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An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees

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Pith's one-line read The paper establishes an H-theorem for the κ-MLFE: the sparse free energy Hκ decreases along every linear-growth solution, with dissipation given by a modified Fisher information Iκ that vanishes exactly on stationary distributions, which…

desk verdict A genuinely new Lyapunov functional for a conditional McKean–Vlasov equation, with a careful H-theorem proof; the global convergence claims, however, lean on an unpublished inequality and are not yet self-contained. read the letter →

arxiv 2412.07710 v2 pith:A53TJZOL submitted 2024-12-10 math.PR math.AP

Kevin Hu , Kavita Ramanan This is my paper · ORCID
classification math.PRmath.AP MSC 60K3560J6060J7082C2235Q8482C31
keywords conditionalMcKean-Vlasovequationkappa-regulartreelocal-fieldsparsefreeenergyH-theoremmodifiedFisherinformationsplittingGibbsmeasureslogarithmicSobolevinequality
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 paper studies the long-time behavior of a conditional McKean-Vlasov equation, the κ-MLFE, which arises as the Markovian version of the local-field equation describing the root neighborhood of interacting diffusions on a κ-regular tree. It identifies a functional it calls the sparse free energy Hκ and proves an H-theorem: along every solution whose conditional drift grows at most linearly, Hκ decreases with time and its rate of decrease is a nonnegative modified Fisher information Iκ. The paper then shows that the zeros of Iκ coincide with the stationary distributions of the flow and with the marginals of splitting Gibbs measures on the tree. Why this matters is that the stationary states are not known in advance and may be multiple, yet Hκ is a global Lyapunov function for the flow, so one can study relaxation to equilibrium without knowing the equilibrium first.

What carries the argument

The load-bearing object is the sparse free energy Hκ together with its dissipation Iκ, a modified Fisher information that subtracts the edge-marginal contribution from the full relative Fisher information. The argument rides on the energy dissipation identity (Theorem 4.1), proved by splitting the evolution of Hκ into a full entropy term and an edge-marginal correction, then using the linear Fokker-Planck equation for the time marginals and the leaf-exchangeability and edge-symmetry of the admissible measures. The zero set of Iκ is then described by a fixed-point recursion on the tree, the Cayley fixed-point equation, which is what ties the stationary states to splitting Gibbs measures.

What would settle it

Take the paper's quadratic example U(x) = $7x^{2}$/4, W(x) = −$3x^{2}$/8 for κ = 2, d = 1, choose a smooth admissible initial condition that is not a 1-MRF, and solve the 2-MLFE numerically; if H2(μt) + ∫_0^t I2(μs) ds is not constant to within numerical error, or if H2 increases on any interval, the H-theorem is false. Alternatively, exhibit a measure with Iκ = 0 that is not a Cayley fixed point, which would break Theorem 4.10.

Watch

Extended reading notes

Core claim

The central discovery is that the correct energy functional for the κ-MLFE is the sparse free energy, defined by an integral of the log-density minus a correction for the edge marginal plus the local potential terms U and W. Theorem 4.1 shows that along every linear-growth solution μt one has the energy dissipation identity Hκ(μt) − Hκ(μr) = −∫_r^t Iκ(μs) ds, where Iκ is the nonnegative modified Fisher information of Eq. (4.3). The zeros of Iκ are characterized as Cayley fixed points, and these are exactly the stationary distributions of the κ-MLFE and the root-neighborhood marginals of automorphism-invariant splitting Gibbs measures on the κ-regular tree. Consequently Hκ − H*κ acts as a global Lyapunov function even when there are multiple stationary states.

Load-bearing premise

The argument assumes that a 'linear growth solution' exists, meaning the conditional drift γ satisfies the uniform bound (3.10), and for unbounded interaction gradients such existence is an open problem rather than a proved fact.

Editorial extensions

If this is right

  • Stationary distributions of the κ-MLFE can be computed by solving the Cayley fixed-point equation (or equivalently the Gibbs-marginal fixed-point equation) instead of by integrating the flow.
  • Every linear-growth solution converges to the zero set of Iκ as t → ∞, so the sparse free energy gives quantitative control over relaxation to equilibrium even in the presence of multiple stationary states.
  • For κ = 2, the sparse free energy is the renormalized limit of relative entropies of lifted 2-MRF measures against finite-tree Gibbs measures, and this representation plus a uniform log-Sobolev inequality yields exponential convergence to the unique stationary distribution.
  • For κ ≥ 3, the same H-theorem and stationarity characterization hold, but rates of convergence are left open because the Gibbs measures on infinite trees may be non-unique.

Reading between the lines

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

  • The paper does not claim the local-field equation shares the Lyapunov function, but were that transfer to hold, the same sparse free energy would control the original sparse-graph dynamics.
  • The paper does not compute rates for κ ≥ 3; a natural way to try is to push the renormalized-entropy representation through a uniform log-Sobolev inequality on the truncated tree, which would be a testable extension.
  • The paper's well-posedness theorem requires bounded interaction gradients ∇W; for unbounded potentials, an explicit check of the linear-growth condition (3.10) would either extend the H-theorem or identify the precise place where it fails.
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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

1 major / 7 minor

Summary. The paper studies the κ-regular Markovian local-field equation (κ-MLFE), a conditional McKean-Vlasov equation on the root neighborhood of a κ-regular tree. It proves well-posedness for bounded interaction gradients via a Schauder fixed-point argument, establishes an H-theorem for linear-growth solutions with a new functional Hκ (the sparse free energy) and a modified Fisher information Iκ, and identifies the zeros of Iκ with stationary distributions, Cayley fixed points, and marginals of continuous Gibbs measures. For κ = 2 it proves a renormalized-entropy representation of H2, uniqueness of stationary distributions via the Lacker-Zhang fixed-point problem, and exponential convergence under a uniform logarithmic Sobolev inequality. The proofs are detailed and largely self-contained in the stochastic-analysis sense, with the central computation being the symmetrization chain in Section 6.2.

Significance. If the results hold, this is a substantial advance: it provides a Lyapunov structure for a class of conditional McKean-Vlasov equations on sparse graphs, connects the dynamics to the Gibbs-measure fixed points of Lacker and Zhang [47], and gives quantitative rates in the κ = 2 case. The H-theorem is genuinely derived from the PDE rather than assumed, and the well-posedness proof is substantial. The main caveat is that Proposition 4.2, which supplies the lower bound and compact level sets used for the global Lyapunov and convergence claims, relies on the unpublished manuscript [20] at Eq. (6.30); until that ingredient is available or proved, the strongest advertised consequences are not fully verifiable.

major comments (1)
  1. [Section 6.3, Proposition 4.2] Equation (6.30) cites the unpublished manuscript [20] for the nonnegativity of ν ↦ H(ν|α_q^{⊗(1+κ)}) − (κ/2)H(ν̄|α_q^{⊗2}) and for the associated finite-entropy and level-set conclusions. This inequality is not a routine consequence of the chain rule: it couples the full density ν to the two-point marginal ν̄ with a delicate κ/2 factor under the symmetry constraints (3.1)-(3.2). It is the sole source of the lower bound (4.9), the containment (4.10), and the compact level sets used in Theorem 4.6 and Remark 4.7. Because [20] is listed as in preparation, the global Lyapunov and convergence-to-stationarity claims are not fully verifiable as submitted. Please supply a self-contained proof of (6.30) and of the level-set claims, or restate the affected theorems as conditional on a supplied inequality.
minor comments (7)
  1. [Section 1.2.1 and Theorem 3.10] The well-posedness theorem is proved only when ∥∇W∥L∞ < ∞, while later results are conditional on existence of a linear-growth solution satisfying (3.10). This is acknowledged in Open Problem 1, but the abstract and Section 1.1 could make clearer that quadratic potentials are not covered by the theorems of this paper.
  2. [Section 8.2, Remark 8.4] The phrase 'Assumption 7.6(2)' appears to refer to Assumption D(2); please correct the reference.
  3. [Theorem 4.17, Eq. (4.29)] The symbol 'H⋆' in Eq. (4.29) should be 'H∗_2' to match the definition in (4.24).
  4. [Section 8.2, proof of Theorem 4.17(3)] The text refers to a 'κ-CMVE' in the proof of part (3); this should be 'κ-MLFE'.
  5. [Title page] The word 'RELA TED' appears with a spurious space in the title; this is a typographical issue.
  6. [Remark 4.5(2)] 'U grows quadratically at infinite' should be 'at infinity'.
  7. [Theorem 4.16] The Rd extension of Theorem 1.9 of [47] is asserted 'by inspection of the proof'; since this is used for uniqueness of the stationary distribution, a sentence outlining the changes needed for Rd would be helpful.

Circularity Check

1 steps flagged · score 4.0 of 10

Global Lyapunov and convergence claims depend on an unpublished self-cited rate-function inequality [20], while the H-theorem identity itself is derived self-containedly.

  1. self citation load bearing [Section 6.3, proof of Proposition 4.2, Eq. (6.30)]
    "By definition of Qκ,d, ν has finite entropy and hence, by the results of [20], the map ν 7→ H(ν|α⊗(1+κ) q ) − κ 2 H(¯ν|α⊗2 q ) is a rate function. Then we have H(ν|α⊗(1+κ) q ) − κ 2 H(¯ν|α⊗2 q ) ≥ 0, ν ∈ Mκ,d. (6.30)"

    Proposition 4.2 is the sole source of the lower bound Hκ ≥ −log Rq (4.9), of the containment (4.10), and of the compact level sets used in Theorem 4.6 and Remark 4.7. Its proof invokes (6.30) — the nonnegativity and lower semicontinuity of ν ↦ H(ν||α_q^{⊗(1+κ)}) − (κ/2)H(barν||α_q^{⊗2}) — citing only the unpublished manuscript [20], whose authors include this paper's coauthor K. Ramanan. The inequality is not proved here, and it forms the core of the lower-bound functional Hκ up to −log Rq plus a nonnegative term. Thus the global Lyapunov and convergence claims reduce at this load-bearing point to a self-citation of unverified work.

full rationale

The H-theorem itself (Theorem 4.1, Eq. (4.4)) is not circular: its proof derives the energy-dissipation identity directly from the Fokker-Planck equation satisfied by linear-growth solutions, using the symmetry lemmas of Section 6.2, and its assumptions are stated explicitly. The characterization of stationary distributions (Theorems 4.4 and 4.10) uses the published Lacker-Zhang fixed-point framework [47] as an external benchmark, not a self-citation, and the proof of the Cayley-fixed-point equivalence is carried out in the paper. The renormalized-entropy representation for κ = 2 (Theorem 4.15) is a direct computation with an explicitly constructed lift map, and the log-Sobolev/convergence claims in Theorem 4.17 follow from the uniform LSI of [61] plus that computation. However, the advertised global Lyapunov property is not fully self-contained: Proposition 4.2's lower bound and compact level sets — which are essential for the LaSalle-type convergence Theorem 4.6 and Remark 4.7 — rest on inequality (6.30), quoted from the unpublished manuscript [20] written with one of the present authors. This is a load-bearing self-citation of unverified material, although it does not affect the derivation of the H-theorem identity itself. The paper also honestly records Open Problem 1: well-posedness for unbounded ∇W is not established, so the H-theorem applies only under the linear-growth-solution assumption (3.10); this is a conditional statement, not a circular one.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The central claim rests on five structural assumptions (A, B, B*, C, D) on the potentials, plus external PDE results and one unpublished rate-function result [20]. There are no fitted free parameters. The new objects (Hκ, Iκ, Cayley fixed points) are defined and proven to have their stated properties inside the paper; none is an unexplained invented entity.

assumptions (7)
  • domain assumption Assumption A: ∇U and ∇W satisfy linear growth and W is even
    Used throughout (Definition 3.4, Theorem 3.10, Theorem 4.1) to ensure SDEs have linear-growth solutions and energies are finite.
  • domain assumption Assumption B: coercivity (4.7) with finite R_q and moments
    Ensures Hκ is bounded below and finite; used in Proposition 4.2 and Theorem 4.15.
  • domain assumption Assumption B*: strong coercivity (4.8)
    Gives compact level sets of Hκ, used in Theorem 4.6.
  • domain assumption Assumption C: linear growth of conditional expectation of ∇W for elements of Sκ
    Needed for Theorem 4.4 (Sκ = stationary distributions) and Proposition 7.2.
  • domain assumption Assumption D: uniform LSI for conditional Gibbs measures and integrability conditions (8.9)-(8.10)
    Used for Theorem 4.17 (modified LSI and exponential convergence).
  • domain assumption Rate-function and level-set claims from [20] (Chen-Ramanan-Yasodharan, in preparation)
    Invoked in proof of Proposition 4.2 to identify ν ↦ H(ν||α) - κ/2 H(ν̄||α) as a rate function; unpublished.
  • standard math External PDE and probability results (Bogachev et al. [7], Karatzas-Shreve [42], Brunick-Shreve [12], Zegarlinski [61], etc.)
    Used for Fokker-Planck regularity, weak solutions, Markov random fields, and uniform LSI criteria.
invented entities (3)
  • Sparse free energy Hκ
    purpose: Lyapunov functional for the κ-MLFE; its decrease is the H-theorem
    Defined in (4.2) with properties proven inside the paper (Theorems 4.1, 4.15); no external confirmation yet.
  • Modified Fisher information Iκ
    purpose: Entropy dissipation functional whose zeros are stationary states
    Defined in (4.3); properties proven in the paper, no external benchmark outside.
  • Cayley fixed points independent evidence
    purpose: Fixed-point characterization of stationary marginals, equivalent to Lacker-Zhang fixed points
    Corollary 7.4 shows equivalence to the fixed point problem in the published [47], giving an external handle.

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Pith. "Pith review of An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees." pith.science (2026). https://pith.science/paper/A53TJZOL

@misc{pith2026241207710,
  author       = {Pith},
  title        = {Pith review of: An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A53TJZOL}},
  note         = {Machine review of arXiv:2412.07710}
}
abstract

We study the long-time behavior of the $\kappa$-Markov local-field equation ($\kappa$-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the $\kappa$-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the $\kappa$-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the $\kappa$-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the $\kappa$-MLFE and are also marginals of splitting Gibbs measures on the $\kappa$-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case $\kappa = 2$ we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the $2$-MLFE measure flow to its unique stationary distribution.

Figures

Figures reproduced from arXiv: 2412.07710 by the authors.

Figure 1.1
Figure 1.1. Convergence diagram for entropy renormalization. Here, S is the set of zeros of I2, which in our setting are also the limit points of µt (see Theorem 4.6). By Theorem 4.10 and Corollary 7.4, the set S is also the set of root marginal distributions of continuous Gibbs measures on regular trees, and therefore can be identified as possible limit points of root marginals of θ n . This fact is reflected by the dashed arr… view at source ↗
Figure 4.1
Figure 4.1. A comparison between Hˆ 2(¯µt) and H2(µt). Here, µt solves the 2-MLFE with d = 1, κ = 2, potentials U(x) = 7x 2/4 and K = −3x 2/8, and an initial condition that is not a 1-MRF. The left column shows the evolution of the 1-MRF renormalized limit Hˆ 2(µt) and the right column shows the evolution of the sparse free energy H2(µt). For both columns, the bottom figure shows the top figure zoomed in on the time interval (0… view at source ↗

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Works this paper leans on

61 extracted references · 57 canonical work pages

  1. [20]

    I-H. Chen, K. Ramanan, and S. Yasodharan, Large deviations of the empirical measure of multi-type configuration models with Polish space marks (2024). In preparation

    work page 2024

  2. [38]

    Hu and K

    K. Hu and K. Ramanan, Marginal dynamics of interacting Ornstein-Uhlenbeck processes on the line: long-time behavior and Markovian approximation (2024). In preparation

    work page 2024

  3. [47]

    Lacker and J

    D. Lacker and J. Zhang, Stationary solutions and local equations for interacting diffusions on regular trees , Electron. J. Probab. 28 (2023), Paper No. 4, 37. MR4529086

    work page 2023

  4. [1]

    Bakry, I

    D. Bakry, I. Gentil, and M. Ledoux, Analysis and geometry of Markov diffusion operators , Grundlehren der math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348, Springer, Cham, 2014. MR3155209

    work page 2014

  5. [2]

    Barbu and M

    V. Barbu and M. R¨ ockner, Nonlinear Fokker-Planck flows and their probabilistic counterparts , Lecture Notes in Mathematics, vol. 2253, Springer Cham, 2024

    work page 2024

  6. [3]

    Bashiri, On the long-time behaviour of McKean-Vlasov paths , Electron

    K. Bashiri, On the long-time behaviour of McKean-Vlasov paths , Electron. Commun. Probab. 25 (2020), Paper No. 52, 14. MR4129676

    work page 2020

  7. [4]

    Billingsley, Convergence of probability measures , John Wiley & Sons, Inc., New York-London-Sydney, 1968

    P. Billingsley, Convergence of probability measures , John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR233396

    work page 1968

  8. [5]

    S. G. Bobkov, Upper bounds for Fisher information , Electron. J. Probab. 27 (2022), Paper No. 115, 44. MR4477011

    work page 2022

Show all 61 references
  1. [6]

    V. I. Bogachev, M. R¨ ockner, and S. V. Shaposhnikov,Distances between transition probabilities of diffusions and appli- cations to nonlinear Fokker-Planck-Kolmogorov equations, J. Funct. Anal. 271 (2016), no. 5, 1262–1300. MR3522009

    2016

  2. [7]

    V. I. Bogachev, N. V. Krylov, M. R¨ ockner, and S. V. Shaposhnikov, Fokker-Planck-Kolmogorov equations, Mathe- matical Surveys and Monographs, vol. 207, American Mathematical Society, Providence, RI, 2015. MR3443169

    2015

  3. [8]

    Bolley and C

    F. Bolley and C. Villani, Weighted Csisz´ ar-Kullback-Pinsker inequalities and applications to transportation inequali- ties, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 3, 331–352. MR2172583

    2005

  4. [9]

    Bossy and J.-F

    M. Bossy and J.-F. Jabir, On the wellposedness of some McKean models with moderated or singular diffusion coeffi- cient, Frontiers in stochastic analysis—BSDEs, SPDEs and their applications, 2019, pp. 43–87. MR4008340

    2019

  5. [10]

    Bossy, J.-F

    M. Bossy, J.-F. Jabir, and D. Talay, On conditional McKean Lagrangian stochastic models , Probab. Theory Related Fields 151 (2011), no. 1-2, 319–351. MR2834721

    2011

  6. [11]

    Brezis, Functional analysis, Sobolev spaces and partial differential equations , Universitext, Springer, New York,

    H. Brezis, Functional analysis, Sobolev spaces and partial differential equations , Universitext, Springer, New York,

  7. [12]

    Brunick and S

    G. Brunick and S. Shreve, Mimicking an Itˆ o process by a solution of a stochastic differential equation , Ann. Appl. Probab. 23 (2013), no. 4, 1584–1628. MR3098443

    2013

  8. [13]

    Buckdahn, J

    R. Buckdahn, J. Li, and J. Ma, A general conditional McKean-Vlasov stochastic differential equation , Ann. Appl. Probab. 33 (2023), no. 3, 2004–2023. MR4583663

    2023

  9. [14]

    Budhiraja, P

    A. Budhiraja, P. Dupuis, M. Fischer, and K. Ramanan, Limits of relative entropies associated with weakly interacting particle systems, Electron. J. Probab. 20 (2015), no. 80, 22. MR3383564 58 KEVIN HU AND KA VITA RAMANAN

    2015

  10. [15]

    , Local stability of Kolmogorov forward equations for finite state nonlinear Markov processes , Electron. J. Probab. 20 (2015), no. 81, 30. MR3383565

    2015

  11. [16]

    A Carrillo, R

    J. A Carrillo, R. J McCann, and C. Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates , Revista Matematica Iberoamericana 19 (2003), no. 3, 971– 1018

    2003

  12. [17]

    J. A. Carrillo, R. S Gvalani, G. A Pavliotis, and A. Schlichting, Long-time behaviour and phase transitions for the Mckean–Vlasov equation on the torus , Archive for Rational Mechanics and Analysis 235 (2020), no. 1, 635–690

    2020

  13. [18]

    Cattiaux, A

    P. Cattiaux, A. Guillin, and F. Malrieu, Probabilistic approach for granular media equations in the non-uniformly convex case, Probab. Theory Related Fields 140 (2008), no. 1-2, 19–40. MR2357669

    2008

  14. [19]

    Cazenave and A

    T. Cazenave and A. Haraux, An introduction to semilinear evolution equations, Oxford Lecture Series in Mathematics and its Applications, vol. 13, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1990 French original by Yvan Martel and revised b...

    1998

  15. [21]

    Cocomello and K

    J. Cocomello and K. Ramanan, Exact description of limiting SIR and SEIR dynamics on locally tree-like graphs (2023), available at 2309.08829

    2023 arXiv

  16. [22]

    Conforti, D

    G. Conforti, D. Lacker, and S. Pal, Projected langevin dynamics and a gradient flow for entropic optimal transport (2023), available at 2309.08598

    2023 arXiv

  17. [23]

    Crimaldi and L

    I. Crimaldi and L. Pratelli, Two inequalities for conditional expectations and convergence results for filters , Statistics & probability letters 74 (2005), no. 2, 151–162

    2005

  18. [24]

    Delarue, E

    F. Delarue, E. Tanr´ e, and R. Maillet, Ergodicity of some stochastic Fokker-Planck equations with additive common noise (2024), available at 2405.09950

    2024 arXiv

  19. [25]

    M. G. Delgadino, R. S. Gvalani, G. A. Pavliotis, and S. A. Smith, Phase transitions, logarithmic Sobolev inequalities, and uniform-in-time propagation of chaos for weakly interacting diffusions , Comm. Math. Phys. 401 (2023), no. 1, 275–323. MR4604897

    2023

  20. [26]

    M. F. Djete, Non–regular Mckean–Vlasov equations and calibration problem in local stochastic volatility models(2022), available at 2208.09986

    2022 arXiv

  21. [27]

    Eberle, A

    A. Eberle, A. Guillin, and R. Zimmer, Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc. 371 (2019), no. 10, 7135–7173. MR3939573

    2019

  22. [28]

    L. C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR1625845

    1998

  23. [29]

    Gamarnik and K

    D. Gamarnik and K. Ramanan, Uniqueness of Gibbs measures for continuous hardcore models , Ann. Probab. 47 (2019), no. 4, 1949–1981. MR3980912

    2019

  24. [30]

    Ganguly, Non-Markovian interacting particle systems on large sparse graphs: Hydrodynamic limits and marginal characterizations, Doctoral Thesis, Brown University (2022)

    A. Ganguly, Non-Markovian interacting particle systems on large sparse graphs: Hydrodynamic limits and marginal characterizations, Doctoral Thesis, Brown University (2022)

    2022

  25. [31]

    Ganguly and K

    A. Ganguly and K. Ramanan, Interacting jump processes preserve semi-global markov random fields on path space , 2022

    2022

  26. [32]

    , Hydrodynamic limits of non-markovian interacting particle systems on sparse graphs , Electronic Journal of Probability, to appear (2024)

    2024

  27. [33]

    In preparation

    , Marginal dynamics of interacting jump processes on regular trees: stationary distributions and Markov local- field equations (2024). In preparation

    2024

  28. [34]

    Georgii, Gibbs measures and phase transitions , De Gruyter Studies in Mathematics, vol

    H.-O. Georgii, Gibbs measures and phase transitions , De Gruyter Studies in Mathematics, vol. 9, Walter de Gruyter & Co., Berlin, 1988. MR956646

    1988

  29. [35]

    Guillin, W

    A. Guillin, W. Liu, L. Wu, and C. Zhang, Uniform Poincar´ e and logarithmic Sobolev inequalities for mean field particle systems, Ann. Appl. Probab. 32 (2022), no. 3, 1590–1614. MR4429996

    2022

  30. [36]

    Gy¨ ongy,Mimicking the one-dimensional marginal distributions of processes having an Itˆ o differential , Probab

    I. Gy¨ ongy,Mimicking the one-dimensional marginal distributions of processes having an Itˆ o differential , Probab. Theory Relat. Fields 71 (1986), no. 4, 501–516. MR833267

    1986

  31. [37]

    J. M. Hammersley, Generalization of the fundamental theorem on sub-additive functions , Proc. Cambridge Philos. Soc. 58 (1962), 235–238. MR137800

    1962

  32. [39]

    Jabin and Z

    P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with W −1,∞ kernels, Invent. Math. 214 (2018), no. 1, 523–591. MR3858403

    2018

  33. [40]

    Jianhai and W

    B. Jianhai and W. Jian, Long time behavior of one-dimensional McKean-Vlasov SDEs with common noise (2024), available at 2401.07665

    2024 arXiv

  34. [41]

    Jordan, D

    R. Jordan, D. Kinderlehrer, and F. Otto, The variational formulation of the Fokker-Planck equation , SIAM J. Math. Anal. 29 (1998), no. 1, 1–17. MR1617171 AN H-THEOREM FOR THE κ-MLFE 59

    1998

  35. [42]

    Karatzas and S

    I. Karatzas and S. E. Shreve, Brownian motion and stochastic calculus , Second, Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR1121940

    1991

  36. [43]

    Lacker, K

    D. Lacker, K. Ramanan, and R. Wu, Locally interacting diffusions as Markov random fields on path space , Stochastic Process. Appl. 140 (2021), 81–114. MR4276494

    2021

  37. [44]

    , Local weak convergence for sparse networks of interacting processes , Ann. Appl. Probab. 33 (2023), no. 2, 643–688. MR4564415

    2023

  38. [45]

    Theory Related Fields 187 (2023), no

    , Marginal dynamics of interacting diffusions on unimodular Galton-Watson trees , Probab. Theory Related Fields 187 (2023), no. 3-4, 817–884. MR4664586

    2023

  39. [46]

    Lacker, M

    D. Lacker, M. Shkolnikov, and J. Zhang, Inverting the Markovian projection, with an application to local stochastic volatility models, Ann. Probab. 48 (2020), no. 5, 2189–2211. MR4152640

    2020

  40. [48]

    L Lauritzen, Graphical models, Vol

    S. L Lauritzen, Graphical models, Vol. 17, Clarendon Press, 1996

    1996

  41. [49]

    Leli` evre, M

    T. Leli` evre, M. Rousset, and G. Stoltz, Long-time convergence of an adaptive biasing force method , Nonlinearity 21 (2008), no. 6, 1155–1181. MR2422373

    2008

  42. [50]

    Maillet, A note on the long-time behaviour of stochastic McKean-Vlasov equations with common noise (2023), available at 2306.16130

    R. Maillet, A note on the long-time behaviour of stochastic McKean-Vlasov equations with common noise (2023), available at 2306.16130

    2023 arXiv

  43. [51]

    Ramanan, Interacting stochastic processes on sparse random graphs , Invited Paper, Proceedings of the 2022 ICM 6 (2023), 4394––4425

    K. Ramanan, Interacting stochastic processes on sparse random graphs , Invited Paper, Proceedings of the 2022 ICM 6 (2023), 4394––4425

    2023

  44. [52]

    U. A. Rozikov, Gibbs measures on Cayley trees , World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. MR3185400

    2013

  45. [53]

    Schw¨ ammle, F

    V. Schw¨ ammle, F. D Nobre, and E. M. Curado, Consequences of the H–theorem from nonlinear Fokker–Planck equations, Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 76 (2007), no. 4, 041123

    2007

  46. [54]

    Spitzer, Markov random fields on an infinite tree , Ann

    F. Spitzer, Markov random fields on an infinite tree , Ann. Probability 3 (1975), no. 3, 387–398. MR378152

    1975

  47. [55]

    Sznitman, Topics in propagation of chaos, ´Ecole d’´Et´ e de Probabilit´ es de Saint-Flour XIX—1989, 1991, pp

    A.-S. Sznitman, Topics in propagation of chaos, ´Ecole d’´Et´ e de Probabilit´ es de Saint-Flour XIX—1989, 1991, pp. 165–

    1989

  48. [56]

    Tamura, Free energy and the convergence of distributions of diffusion processes of McKean type , J

    Y. Tamura, Free energy and the convergence of distributions of diffusion processes of McKean type , J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987), no. 2, 443–484. MR914029

    1987

  49. [57]

    Trevisan, Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients , Electron

    D. Trevisan, Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients , Electron. J. Probab. 21 (2016), Paper No. 22, 41. MR3485364

    2016

  50. [58]

    Villani, A review of mathematical topics in collisional kinetic theory , Handbook of mathematical fluid dynamics 1 (2002), no

    C. Villani, A review of mathematical topics in collisional kinetic theory , Handbook of mathematical fluid dynamics 1 (2002), no. 71-305, 3–8

    2002

  51. [59]

    Wu, Gradient estimates of Poisson equations on Riemannian manifolds and applications , J

    L. Wu, Gradient estimates of Poisson equations on Riemannian manifolds and applications , J. Funct. Anal. 257 (2009), no. 12, 4015–4033. MR2557733

    2009

  52. [60]

    Zachary, Countable state space Markov random fields and Markov chains on trees , Ann

    S. Zachary, Countable state space Markov random fields and Markov chains on trees , Ann. Probab. 11 (1983), no. 4, 894–903. MR714953

    1983

  53. [61]

    B. a. Zegarli´ nski, Dobrushin uniqueness theorem and logarithmic Sobolev inequalities , J. Funct. Anal. 105 (1992), no. 1, 77–111. MR1156671 Kevin Hu: Division of Applied Mathematics, Brown University. Email address : kevin hu@brown.edu Kavita Ramanan: Division of Applied Mat...

    1992

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