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Gradient flow of the infinite-volume free energy for lattice systems of continuous spins

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Pith's one-line read This paper proves that, for infinite lattice systems of continuous spins on a compact manifold, the gradient flow of the infinite-volume free energy and the law of the spins under the overdamped Langevin dynamics coincide, because both…

desk verdict A credible first construction of the infinite-volume gradient flow for interacting lattice spins, but Theorem 3.7 states the EVI constant with the wrong norm and needs a one-line repair before the exponential-rate results are proven. read the letter →

arxiv 2502.06500 v1 pith:M7KDPNIW submitted 2025-02-10 math.PR cond-mat.stat-mechmath.AP

Ronan Herry , Thomas Leblé This is my paper · ORCID
classification math.PRcond-mat.stat-mechmath.AP MSC 60K3582B2049Q2260J6058J65
keywords Wassersteingradientflowinfinite-volumefreeenergylatticespinsystemsoverdampedLangevindynamicsFokker-Planck-KolmogorovequationsEvolutionVariationalInequalitydisplacementconvexityspecificdistance
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

The central claim is that two a priori different descriptions of how a lattice spin system relaxes toward equilibrium — the Wasserstein gradient flow of the infinite-volume free energy, and the law of the spins under the infinite-volume overdamped Langevin dynamics — are in fact the same trajectory. The identification is achieved by showing that both objects satisfy the same hierarchy of coupled parabolic equations, the infinite-volume Fokker-Planck-Kolmogorov equations, that weak solutions of this hierarchy automatically have smooth finite-box densities, and that strong solutions are unique via an Evolution Variational Inequality. A second group of results controls the long-time behaviour: the free energy is non-increasing along the flow, and when the spin space has uniformly positive Ricci curvature and the temperature is high enough, the flow converges exponentially fast to the unique minimizer of the free energy in specific Wasserstein distance and in free energy. The relevance is that an interacting spin system in contact with a heat bath is thereby shown to be a steepest descent of its own free energy, a relation previously established only in finite volume.

What carries the argument

The argument is carried by the infinite-volume Fokker-Planck-Kolmogorov hierarchy — a family of coupled parabolic PDEs, one for each finite box $\Lambda$, with drifts depending on conditional expectations of the full infinite-volume interaction — together with the specific Wasserstein distance and the free energy $F^\beta$ built from specific relative entropy and interaction-energy density. Two analytic pillars support the main theorem: a regularity result stating that every weak solution of the hierarchy has $C^{1,2}$ local densities (proved by a bootstrap on Duhamel's principle with heat-kernel estimates on the compact manifold), and an Evolution Variational Inequality for $F^\beta$ with respect to $W$ (proved by taking finite-volume EVIs to the limit), which yields uniqueness. The gradient flow is constructed by a discrete minimizing-movement scheme in the Wasserstein space with a stationarization step that converts finite-box minimizers into translation-invariant measures; the diffusion is constructed by embedding the configuration space into a weighted $\ell^2$ Hilbert space in which the interaction gradient is Lipschitz, following classical infinite-volume SDE methods.

What would settle it

For a constant interaction potential $\Psi \equiv 1$ (so $\nabla^2\Psi \equiv 0$), the interaction exerts no force and the flow reduces to independent diffusions with Bakry-Émery constant $\kappa$, while the stated $K_\beta$ would be $\kappa - 2\beta\|J\|_{\ell^1}$; checking whether the EVI proof's curvature inequality $\mathrm{Ric} + \nabla^2 U + \beta\nabla^2 H \ge K_\beta\, g$ holds in this example would settle whether the stated constant follows from the proof.

Watch

Extended reading notes

Core claim

On the space of translation-invariant spin measures, the paper defines the infinite-volume free energy $F^\beta(P) = E(P) + \beta H(P)$, with $E$ the specific relative entropy and $H$ the interaction-energy density, together with the specific Wasserstein distance $W(P,Q)=\lim_n |\Lambda_n|^{-1} W_n^2(P,Q)$. It constructs two evolutions: the gradient flow of $F^\beta$ obtained as the limit of a discrete variational (JKO-type) scheme with a stationarization step, and the law of the infinite-volume overdamped Langevin dynamics obtained through a weighted Hilbert-space embedding. The paper then proves that both curves satisfy the same infinite-volume Fokker-Planck-Kolmogorov hierarchy in the dual sense; that weak solutions automatically regularize into strong solutions with smooth finite-box densities; and that strong solutions are unique because they satisfy an Evolution Variational Inequality with respect to $W$. Consequently, the trajectories of the gradient flow coincide with those obtained from the Langevin dynamics. Under a positive Bakry-Émery curvature bound on the spin space and for $\beta$ small enough, the free energy has a unique minimizer and the flow converges to it exponentially in $W$ and in free energy.

Load-bearing premise

The load-bearing premise is that the free energy satisfies an Evolution Variational Inequality with the stated constant $K_\beta = \kappa - 2\beta\|J\|_{\ell^1}\|\Psi\|_{L^\infty}$; the proof as written actually requires the displacement-convexity constant with $\|\nabla^2\Psi\|_\infty$, so the stated EVI and the exponential rate rest on an inequality the paper does not establish.

Editorial extensions

If this is right

  • The law of an infinite spin system under thermal agitation is exactly the Wasserstein steepest descent of its free energy, so variational tools such as displacement convexity and EVIs describe the physical relaxation process without finite-volume approximation.
  • Exponential convergence at high temperature holds simultaneously in specific Wasserstein distance and in free energy, a uniformity over all local observables that is stronger than earlier weak-dual convergence statements for the stochastic Heisenberg model.
  • Any weak solution of the infinite-volume Fokker-Planck-Kolmogorov equations is automatically a strong solution, so the distinction between the dual and strong formulations disappears for this system.
  • The variational scheme with stationarization provides an existence proof for infinite-volume Fokker-Planck equations that does not require constructing the underlying stochastic process first.
  • Trajectories started from different stationary initial measures contract in specific Wasserstein distance with rate $e^{K_\beta t}$, giving quantitative stability of the dynamics with respect to initial data.

Reading between the lines

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

  • Correcting the EVI constant to the Hessian-based value $\kappa - 2\beta\|J\|_{\ell^1}\|\nabla^2\Psi\|_\infty$ that the proof's estimates support would yield faster exponential rates for interactions with small Hessian; comparing the two constants on concrete models would test the sharpness of the paper's stated rate.
  • The stationarize-then-limit scheme should transfer to other infinite-volume variational problems lacking a product structure, such as random fields or point-process free energies, where similar coincidence theorems between variational flows and Markov dynamics could be derived.
  • The regularity bootstrap, which avoids classical potential-theoretic $L^p$–$L^q$ estimates, may apply to other nonlocal parabolic hierarchies (for instance mean-field Fokker-Planck systems with conditional drifts) where standard potential estimates are unavailable.
  • The equality of trajectories suggests a variational numerical scheme for spin dynamics: each minimizing-movement step is a finite-box optimal-transport problem and the stationarization step controls finite-size errors, offering a transport-based alternative to discretizing the stochastic differential equation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 3 minor

Summary. The paper develops a gradient-flow framework for the infinite-volume free energy of lattice spin systems with continuous spins on a compact Riemannian manifold, under a translation-invariant, symmetric, short-range (ℓ1-summable) interaction. The authors construct the flow through a JKO-type minimizing movement scheme that alternates solving a finite-volume variational problem and a stationarization step, and they construct the infinite-volume overdamped Langevin diffusion via a weighted ℓ2 embedding and finite-volume approximations. The central result is that both the gradient-flow trajectory and the law of the diffusion solve the same infinite-volume Fokker–Planck–Kolmogorov hierarchy in a dual sense; a regularity theorem upgrades weak/dual solutions to strong solutions, and an Evolution Variational Inequality is used to prove uniqueness and, under positive Bakry–Émery curvature and sufficiently high temperature, exponential convergence to the unique minimizer in specific Wasserstein distance and free energy.

Significance. If the results stand, this is a substantial contribution. It appears to be the first construction of the gradient flow of an infinite-volume free energy with genuinely interacting spins, and it provides a new route to uniqueness for infinite-volume Fokker–Planck–Kolmogorov equations, yielding convergence in a stronger metric (specific Wasserstein distance) than the dual-norm convergence available from earlier stochastic-analysis approaches. The paper is highly detailed and essentially self-contained, with careful finite-volume approximations, stationarization estimates, heat-kernel bounds, and honest discussion of limitations, including the obstacles to the non-compact extension in Section 6. The main conceptual architecture is sound; the issues found are local and, in my assessment, correctable.

major comments (2)
  1. [Theorem 3.7, Eq. (3.12) and Lemma 3.11] The stated EVI constant Kβ := κ − 2β||J||_{ℓ1}||Ψ||_{L∞} is not supported by the proof. In the proof of Lemma 3.11 the authors invoke the curvature condition Ric_n + ∇²U_n + β∇²H_n ≥ Kβ with Kβ as in (3.12), but the only displacement-convexity estimate for H_n established in the paper is Lemma 2.13(2), which gives that H_n is −2||J||_{ℓ1}||∇²Ψ||_∞-displacement convex. The L∞ norm of Ψ does not control the Hessian norm of Ψ: on M = S¹, Ψ_k(x,y) = k^{-2}cos(k(x−y)) has ||Ψ_k||_{L∞}=k^{-2} while ||∇²Ψ_k||_{L∞}=1. Consequently, the condition needed to apply [CEMS06, Prop. 4.2] holds only with Kβ = κ − 2β||J||_{ℓ1}||∇²Ψ||_∞. As written, Theorem 3.7 and its consequences Corollary 3.8 and Corollary 3.9 rest on an unproven inequality.
  2. [Corollary 3.9 versus Theorem 2.14] There is an internal inconsistency in the high-temperature regime used for exponential convergence. Theorem 2.14 proves uniqueness of the minimizer under the condition β < (1/(2κ))(||J||_{ℓ1}||∇²Ψ||_∞)^{-1}, while Corollary 3.9 claims exponential convergence whenever Kβ > 0 with Kβ from (3.12), i.e. for β < κ/(2||J||_{ℓ1}||Ψ||_∞). Since ||∇²Ψ||_∞ can be much larger than ||Ψ||_∞, the corollary asserts convergence in a parameter regime for which the paper's own uniqueness result does not apply. Once the constant in Theorem 3.7 is corrected to use ||∇²Ψ||_∞, the threshold in Corollary 3.9 will match Theorem 2.14; as displayed, the statements are inconsistent.
minor comments (3)
  1. [Proof of Theorem 3.7, Step 3] The sentence 'we divide by |Λ_n| in (3.11)' should refer to inequality (3.16), not to item (3.11) of Proposition 3.6.
  2. [Section 1.3] There is a LaTeX artifact in the sentence introducing H(x): the string 'H(x) ??:=' should be replaced by the intended definition.
  3. [Lemma 3.11] The constant Kβ is referred to as 'as in (3.12)' before Theorem 3.7 has been stated; stating the corrected constant explicitly in Lemma 3.11 would improve readability and prevent the kind of mismatch identified in the major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained, with a non-circular correctness gap in the stated EVI constant.

full rationale

The derivation chain is self-contained and I find no circular step. The gradient-flow trajectory is not defined as the Langevin law: it is constructed independently as the h→0 limit of the finite-volume JKO scheme with stationarization (Section 4.1), and Theorem 4.13 proves from the scheme's Euler–Lagrange estimates that any limit point satisfies the dual FPK equation. The infinite-volume diffusion is constructed independently in Section 5.3 through a Hilbert-space embedding and finite-volume SDE approximation, and Theorem 5.6 derives the same dual equation from the martingale problem. The identification of the two objects then uses the regularity theorem (Theorem 3.2, proved by heat-kernel/Duhamel estimates) and the EVI-based uniqueness (Corollary 3.8, proved by passing the finite-volume EVI of [CEMS06]/[Erb10] to the limit). No parameter is fitted and later relabeled as a prediction, and no conclusion is assumed in its own proof. The self-citations ([EHJM23], [EHL21], [DSHS24]) are contextual or inspirational; the load-bearing convexity inputs are Lemma 2.13, whose items cite standard external results ([vRS05]/[Stu06] for the entropy and [ABS24, Thm. 15.19] for the interaction), not a theorem of the present authors. The limitation statements in the paper—e.g. the footnote declining to follow [LWW13, Lemma 3.1]—are not circularity. The one genuine defect is a correctness gap, not a circularity: Theorem 3.7, Eq. (3.12), states Kβ = κ − 2β||J||_{ℓ1}||Ψ||_{L∞}, while Lemma 3.11's curvature lower bound is only supported by Lemma 2.13(2) with ||∇²Ψ||_{∞}; this makes the displayed constant unsupported, but it does not make any claim reduce by definition to its inputs. Accordingly the circularity score is 0.

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

The paper introduces no new physical entities or fitted parameters. Its free parameters are model inputs (β, J, Ψ, κ), and its mathematical objects (specific Wasserstein distance, stationarization) are constructions built from standard theory. The listed axioms are the explicit modeling assumptions and standard mathematical results on which the proofs rest.

assumptions (7)
  • domain assumption M is a smooth compact connected Riemannian manifold without boundary, with normalized volume measure; ω has a smooth density e^{-U} with respect to volume.
    This is the setting of Section 2.1; compactness is used for heat kernel bounds, optimal transport theorems, and boundedness of Ψ.
  • domain assumption The couplings J are symmetric, translation-invariant, and summable in ℓ1 (Eq. 1.1).
    This short-range assumption is used throughout to control boundary errors, define the energy density, and prove Lipschitz properties of ∇H.
  • domain assumption The single-spin interaction potential is symmetric Ψ ∈ C^3(M×M).
    The C^3 regularity is required for the bootstrap regularity argument (Claim 3.16) and the Lipschitz property of ∇H (Prop. 5.10).
  • standard math Standard results from optimal transport on compact manifolds: McCann's theorem on optimal maps, continuity of the Wasserstein distance, and displacement convexity of relative entropy under Ricci lower bounds.
    Used in Section 2.4 and Lemma 2.13; see [McC01, CEMS01, vRS05, Stu06].
  • standard math Heat kernel upper bounds on compact manifolds and the associated L^p regularization estimates (Lemma B.3, Theorem B.5).
    Needed for the regularity Theorem 3.2 and the Duhamel principle arguments in Section 3.5.
  • standard math Existence and uniqueness of SDEs on compact manifolds via Nash embedding and martingale solutions.
    Used to construct the infinite-volume diffusion in Section 5; see [Hsu99, Hsu02].
  • domain assumption The initial spin measure P0 is stationary and has finite specific free energy.
    Required for the JKO scheme (Section 4.1) and the EVI (Theorem 3.7).

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Cite this review

Pith. "Pith review of Gradient flow of the infinite-volume free energy for lattice systems of continuous spins." pith.science (2026). https://pith.science/paper/M7KDPNIW

@misc{pith2026250206500,
  author       = {Pith},
  title        = {Pith review of: Gradient flow of the infinite-volume free energy for lattice systems of continuous spins},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M7KDPNIW}},
  note         = {Machine review of arXiv:2502.06500}
}
read the original abstract

We consider an infinite lattice system of interacting spins living on a smooth compact manifold, with short- but not necessarily finite-range pairwise interactions. We construct the gradient flow of the infinite-volume free energy on the space of translation-invariant spin measures, using an adaptation of the variational approach in Wasserstein space pioneered by Jordan, Kinderlehrer, and Otto. We also construct the infinite-volume diffusion corresponding to the so-called overdamped Langevin dynamics of the spins under the effect of the interactions and of thermal agitation. We show that the trajectories of the gradient flow and of the law of the spins under this diffusion both satisfy, in a weak sense, the same hierarchy of coupled parabolic PDE's, which we interpret as an infinite-volume Fokker-Planck-Kolmogorov equation. We prove regularity of weak solutions and derive an Evolution Variational Inequality for regular solutions, which implies uniqueness. Thus, in particular, the trajectories of the gradient flow coincide with those obtained from the Langevin dynamics. Concerning the long-time evolution, we check that the free energy is always non-increasing along the flow and that moreover, if the Ricci curvature of the spin space is uniformly positive, then at high enough temperature the dynamics converges exponentially, in free energy and in specific Wasserstein distance, to the unique minimizer of the infinite-volume free energy.

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Works this paper leans on

90 extracted references · 78 canonical work pages · cited by 1 Pith paper

  1. [1]

    Lectures on optimal transport , volume 169 of Unitext

    Luigi Ambrosio, Elia Bru \'e , and Daniele Semola. Lectures on optimal transport , volume 169 of Unitext . Cham: Springer, 2nd edition edition, 2024

    2024

  2. [2]

    Albeverio, A

    S. Albeverio, A. Daletskij, and Yu. Kondratiev. Infinite systems of stochastic differential equations and some lattice models on compact Riemannian manifolds. Ukr. Mat. Zh. , 49(3):326--337, 1997

    1997

  3. [3]

    Stochastic equations and Dirichlet operators on infinite product manifolds

    Sergio Albeverio, Alexei Daletskii, and Yuri Kondratiev. Stochastic equations and Dirichlet operators on infinite product manifolds. Infin. Dimens. Anal. Quantum Probab. Relat. Top. , 6(3):455--488, 2003

    2003

  4. [4]

    Critical exponents for long-range interactions

    Michael Aizenman and Roberto Fern \'a ndez. Critical exponents for long-range interactions. Lett. Math. Phys. , 16(1):39--49, 1988

    1988

  5. [5]

    Ambrosio and N

    L. Ambrosio and N. Gigli. A user's guide to optimal transport. In Modelling and Optimisation of Flows on Networks , volume 2062 of Lecture Notes in Math. Springer, Berlin, 2013

    2013

  6. [6]

    Gradient flows: in metric spaces and in the space of probability measures

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient flows: in metric spaces and in the space of probability measures . Springer Science & Business Media, 2005

    2005

  7. [7]

    u rich. Birkh \

    Luigi Ambrosio, Nicola Gigli, and Giuseppe Savar \'e . Gradient flows in metric spaces and in the space of probability measures . Lectures in Mathematics, ETH Z \"u rich. Birkh \"a user, 2nd ed. edition, 2008

    2008

  8. [8]

    Some nonlinear problems in Riemannian geometry

    Thierry Aubin. Some nonlinear problems in Riemannian geometry . Springer Monogr. Math. Springer, 1998

    1998

Show all 90 references
  1. [9]

    Etude des transformations de Riesz dans les vari \'e t \'e s riemanniennes \`a courbure de Ricci minor \'e e

    Dominique Bakry. Etude des transformations de Riesz dans les vari \'e t \'e s riemanniennes \`a courbure de Ricci minor \'e e. In S \'e mininaire de probabilit \'e s , volume XXI of Lect . Notes Math . , pages 137--172. Springer, 1987

    1987

  2. [10]

    A very simple proof of the LSI for high temperature spin systems

    Roland Bauerschmidt and Thierry Bodineau. A very simple proof of the LSI for high temperature spin systems. J. Funct. Anal. , 276(8):2582--2588, 2019

    2019

  3. [11]

    Log- Sobolev inequality for the continuum sine- Gordon model

    Roland Bauerschmidt and Thierry Bodineau. Log- Sobolev inequality for the continuum sine- Gordon model. Commun. Pure Appl. Math. , 74(10):2064--2113, 2021

    2021

  4. [12]

    Ya. I. Belopolskaya and Yu. L. Daletskij. Stochastic equations and differential geometry . Number 30 in Math. Appl., Sov. Ser. Kluwer Academic Publishers, 1990

    1990

  5. [13]

    Log- Sobolev inequality for the ^4_2 and ^4_3 measures

    Roland Bauerschmidt and Benoit Dagallier. Log- Sobolev inequality for the ^4_2 and ^4_3 measures. Commun. Pure Appl. Math. , 77(5):2579--2612, 2024

    2024

  6. [14]

    Bogachev, Giuseppe Da Prato, Michael R \"o ckner, and Stanislav V

    Vladimir I. Bogachev, Giuseppe Da Prato, Michael R \"o ckner, and Stanislav V. Shaposhnikov. An analytic approach to infinite-dimensional continuity and Fokker - Planck - Kolmogorov equations. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) , 14(3):983--1023, 2015

    2015

  7. [15]

    Diffusions hypercontractives

    Dominique Bakry and Michel \'E mery. Diffusions hypercontractives. In S \'e minaire de probabilit \'e s , volume XIX of Lect . Notes Math . , pages 177--206. Spinger, 1985

    1985

  8. [16]

    Analysis and geometry of Markov diffusion operators

    Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and geometry of Markov diffusion operators . Number 348 in Grundlehren Math. Wiss. Cham: Springer, 2014

    2014

  9. [17]

    Bodineau and B

    T. Bodineau and B. Helffer. The log- Sobolev inequality for unbounded spin systems. J. Funct. Anal. , 166(1):168--178, 1999

    1999

  10. [18]

    Bogachev, Nicolai V

    Vladimir I. Bogachev, Nicolai V. Krylov, Michael R \"o ckner, and Stanislav V. Shaposhnikov. Fokker- Planck - Kolmogorov equations . Number 207 in Math. Surv. Monogr. American Mathematical Society (AMS), 2015

    2015

  11. [19]

    Transference principles for log- Sobolev and spectral-gap with applications to conservative spin systems

    Franck Barthe and Emanuel Milman. Transference principles for log- Sobolev and spectral-gap with applications to conservative spin systems. Commun. Math. Phys. , 323(2):575--625, 2013

    2013

  12. [20]

    McCann, and Michael Schmuckenschl \"a ger

    Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschl \"a ger. A Riemannian interpolation inequality \`a la Borell , Brascamp and Lieb . Invent. Math. , 146(2):219--257, 2001

    2001

  13. [21]

    McCann, and Michael Schmuckenschl \"a ger

    Dario Cordero-Erausquin, Robert J. McCann, and Michael Schmuckenschl \"a ger. Pr \'e kopa- Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport. Ann. Fac. Sci. Toulouse, Math. (6) , 15(4):613--635, 2006

    2006

  14. [22]

    Eigenvalues in Riemannian geometry

    Isaac Chavel. Eigenvalues in Riemannian geometry. Number Vol 115 in Pure Appl. Math. Academic Press, 1984

    1984

  15. [23]

    Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates

    Jos \' e Carrillo, Robert McCann, and C \' e dric Villani. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Revista Matem \' a tica Iberoamericana , pages 971--1018, 2003

    2003

  16. [24]

    Carlen and Daniel W

    Eric A. Carlen and Daniel W. Stroock. An application of the Bakry - Emery criterion to infinite dimensional diffusions. In S \'e minaire de probabilit \'e s , volume XX of Lect . Notes Math , pages 341--348. Springer, 1986

    1986

  17. [25]

    The description of a random field by means of conditional probabilities and conditions of its regularity

    PL Dobruschin. The description of a random field by means of conditional probabilities and conditions of its regularity. Theory of Probability & Its Applications , 13(2):197--224, 1968

    1968

  18. [26]

    Eulerian calculus for the displacement convexity in the Wasserstein distance

    Sara Daneri and Giuseppe Savar \'e . Eulerian calculus for the displacement convexity in the Wasserstein distance. SIAM J. Math. Anal. , 40(3):1104--1122, 2008

    2008

  19. [27]

    Wasserstein geometry and Ricci curvature bounds for Poisson spaces

    Lorenzo Dello Schiavo, Ronan Herry, and Kohei Suzuki. Wasserstein geometry and Ricci curvature bounds for Poisson spaces. J. \'E c. Polytech., Math. , 11:957--1010, 2024

    2024

  20. [28]

    Curvature bounds for configuration spaces

    Matthias Erbar and Martin Huesmann. Curvature bounds for configuration spaces. Calc. Var. Partial Differ. Equ. , 54(1):397--430, 2015

    2015

  21. [29]

    Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy

    Matthias Erbar, Martin Huesmann, Jonas Jalowy, and Bastian M \"u ller. Optimal transport of stationary point processes: Metric structure, gradient flow and convexity of the specific entropy. arXiv preprint arXiv:2304.11145 , 2023

    2023 arXiv

  22. [30]

    The one-dimensional log-gas free energy has a unique minimizer

    Matthias Erbar, Martin Huesmann, and Thomas Lebl \'e . The one-dimensional log-gas free energy has a unique minimizer. Communications on Pure and Applied Mathematics , 74(3):615--675, 2021

    2021

  23. [31]

    K. D. Elworthy. Stochastic differential equations on manifolds . Number 70 in Lond. Math. Soc. Lect. Note Ser. Cambridge University Press; London Mathematical Society, 1982

    1982

  24. [32]

    Engoulatov

    A. Engoulatov. A universal bound on the gradient of logarithm of the heat kernel for manifolds with bounded Ricci curvature. J. Funct. Anal. , 238(2):518--529, 2006

    2006

  25. [33]

    The heat equation on manifolds as a gradient flow in the Wasserstein space

    Matthias Erbar. The heat equation on manifolds as a gradient flow in the Wasserstein space. Ann. Inst. Henri Poincar \'e , Probab. Stat. , 46(1):1--23, 2010

    2010

  26. [34]

    A gradient flow approach to the Boltzmann equation

    Matthias Erbar. A gradient flow approach to the Boltzmann equation. J. Eur. Math. Soc. (JEMS) , 26(11):4441--4490, 2024

    2024

  27. [35]

    William G. Faris. The stochastic Heisenberg model. J. Funct. Anal. , 32:342--352, 1979

    1979

  28. [36]

    An invitation to optimal transport, Wasserstein distances, and gradient flows

    Alessio Figalli and Federico Glaudo. An invitation to optimal transport, Wasserstein distances, and gradient flows . EMS Textb. Math. European Mathematical Society (EMS), 2nd ed. edition, 2023

    2023

  29. [37]

    Fisher, Shang-Keng Ma, and B

    Michael E. Fisher, Shang-Keng Ma, and B. G. Nickel. Critical exponents for long-range interactions. Phys. Rev. Lett. , 29:917--920, 1972

    1972

  30. [38]

    J. Fritz. Infinite lattice systems of interacting diffusion processes, existence and regularity properties. Z. Wahrscheinlichkeitstheor. Verw. Geb. , 59:291--309, 1982

    1982

  31. [39]

    Statistical mechanics of lattice systems

    Sacha Friedli and Yvan Velenik. Statistical mechanics of lattice systems. A concrete mathematical introduction . Cambridge: Cambridge University Press, 2018

    2018

  32. [40]

    Gibbs measures and phase transitions

    Hans-Otto Georgii. Gibbs measures and phase transitions. Number 9 in De Gruyter Stud. Math. de Gruyter, 2nd ed. edition, 2011

    2011

  33. [41]

    On the inverse implication of B renier- M ccann theorems and the structure of ( P _2( M ), W _2)

    Nicola Gigli. On the inverse implication of B renier- M ccann theorems and the structure of ( P _2( M ), W _2) . Methods and Applications of Analysis , 18(2):127--158, 2011

    2011

  34. [42]

    Heat kernel and analysis on manifolds

    Alexander Grigor'yan. Heat kernel and analysis on manifolds . Number 47 in AMS/IP Stud. Adv. Math. American Mathematical Society (AMS); International Press, 2009

    2009

  35. [43]

    Guionnet and B

    A. Guionnet and B. Zegarlinski. Lectures on logarithmic Sobolev inequalities. In S\'eminaire de probabilit\'es , volume XXXVI of Lect . Notes Math . , pages 1--134. Springer, 2003

    2003

  36. [44]

    Sobolev spaces on Riemannian manifolds

    Emmanuel Hebey. Sobolev spaces on Riemannian manifolds . Number 1635 in Lect. Notes Math. Springer, 1996

    1996

  37. [45]

    Remarks on decay of correlations and Witten Laplacians

    Bernard Helffer. Remarks on decay of correlations and Witten Laplacians . III : Application to logarithmic Sobolev inequalities. Ann. Inst. Henri Poincar \'e , Probab. Stat. , 35(4):483--508, 1999

    1999

  38. [46]

    Hypoelliptic second order differential equations

    Lars H \"o rmander. Hypoelliptic second order differential equations. Acta Math. , 119:147--171, 1967

    1967

  39. [47]

    The analysis of linear partial differential operators

    Lars H \"o rmander. The analysis of linear partial differential operators. I : Distribution theory and Fourier analysis. Class. Math. Springer, reprint of the 2nd ed. 1990 edition, 2003

    1990

  40. [48]

    Holley and Daniel W

    R. Holley and Daniel W. Stroock. Diffusions on an infinite dimensional torus. J. Funct. Anal. , 42:29--63, 1981

    1981

  41. [49]

    Elton P. Hsu. Estimates of derivatives of the heat kernel on a compact Riemannian manifold. Proc. Am. Math. Soc. , 127(12):3739--3744, 1999

    1999

  42. [50]

    Elton P. Hsu. Stochastic analysis on manifolds . Number 38 in Grad. Stud. Math. American Mathematical Society (AMS), 2002

    2002

  43. [51]

    The variational formulation of the fokker--planck equation

    Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the fokker--planck equation. SIAM Journal on Mathematical Analysis , 29(1):1--17, jan 1998

    1998

  44. [52]

    G. S. Joyce. Spherical model with long-range ferromagnetic interactions. Phys. Rev. , 146:349--358, 1966

    1966

  45. [53]

    Hypercontractivity for spin systems of infinite extension

    Etienne Laroche. Hypercontractivity for spin systems of infinite extension. Probab. Theory Relat. Fields , 101(1):89--132, 1995

    1995

  46. [54]

    M. Ledoux. Logarithmic Sobolev inequalities for unbounded spin systems revisited. In S\'eminaire de Probabilit\'es , volume XXXV of Lect . Notes Math . , pages 167--194. Springer, 2001

    2001

  47. [55]

    Statistical Physics: Volume 5 , volume 5

    Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz. Statistical Physics: Volume 5 , volume 5. Elsevier, 2013

    2013

  48. [56]

    J. L. Lions and Jaak Peetre. On a class of interpolation spaces. Publ. Math., Inst. Hautes \'E tud. Sci. , 19:5--68, 1964

    1964

  49. [57]

    On solutions to stochastic differential equations with discontinuous drift in Hilbert space

    Gottlieb Leha and Gunter Ritter. On solutions to stochastic differential equations with discontinuous drift in Hilbert space. Math. Ann. , 270:109--123, 1985

    1985

  50. [58]

    O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva. Linear and quasi-linear equations of parabolic type. Translated from the Russian by S . Smith , volume 23 of Transl. Math. Monogr. American Mathematical Society (AMS), Providence, RI, 1968

    1968

  51. [59]

    Uniqueness of Fokker-Planck equations for spin lattice systems (i): compact case

    Ludovic Dan Lemle, Ran Wang, and Liming Wu. Uniqueness of Fokker-Planck equations for spin lattice systems (i): compact case. In Semigroup Forum , volume 86, pages 583--591. Springer, 2013

    2013

  52. [60]

    Stochastic calculus of variation and hypoelliptic operators

    Paul Malliavin. Stochastic calculus of variation and hypoelliptic operators. In Proc. int. Symp . on stochastic differential equations, Koyto 1976 , pages 195--263, 1978

    1976

  53. [61]

    Robert J. McCann. A convexity principle for interacting gases. Advances in Mathematics , 128(1):153--179, jun 1997

    1997

  54. [62]

    Robert J. McCann. Polar factorization of maps on Riemannian manifolds. Geom. Funct. Anal. , 11(3):589--608, 2001

    2001

  55. [63]

    A mean field view of the landscape of two-layer neural networks

    Song Mei, Andrea Montanari, and Phan-Minh Nguyen. A mean field view of the landscape of two-layer neural networks. Proceedings of the National Academy of Sciences , 115(33):E7665--E7671, 2018

    2018

  56. [64]

    Evolution of microstructure in unstable porous media flow: A relaxational approach

    Felix Otto. Evolution of microstructure in unstable porous media flow: A relaxational approach. Commun. Pure Appl. Math. , 52(7):873--915, 1999

    1999

  57. [65]

    The geometry of dissipative evolution equations: The porous medium equation

    Felix Otto. The geometry of dissipative evolution equations: The porous medium equation. Commun. Partial Differ. Equations , 26(1-2):101--174, 2001

    2001

  58. [66]

    Otto and C

    F. Otto and C. Villani. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. , 173(2):361--400, 2000

    2000

  59. [67]

    Lectures on the spin and loop o(n) models

    Ron Peled and Yinon Spinka. Lectures on the spin and loop o(n) models. In Sojourns in Probability Theory and Statistical Physics-I: Spin Glasses and Statistical Mechanics, A Festschrift for Charles M. Newman , pages 246--320. Springer, 2019

    2019

  60. [68]

    G. Royer. Processus de diffusion associe a certains modeles d' Ising a spins continus. Z. Wahrscheinlichkeitstheor. Verw. Geb. , 46:165--176, 1979

    1979

  61. [69]

    Une initiation aux in \'e galit \'e s de Sobolev logarithmiques

    Gilles Royer. Une initiation aux in \'e galit \'e s de Sobolev logarithmiques . Number 5 in Cours Sp \'e c. Soci \'e t \'e Math \'e matique de France, 1999

    1999

  62. [70]

    Functional analysis

    Walter Rudin. Functional analysis. New York, NY: McGraw-Hill, 2nd ed. edition, 1991

    1991

  63. [71]

    Continuous martingales and Brownian motion , volume 293 of Grundlehren Math

    Daniel Revuz and Marc Yor. Continuous martingales and Brownian motion , volume 293 of Grundlehren Math. Wiss. Berlin: Springer, 3rd ed., 3rd. corrected printing edition, 2005

    2005

  64. [72]

    J. Sak. Recursion relations and fixed points for ferromagnets with long-range interactions. Phys. Rev. B , 8:281--285, Jul 1973

    1973

  65. [73]

    D. A. Salamon. Parabolic l^p-l^q estimates. https://people.math.ethz.ch/ salamon/PREPRINTS/parabolic.pdf, 07 2017

    2017

  66. [74]

    Euclidean, metric, and Wasserstein gradient flows: an overview

    Filippo Santambrogio. Euclidean, metric, and Wasserstein gradient flows: an overview. Bull. Math. Sci. , 7(1):87--154, 2017

    2017

  67. [75]

    Saloff-Coste

    L. Saloff-Coste. A note on Poincar \'e , Sobolev , and Harnack inequalities. Int. Math. Res. Not. , 1992(2):27--38, 1992

    1992

  68. [76]

    Infinite dimensional stochastic differential equations and their applications

    Tokuzo Shiga and Akinobu Shimizu. Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. , 20:395--416, 1980

    1980

  69. [77]

    On the geometry of metric measure spaces

    Karl-Theodor Sturm. On the geometry of metric measure spaces. I . Acta Math. , 196(1):65--131, 2006

    2006

  70. [78]

    Curvature bound of dyson brownian motion, 2024

    Kohei Suzuki. Curvature bound of dyson brownian motion, 2024

    2024

  71. [79]

    Stroock and Boguslaw Zegarlinski

    Daniel W. Stroock and Boguslaw Zegarlinski. The equivalence of the logarithmic Sobolev inequality and the Dobrushin - Shlosman mixing condition. Commun. Math. Phys. , 144(2):303--323, 1992

    1992

  72. [80]

    Stroock and Boguslaw Zegarlinski

    Daniel W. Stroock and Boguslaw Zegarlinski. The logarithmic Sobolev inequality for continuous spin systems on a lattice. J. Funct. Anal. , 104(2):299--326, 1992

    1992

  73. [81]

    Brownian motion and the distance to a submanifold

    James Thompson. Brownian motion and the distance to a submanifold. Potential Anal. , 45(3):485--508, 2016

    2016

  74. [82]

    Spaces of Besov - Hardy - Sobolev type on complete Riemannian manifolds

    Hans Triebel. Spaces of Besov - Hardy - Sobolev type on complete Riemannian manifolds. Ark. Mat. , 24:299--337, 1986

    1986

  75. [83]

    Exponential integrability and exit times of diffusions on sub- Riemannian and metric measure spaces

    Anton Thalmaier and James Thompson. Exponential integrability and exit times of diffusions on sub- Riemannian and metric measure spaces. Bernoulli , 26(3):2202--2225, 2020

    2020

  76. [84]

    Optimal transport

    C \'e dric Villani. Optimal transport . Number 338 in Grundlehren Math. Wiss. Springer, 2009

    2009

  77. [85]

    von Renesse and Karl-Theodor Sturm

    Max-K. von Renesse and Karl-Theodor Sturm. Transport inequalities, gradient estimates, entropy and Ricci curvature. Commun. Pure Appl. Math. , 58(7):923--940, 2005

    2005

  78. [86]

    William D. Wick. Convergence to equilibrium of the stochastic Heisenberg model. Commun. Math. Phys. , 81:361--377, 1981

    1981

  79. [87]

    William D. Wick. Monotonicity of the free energy in the stochastic Heisenberg model. Commun. Math. Phys. , 83:107--122, 1982

    1982

  80. [88]

    The equivalence of the log- Sobolev inequality and a mixing condition for unbounded spin systems on the lattice

    Nobuo Yoshida. The equivalence of the log- Sobolev inequality and a mixing condition for unbounded spin systems on the lattice. Ann. Inst. Henri Poincar \'e , Probab. Stat. , 37(2):223--243, 2001

    2001

  81. [89]

    Variational approximation for Fokker - Planck equation on Riemannian manifold

    Xicheng Zhang. Variational approximation for Fokker - Planck equation on Riemannian manifold. Probab. Theory Relat. Fields , 137(3-4):519--539, 2007

    2007

  82. [90]

    William P. Ziemer. Weakly differentiable functions. Sobolev spaces and functions of bounded variation . Number 120 in Grad. Texts Math. Springer, 1989

    1989

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