should be noted: Lemma 3.1 verifies the load-bearing boundary-regularity Assumption 2.3 for Langevin dynamics by citing [22, Theorem 8.4] and asserting that the time-reversed case 'follows in a nearly identical way' after changing the sign of the drift. This is a deferred verification and a potential correctness gap, but it is not a circular reduction: the reversed-process claim is not obtained by assuming the conclusion of Theorem 2.1, and the forward case is attributed to an independent prior theorem. Overall, no step in the derivation chain is equivalent by construction to its inputs.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central theorem rests on broad domain assumptions about the Hamiltonian, the vector fields, and boundary regularity, together with imported technical estimates from prior literature. No free constants are fitted to data and no new physical entities are introduced.

free parameters (2)

eta = assumed in Assumption 2.2(V4); examples: eta=1 for Langevin, 0<eta<=1/Tmax for oscillator chains, eta<=min{lambda1/(2sig
Lyapunov exponent in the subinvariance inequality. Existence is assumed independent of epsilon; in examples it is chosen by hand, not fitted to data.
d* = arbitrary positive constant in Assumption 2.2(V4)
Upper bound constant in the inequality eta sum (Z_j H)^2 <= ZH + sum Z_j^2 H <= ZH + d*/2. The value is not specified; only existence matters.

assumptions (7)

domain assumption Assumption 2.1 (H1-H4): H independent of epsilon, H_R open with compact closure, H smooth on X, X connected.
Defines the state space and level-set geometry used throughout the paper.
domain assumption Assumption 2.2 (V1-V5): divergence conditions, conservation Z0 H = 0, subinvariance inequalities involving eta and d*, and uniform parabolic Hormander condition on each H_R.
This is the quantitative hypoellipticity and Lyapunov structure that the main theorem requires.
domain assumption Assumption 2.3: for each R there exists a boundary-regular open set O_R containing H_R for both the forward and time-reversed processes.
Needed for classical well-posedness of Poisson/Dirichlet problems in the Moser iteration, and verified separately in examples.
domain assumption Assumption 2.4 (LF1-LF2): epsilon-uniform Lyapunov function with exponential drift, used only in Theorem 2.2.
Converts minorization into exponential convergence; not needed for the main minorization theorem.
standard math Quantitative Hormander smoothing estimates from [8] and Hormander [33], stated as Theorems 6.3 and 6.4.
Imported rather than reproved; they provide uniform Hs control of densities in epsilon.
standard math Boundary regularity results from [13] and [22] used to verify Assumption 2.3 in examples.
Carfagnini-Foldes-Herzog [13] supplies the convex-hull boundary regularity criterion; Foldes-Herzog [22] supplies the Langevin level-set regularity.
standard math De Giorgi iteration, Moser iteration, and Meyn-Tweedie small-set arguments.
Template for converting quantitative upper bounds into lower bounds and for extracting fixed-time minorization.

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Cite this review

Pith. "Pith review of Quantitative positivity of transition densities for random perturbations of Hamiltonian systems." pith.science (2026). https://pith.science/paper/CHZLP2K5

@misc{pith2026250902448,
  author       = {Pith},
  title        = {Pith review of: Quantitative positivity of transition densities for random perturbations of Hamiltonian systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CHZLP2K5}},
  note         = {Machine review of arXiv:2509.02448}
}

abstract

We study a class of diffusion processes arising from random perturbations of conservative Hamiltonian systems. Under a set of abstract hypotheses -- including basic structural assumptions on the Hamiltonian, a weak Lyapunov structure, and a quantitative notion of hypoellipticity -- we prove that transition densities satisfy a sharp, uniform pointwise lower bound over Hamiltonian sublevel sets in the small noise limit $\epsilon \to 0$. By applying our general theorem, we obtain quantitative minorization estimates for a variety of models including Langevin dynamics, chains of oscillators coupled to heat bathes at different temperatures, and finite-dimensional fluid models such as stochastically forced Galerkin truncations of the Navier-Stokes equations and the Lorenz '96 system. As a corollary, assuming a stronger Lyapunov structure, our main result yields a sharp exponential rate of convergence to equilibrium for $0 < \epsilon \ll 1$ in a weighted total variation norm. A central feature of our approach is that it does not require knowledge of the explicit form of the invariant measure, nor even its existence, and hence is broadly applicable to deduce minorization for physically relevant systems where invariant measures are inaccessible.

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Works this paper leans on

52 extracted references · 52 canonical work pages · cited by 1 Pith paper

[8]

Bedrossian and K

J. Bedrossian and K. Liss. Quantitative spectral gaps for hypoelliptic stochastic differential equations with small noise. Prob. Math. Phys., 2(3):477–532, 2021

work page 2021
[1]

Albritton, S

D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack. Variational methods for the kinetic fokker– planck equation. Anal. PDE, 17(6):1953–2010, 2024

work page 1953
[2]

Albritton, R

D. Albritton, R. Beekie, and M. Novack. Enhanced dissipation and Hörmander’s hypoellipticity. J. Funct. Anal., 283(3):109522, 2022

work page 2022
[3]

Arnold, J

A. Arnold, J. Dolbeault, C. Schmeiser, and T. Wöhrer. Sharpening of decay rates in fourier based hypocoercivity methods. In Recent advances in kinetic equations and applications , pages 1–50. Springer, 2021

work page 2021
[4]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations. Springer, 2011

work page 2011
[5]

Baudoin, M

F. Baudoin, M. Gordina, and D. P. Herzog. Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials. Arch. Ration. Mech. Anal., 241(2):765–804, 2021

work page 2021
[6]

Baudoin, M

F. Baudoin, M. Gordina, D.P. Herzog, J. Kim, and T. Melcher. Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise. J. Funct. Anal, 288(6):110814, 2025

work page 2025
[7]

Bedrossian and M

J. Bedrossian and M. Coti-Zelati. Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Archi. Ration. Mech., 224(3):1161–1204, 2017

work page 2017

Show all 52 references

[9]

Ergodic properties of Markov processes

L-R Bellet. Ergodic properties of Markov processes. In Open Quantum Systems II: The Markovian Approach, pages 1–39. Springer, 2006

2006
[10]

Brigati and G

G. Brigati and G. Stoltz. How to construct explicit decay rates for kinetic Fokker–Planck equations? SIAM J. on Math. Anal., 57(4):3587–3622, 2025

2025
[11]

Camrud, D

E. Camrud, D. P. Herzog, G. Stoltz, and M. Gordina. Weighted L2-contractivity of Langevin dynamics with singular potentials. Nonlinearity, 35(2):998, 2021. 59

2021
[12]

Y . Cao, J. Lu, and L. Wang. On explicit L2-convergence rate estimate for underdamped Langevin dynamics. Arch. Ration. Mech. Anal., 247(5):90, 2023

2023
[13]

Carfagnini, J

M. Carfagnini, J. Földes, and D. P. Herzog. A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero. Stoc. Process. Appl., 149:188–223, 2022

2022
[14]

Coti-Zelati and M

M. Coti-Zelati and M. Dolce. Separation of time-scales in drift-diffusion equations on r2. J. Math. Pure. Appl., 142:58–75, 2020

2020
[15]

Cuneo, J.-P

N. Cuneo, J.-P. Eckmann, M. Hairer, and L. Rey-Bellet. Non-equilibrium steady states for networks of oscillators. Electron. J. Probab., 23:1 – 28, 2018

2018
[16]

De Giorgi

E. De Giorgi. Sulla differenziabilitae l’analiticita delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Scr. Fis. Mat. Nat., 3(3):25–43, 1957

1957
[17]

Dolbeault, C

J. Dolbeault, C. Mouhot, and C. Schmeiser. Hypocoercivity for linear kinetic equations conserving mass. Trans. Am. Math. Soc., 367(6):3807–3828, 2015

2015
[18]

R. Douc, G. Fort, and A. Guillin. Subgeometric rates of convergence of f-ergodic strong Markov pro- cesses. Stoch. Process. Appl., 119(3):897–923, 2009

2009
[19]

Eberle, A

A. Eberle, A. Guillin, and R. Zimmer. Couplings and quantitative contraction rates for Langevin dy- namics. Ann. of Probab., 47(4):1982 – 2010, 2019

1982
[20]

J. Evans. Hypocoercivity in Wasserstein-1 for the kinetic Fokker-Planck equation via Malliavin calcu- lus. arXiv preprint arXiv:1810.01324, 2018

2018 arXiv
[21]

Földes and D

J. Földes and D. P. Herzog. The method of stochastic characteristics for linear second-order hypoelliptic equations. Probab. Surv., 20:113–169, 2023

2023
[22]

Földes and D

J. Földes and D. P. Herzog. Small-time asymptotics for hypoelliptic diffusions. arXiv preprint arXiv:2412.11323, 2024

2024 arXiv
[23]

Gardner, K.L

V . Gardner, K.L. Liss, and J.C. Mattingly. A pathwise approach to the enhanced dissipation of passive scalars advected by shear flows. arXiv preprint arXiv:2410.05657, 2024

2024 arXiv
[24]

N. E. Glatt-Holtz, D. P. Herzog, and J. C. Mattingly. Scaling and saturation in infinite-dimensional control problems with applications to stochastic partial differential equations. Ann. PDE, 4(2):1–103, 2018

2018
[25]

Golse, C

F. Golse, C. Imbert, C. Mouhot, and A. Vasseur. Harnack inequality for kinetic Fokker-Planck equa- tions with rough coefficients and application to the Landau equation. Ann. Scuola Norm-Sci. , pages 253–295, 2019

2019
[26]

H\"{o} lder regularity for hypoelliptic kinetic equations with rough diffusion coefficients

François Golse and Alexis Vasseur. H\"{o} lder regularity for hypoelliptic kinetic equations with rough diffusion coefficients. arXiv preprint arXiv:1506.01908, 2015

2015 arXiv
[27]

Grothaus and P

M. Grothaus and P. Stilgenbauer. Hypocoercivity for Kolmogorov backward evolution equations and applications. J. Funct. Anal., 267(10):3515–3556, 2014

2014
[28]

M. Hairer. Convergence of Markov processes. Lecture notes, 18(26):11, 2010

2010
[29]

Hairer and J.C

M. Hairer and J.C. Mattingly. Yet another look at Harris’ ergodic theorem for Markov chains. In Sem- inar on Stochastic Analysis, Random Fields and Applications VI: Centro Stefano Franscini, Ascona, May 2008, pages 109–117. Springer, 2011

2008
[30]

F. Hérau. Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltz- mann equation. Asymptot. Anal., 46(3-4):349–359, 2006

2006
[31]

D. P. Herzog and J. C. Mattingly. A practical criterion for positivity of transition densities.Nonlinearity, 28(8):2823–2845, 2015

2015
[32]

Herzog and J

D.P. Herzog and J. C. Mattingly. Ergodicity and Lyapunov functions for Langevin dynamics with singular potentials. Comm. Pure. Appl. Math., 72(10):2231–2255, 2019

2019
[33]

Hörmander

L. Hörmander. Hypoelliptic second order differential equations. Acta Math., 119(1):147–171, 1967. 60

1967
[34]

Hörmander

L. Hörmander. The analysis of linear partial differential operators III: Pseudo-differential operators . Springer Science & Business Media, 2007

2007
[35]

Leimkuhler, M

B. Leimkuhler, M. Sachs, and G. Stoltz. Hypocoercivity properties of adaptive Langevin dynamics. SIAM J. on Appl. Math., 80(3):1197–1222, 2020

2020
[36]

Leliévre, G

T. Leliévre, G. Stoltz, and M. Rousset. Free energy computations: A mathematical perspective. World Scientific, 2010

2010
[37]

V . F. Lev. Consecutive integers in high-multiplicity sumsets.Acta Math. Hung., 129(3):245–253, 2010

2010
[38]

Mattingly, A.M

J.C. Mattingly, A.M. Stuart, and D.J. Higham. Ergodicity for SDEs and approximations: locally Lips- chitz vector fields and degenerate noise. Stoch. Process. Appl., 101(2):185–232, 2002

2002
[39]

Meyn and R

S.P. Meyn and R. L. Tweedie. Markov chains and stochastic stability . Springer Science & Business Media, 2012

2012
[40]

J. Moser. On Harnack’s theorem for elliptic differential equations.Comm. Pure Appl. Math., 14(3):577– 591, 1961

1961
[41]

Oksendal

B. Oksendal. Stochastic differential equations: an introduction with applications. Springer Science & Business Media, 2013

2013
[42]

Prochno, C

J. Prochno, C. Schütt, M. Sonnleitner, and E. M. Werner. Random approximation of convex bodies in Hausdorff metric. arXiv preprint arXiv:2404.02870, 2024

2024 arXiv
[43]

Roussel and G

J. Roussel and G. Stoltz. Spectral methods for langevin dynamics and associated error estimates. ESAIM: M2AN, 52(3):1051–1083, 2018

2018
[44]

P. R. Stinga. Regularity Techniques for Elliptic PDEs and the Fractional Laplacian . Chapman and Hall/CRC, 2024

2024
[45]

Stroock and S.R.S

D. Stroock and S.R.S. Varadhan. On degenerate elliptic-parabolic operators of second order and their associated diffusions. Comm. Pure Appl. Math., 25(6):651–713, 1972

1972
[46]

Stroock and S.R.S

D. Stroock and S.R.S. Varadhan. On the support of diffusion processes with applications to the strong maximum principle. In Proceedings of the Berkeley symposium on mathematical statistics and proba- bility, volume 1, page 333, 1972

1972
[47]

D. Talay. Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and dis- cretization by the implicit Euler scheme. Markov Process. Related Fields, 8(2):163–198, 2002

2002
[48]

C. Villani. Hypocoercive diffusion operators. In International Congress of Mathematicians, volume 3, pages 473–498, 2006

2006
[49]

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

2009
[50]

Villringer

D. Villringer. Enhanced dissipation via the Malliavin calculus. Electron. Comm. Probab. , 30:1–11, 2025

2025
[51]

Vukadinovic

J. Vukadinovic. The limit of vanishing diffusivity for passive scalars in hamiltonian flows. Arch. for Ration. Mech., 242(3):1395–1444, 2021

2021
[52]

C Mattingly

E Weinan and J. C Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forc- ing: finite-dimensional approximation. Comm. Pure Appl. Math, 54(11):1386–1402, 2001. 61

2001

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [8]

Bedrossian and K

J. Bedrossian and K. Liss. Quantitative spectral gaps for hypoelliptic stochastic differential equations with small noise. Prob. Math. Phys., 2(3):477–532, 2021

work page 2021

[2] [1]

Albritton, S

D. Albritton, S. Armstrong, J.-C. Mourrat, and M. Novack. Variational methods for the kinetic fokker– planck equation. Anal. PDE, 17(6):1953–2010, 2024

work page 1953

[3] [2]

Albritton, R

D. Albritton, R. Beekie, and M. Novack. Enhanced dissipation and Hörmander’s hypoellipticity. J. Funct. Anal., 283(3):109522, 2022

work page 2022

[4] [3]

Arnold, J

A. Arnold, J. Dolbeault, C. Schmeiser, and T. Wöhrer. Sharpening of decay rates in fourier based hypocoercivity methods. In Recent advances in kinetic equations and applications , pages 1–50. Springer, 2021

work page 2021

[5] [4]

Bahouri, J

H. Bahouri, J. Chemin, and R. Danchin. Fourier analysis and nonlinear partial differential equations. Springer, 2011

work page 2011

[6] [5]

Baudoin, M

F. Baudoin, M. Gordina, and D. P. Herzog. Gamma calculus beyond Villani and explicit convergence estimates for Langevin dynamics with singular potentials. Arch. Ration. Mech. Anal., 241(2):765–804, 2021

work page 2021

[7] [6]

Baudoin, M

F. Baudoin, M. Gordina, D.P. Herzog, J. Kim, and T. Melcher. Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise. J. Funct. Anal, 288(6):110814, 2025

work page 2025

[8] [7]

Bedrossian and M

J. Bedrossian and M. Coti-Zelati. Enhanced dissipation, hypoellipticity, and anomalous small noise inviscid limits in shear flows. Archi. Ration. Mech., 224(3):1161–1204, 2017

work page 2017