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Quantitative propagation of chaos for the Boltzmann equation with moderately soft potentials

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proves an explicit convergence rate for the Kac particle system to the non-cutoff Boltzmann equation in the moderately soft potential range, the first quantitative propagation of chaos result in this setting.

desk verdict First quantitative POC for the Kac system in moderately soft potentials, with a mostly sound proof that leans hard on a possibly unverified Fisher-information monotonicity for the cutoff kernel. read the letter →

arxiv 2607.13825 v1 pith:W562VUFL submitted 2026-07-15 math.AP math.PR

Chenguang Liu , Liping Xu , An Zhang This is my paper · ORCID
classification math.APmath.PR MSC 82C4060K3565C05
keywords BoltzmannequationKacparticlesystempropagationofchaosFisherinformationnon-cutoffkernelsmoderatelysoftpotentialsWassersteindistancepathwiseuniqueness
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 establishes quantitative propagation of chaos for the Kac N-particle system associated with the spatially homogeneous non-cutoff Boltzmann equation when the collision kernel is moderately soft (-1 < gamma < 0, 0 < nu < 1, gamma + nu < 1). It shows that the expected squared Wasserstein-2 distance between the empirical measure of the particles and the Boltzmann solution decays at the explicit rate N^{-1/3} + N^{-ℓ(q,γ)} on any finite time interval, provided the initial datum has finite Fisher information and a polynomial moment of order q > 8/(1+gamma). It also proves pathwise uniqueness for the particle system. A sympathetic reader would care because this is the first quantitative convergence rate for the Kac particle system in the soft potential regime, where singular relative velocities and grazing collisions make the problem substantially harder than the hard potential case.

What carries the argument

The central object is a two-layer coupling. First, a measurable optimal transport map pulls the Boltzmann solution f_t back to the empirical measure of the evolving particles, producing a family of f_t-distributed random variables used in cutoff Boltzmann processes. Second, a block partition of the N particles restores partial exchangeability, and a suite of coupled and decoupled processes allows estimation of the Wasserstein error. The singular terms |v - w|^γ are handled through uniform bounds on ∫|v-w|^{-α} f_t^K(w) dw, which follow from Fisher information monotonicity, and through an L² bound on a blob-regularized empirical measure of the decoupled processes.

What would settle it

Take a Gaussian initial condition with finite Fisher information, run the cutoff Boltzmann equation for a fixed large K, and monitor sup_{t,w} ∫ |v-w|^{-2} f_t^K(v) dv over long times. If this quantity ever exceeds 1 + I_1(f0) by a non-negligible margin, the uniform singular-moment bound fails and the coupling estimate in Section 7 would break down; if it stays bounded, the mechanism behind the theorem is consistent.

Watch

Extended reading notes

Core claim

For moderately soft potentials, the Kac particle system is pathwise unique, and its empirical measure converges to the unique weak solution of the Boltzmann equation in expected squared Wasserstein-2 distance with the explicit bound sup_{0≤t≤T} E[W_2^2(μ^N_{V_t}, f_t)] ≤ C_{T,q,f0}(N^{-1/3} + N^{-ℓ(q,γ)}), where ℓ(q,γ) is given by an explicit formula. The proof combines a coupling between the particle system and cutoff Boltzmann processes, a block-based decoupling argument that restores partial exchangeability, and a blob regularization of empirical measures to control the singular negative powers of relative velocities. The authors state that this is the first quantitative propagation of ch

Load-bearing premise

The load-bearing premise is the uniform-in-time and uniform-in-K bound sup_w ∫ |v-w|^{-α} f_t^K(v) dv ≤ 1 + I_1(f0), which is cited from the Fisher-information monotonicity result and not proved in the paper; all singular coupling estimates rest on this bound.

Editorial extensions

If this is right

  • If the central estimate is correct, propagation of chaos for the Kac particle system holds quantitatively across the whole moderate soft potential range, not just for hard potentials and Maxwell molecules.
  • When the potential is not too soft and the initial datum has enough moments, the N^{-1/3} term dominates, matching the known rate for Maxwell molecules.
  • Pathwise uniqueness of the Kac particle system now has a proof for every N under finite Fisher information, a necessary starting point for any strong quantitative analysis.
  • The explicit rate ℓ(q,γ) improves as γ approaches 0 and as the polynomial moment q grows, giving a concrete trade-off between moment assumptions and convergence speed.
  • The finite Fisher information assumption removes the need to partition time and separately analyze small and large jumps, simplifying the proof strategy relative to earlier soft-potential results.

Reading between the lines

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

  • Editorial inference: the N^{-1/3} exponent is likely not sharp; the natural fluctuation rate for empirical measures in Wasserstein-2 distance is N^{-1/2}, and obtaining it in the soft potential setting would require a stronger control of negative relative-velocity moments than the current method provides.
  • Editorial inference: the same two-coupling, block-decoupling, blob-regularization scheme could plausibly yield quantitative propagation of chaos for the Landau equation with moderately soft potentials, where existing results are mostly qualitative.
  • Editorial inference: the paper's dependence on the uniform singular-moment bound suggests that relaxing the finite Fisher information assumption would force a local-in-time analysis and likely worsen the rate; a testable extension is to replace Fisher information by a local-in-time L² bound and check whether a rate of the same form survives.
  • Editorial inference: the restriction to γ ∈ (-1,0) is pinned down by the inequality |a^γ - b^γ| ≤ C|a^{-1} - b^{-1}| min{a^{γ+1}, b^{γ+1}}; finding a substitute for this inequality in the range γ ∈ (-2,-1] would be the natural route to extending the quantitative rate to very soft potentials.
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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 / 4 minor

Summary. The paper studies the Kac particle system associated with the spatially homogeneous non-cutoff Boltzmann equation in the moderately soft potential range −1<γ<0, 0<ν<1, γ+ν<1. The two main results are: (Theorem 1.2) pathwise uniqueness for the N-particle SDE system, assuming finite initial Fisher information; and (Theorem 1.3) a quantitative propagation of chaos estimate, sup_{0≤t≤T} E[W_2^2(μ^N_{V_t}, f_t)] ≤ C_{T,q,f0}(N^{-1/3}+N^{-ℓ(q,γ)}), under an additional polynomial moment condition q>8/(1+γ). The proof combines a cutoff approximation of the collision kernel, a coupling between the particle system and cutoff Boltzmann processes, a block-wise decoupling to recover independence, and a blob-regularization estimate for empirical measures. The exponent ℓ(q,γ) is explicit and is obtained by optimizing auxiliary parameters.

Significance. If the proof is completed as written, this would be the first quantitative propagation of chaos result for the Kac particle system in the soft-potential regime, a genuinely open and significant step beyond the hard-potential and Maxwellian cases. The paper is technically substantial: the coupling construction is intricate, the estimates are detailed, and the final rate is derived from explicit error terms rather than fitted. The use of Fisher-information regularity as a substitute for exchangeability is original and promising. However, the main result rests on an unverified application of an external Fisher-information monotonicity theorem to a cutoff kernel, and there is a sign/exponent error in the final K-optimization step. These are local but load-bearing issues; they do not appear to be fatal, but they must be fixed before the theorem can be accepted.

major comments (2)
  1. [Section 4, Lemma 4.1(i)] The uniform singular-moment bound sup_w ∫ |v−w|^{−α} f^K_t(v)dv ≤ 1+I_1(f0) is load-bearing: it is used in Lemma 4.3, Lemma 5.5, and in the Grönwall estimates of Section 7 (e.g. (7.19), (7.22)). Its proof invokes the Fisher-information monotonicity of [35, Thm 1.5] for the cutoff solution f^K_t. But B^K in (1.14) vanishes for θ<G(K), so f^K_t solves a cutoff equation, not the non-cutoff equation to which [35, Thm 1.5] is stated to apply. The sentence 'since the condition Λ_b ≥1 is satisfied for any angular kernel b' does not demonstrate applicability to the truncated kernel. The authors must either prove I_1(f^K_t)≤I_1(f0) uniformly in K for this cutoff equation, or replace Lemma 4.1(i) by an alternative estimate. Without this estimate, the singular factors |W^1,K−Π|^γ in Section 7 are not controlled.
  2. [Section 7, after (7.20) and final estimate] The term N^{4/((1+γ)q−4)} K^{1/ν−1} appears in the bound for B^K_{30} and again in the final estimate before 'Letting K→∞'. Since ν<1, we have 1/ν−1>0, so this term diverges as K→∞, making the subsequent 'Letting K→∞' impossible. Tracing the optimization in Step 3, the correct expression is (N/k)^{4/((1+γ)q−4)} K^{1−1/ν}; the exponent should be K^{1−1/ν} (or equivalently K^{−(1/ν−1)}). The error appears to be a sign typo, but it is in a load-bearing limiting step and must be corrected in (7.20) and in all subsequent lines.
minor comments (4)
  1. [Section 5.2, after (5.10)] Typo: 'compare compare' should be 'compare'.
  2. [Section 7, final optimization paragraph] The phrase 'optimizing by balancing the last two terms' is inaccurate. The displayed ℓ* is chosen so that the k^{−B} term has exponent exactly ℓ(q,γ); the (k/N)^A term is then smaller. This works, but the wording should be corrected to avoid implying a balance that is not performed.
  3. [Lemma 4.1(iii)] The identity R=√(3m_2(f^K_t))=3 uses the normalization m_2(f0)=3; this should be stated explicitly.
  4. [Eq. (1.14)] The expression r^γ∧K is undefined at r=0; add a convention for r=0 or state the estimates are for r>0.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the soft-potential convergence rate is derived from external Fisher-information decay, couplings, and empirical-measure estimates; self-citations are methodological only.

full rationale

The central claim (Theorem 1.3) is not obtained by fitting or by importing the target result. The rate N^{-1/3}+N^{-\ell} is produced by a chain of estimates: Lemma 4.2 gives moment propagation; Lemma 4.3 bounds the cutoff error W2(f^K_t, f_t) via the non-cutoff/cutoff coupling and Lemma 2.2; Proposition 5.7 bounds E W2^2(\mu^N_{W^K_t}, f^K_t) using Lemma 5.5 and the i.i.d. empirical-measure bound [22]; Section 6 controls the regularized L^2 norm of the empirical measure; Section 7 closes a Grönwall argument on E|V^1_t - W^{1,K}_t|^2 and optimizes k and K. The exponent \ell(q,\gamma) is literally obtained by balancing the displayed k-terms, not by assuming (1.24). The only potentially fragile input, Lemma 4.1(i), invokes the external Fisher-information monotonicity [35, Thm 1.5] for the cutoff kernel; whether the hypotheses of [35] apply to B^K is a correctness concern, not a circularity concern: the paper does not define f^K in terms of the target rate, and no fitted parameter is renamed as a prediction. Self-citations to [40] are used as a methodological precursor (e.g., Lemma 5.8 is attributed primarily to [21, Lemma 21]) and are not load-bearing for the soft-potential result. No circular step can be exhibited by equation-to-equation reduction.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No fitted parameters appear in the final theorem; the auxiliary constants K, k, δ, ϵ, M are internal proof parameters that are optimized or sent to limits and do not survive in the statement. The central claim is built on standard inequalities and on previously established kinetic-regularity theorems (Fisher decay, particle-system existence, empirical-measure concentration), not on new postulated physical entities.

assumptions (8)
  • standard math Hardy inequality in R3
    Used to control ∫|v-w|^{-2}g and related singular moments in Lemmas 3.2, 3.3, 4.1.
  • standard math Sobolev embedding H^1(R3)⊂L^6(R3)
    Used in Lemma 4.1(ii) to bound ||f^K_t||_{L^3} via the Fisher information.
  • standard math Measurable optimal-transport selection theorem (Villani, Cor 5.22)
    Used in Lemma 5.1 to construct a jointly measurable optimal map Π between f_t and the empirical measure.
  • standard math Itô formula for Poisson-driven jump SDEs and martingale identification
    Used throughout Sections 3–7 for the particle system, Boltzmann processes, and couplings.
  • domain assumption Fisher-information monotonicity for the Boltzmann equation (non-cutoff and cutoff) [35, Thm 1.5]
    Basis for Lemma 4.1 uniform singular-moment bounds and Lemma 4.3; the soft-potential machinery depends on this imported regularity fact.
  • domain assumption Existence, exchangeability, conservation laws, and Fisher bound for the Kac particle system [27, Thm 4.1]
    Theorem 3.1 is taken as given; it supplies the solution whose pathwise uniqueness is proved in Theorem 1.2.
  • domain assumption Well-posedness and moment bounds for the cutoff Boltzmann equation, and existence of the non-cutoff solution f_t in C([0,T], L3∩P2)
    Assumed in Theorem 1.3 and Section 4; standard from [28,35] but not reproved.
  • standard math Empirical measure Wasserstein-2 concentration bound for i.i.d. samples [22, Thm 1]
    Used in Proposition 5.7 to control the distance from the independent decoupled processes to f^K_t.

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Pith. "Pith review of Quantitative propagation of chaos for the Boltzmann equation with moderately soft potentials." pith.science (2026). https://pith.science/paper/W562VUFL

@misc{pith2026260713825,
  author       = {Pith},
  title        = {Pith review of: Quantitative propagation of chaos for the Boltzmann equation with moderately soft potentials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W562VUFL}},
  note         = {Machine review of arXiv:2607.13825}
}
abstract

We study the Kac particle system associated with the spatially homogeneous Boltzmann equation with non-cutoff collision kernels in the moderately soft potential regime $-1<\gamma<0$. We prove uniqueness in law for the particle system and, in particular, establish an explicit convergence rate from the empirical measure of the Kac particle system to the weak solution of the Boltzmann equation in Wasserstein-2 distance, assuming that the initial datum $f_0$ has finite Fisher information and polynomial moments. To the best of our knowledge, this provides the first quantitative convergence rate for the Kac particle system in the soft potential setting.

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Works this paper leans on

54 extracted references · 7 linked inside Pith

  1. [13]

    Cortez and J

    R. Cortez and J. Fontbona. Quantitative uniform propagation of chaos for Maxwell molecules.Comm. Math. Phys., 357(3):913–941, 2018

    2018

  2. [40]

    C. Liu, L. Xu and A. Zhang. Rate of convergence of the Kac particle system for the Boltzmann equation with hard potentials.arXiv:2409.04031

    arXiv

  3. [55]

    L. Xu. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials.Ann. Appl. Probab., 28(2):1136–1189, 2018. Delft Institute of Applied Mathematics, EEMCS, TU Delft, 2628 Delft, The Netherlands Email address:liucg92@gmail.com School of Mathematical Sciences, Beihang University, PR China Email address:xuliping.p6@gmail.c...

    2018

  4. [35]

    Imbert, L

    C. Imbert, L. Silvestre and C. Villani. On the monotonicity of the Fisher information for the Boltzmann equationInvent. math. (2025).,

    2025

  5. [27]

    Fournier and S

    N. Fournier and S. Mischler. Propagation of chaos for the homogeneous Boltzmann equation with moderately soft potentials.hal-05398962

  6. [1]

    D. Aldous. Stopping times and tightness.Ann. Probability, 6(2):335–340, 1978

    1978

  7. [2]

    Alexandre

    R. Alexandre. A review of Boltzmann equation with singular kernels.Kinet. Relat. Models, 2(4):551–646, 2009

    2009

  8. [3]

    Alexandre, L

    R. Alexandre, L. Desvillettes, C. Villani, and B. Wennberg. Entropy dissipation and long-range interactions. Arch. Ration. Mech. Anal., 152(4):327–355, 2000

    2000

Show all 54 references
  1. [4]

    L. Arkeryd. On the Boltzmann equation. I. Existence.Arch. Rational Mech. Anal., 45:1–16, 1972

    1972

  2. [5]

    L. Arkeryd. On the Boltzmann equation. II. The full initial value problem.Arch. Rational Mech. Anal., 45:17– 34, 1972

    1972

  3. [6]

    Boltzmann

    L. Boltzmann. Weitere studien ¨ uber das w¨ armegleichgewicht unter gasmolek¨ ulen. Sitzungsberichte Akademie der Wissenschaften, 66:275–370, 1872

  4. [7]

    E. A. Carlen. Superadditivity of Fisher’s information and logarithmic Sobolev inequalities.J. Funct. Anal., 101, 1 (1991), 194–211

    1991

  5. [8]

    C. Cao, D. Deng, and X. Li. The Vlasov-Poisson-Boltzmann/Landau system with polynomial perturbation near Maxwellian.SIAM J. Math. Anal., 56(1):820–876, 2024

    2024

  6. [9]

    J. A. Carrillo, X. Feng, S. Guo, P. -E. Jabin, and Z. Wang. Relative entropy method for particle approximation of the Landau equation for Maxwellian molecules.Journal de Math´ ematiques Pures et Appliqu´ ees, to appear. KAC PARTICLE SYSTEMS FOR THE BOLTZMANN EQUATION 47

  7. [10]

    J. A. Carrillo and S. Guo. From Fisher information decay for the Kac model to the Landau-Coulomb hierarchy. Arxiv: 2502.18606

    arXiv

  8. [11]

    Cercignani

    C. Cercignani. The Boltzmann equation and its applications, volume 67 ofApplied Mathematical Sciences. Springer-Verlag, New York, 1988

    1988

  9. [12]

    Cortez and J

    R. Cortez and J. Fontbona. Quantitative propagation of chaos for generalized Kac particle systems.Ann. Appl. Probab., 26(2):892–916, 2016

    2016

  10. [14]

    Desvillettes

    L. Desvillettes. Boltzmann’s kernel and the spatially homogeneous Boltzmann equation.Riv. Mat. Univ. Parma (6), 4*:1–22, 2001. Fluid dynamic processes with inelastic interactions at the molecular scale (Torino, 2000)

    2001

  11. [15]

    Desvillettes, C

    L. Desvillettes, C. Graham, and S. M´ el´ eard. Probabilistic interpretation and numerical approximation of a Kac equation without cutoff.Stochastic Process. Appl., 84(1):115–135, 1999

    1999

  12. [16]

    Desvillettes and C

    L. Desvillettes and C. Mouhot. Stability and uniqueness for the spatially homogeneous Boltzmann equation with long-range interactions.Arch. Ration. Mech. Anal., 193(2):227–253, 2009

    2009

  13. [17]

    Y. Deng, Z. Hani, and X. Ma. Long time derivation of the Boltzmann equation from hard sphere dynamics, Annals of Mathematics (2025),to appear

    2025

  14. [18]

    Feng and Z

    X. Feng and Z. Wang. Kac’s Program for the Landau EquationarXiv:2506.14309

    arXiv

  15. [19]

    Fournier

    N. Fournier. Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition. Ann. Appl. Probab., 25(2):860–897, 2015

    2015

  16. [20]

    Fournier and H

    N. Fournier and H. Gu´ erin. On the uniqueness for the spatially homogeneous Boltzmann equation with a strong angular singularity.J. Stat. Phys., 131(4):749–781, 2008

    2008

  17. [21]

    Fournier and A

    N. Fournier and A. Guillin. From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules.Ann. Sci. ´Ec. Norm. Sup´ er. (4), 50(1):157–199, 2017

    2017

  18. [22]

    Fournier and A

    N. Fournier and A. Guillin. On the rate of convergence in Wasserstein distance of the empirical measure.Probab. Theory Related Fields 143(2015), Vol. 162, 707-738

    2015

  19. [23]

    Fournier and M

    N. Fournier and M. Hauray. Propagation of chaos for the Landau equation with moderately soft potentials. Ann. Probab., 44(6):3581–3660, 2016

    2016

  20. [24]

    Fournier, M

    N. Fournier, M. Hauray and S. Mischler. Propagation of chaos for the 2D viscous vortex model.J. Eur. Math. Soc. (JEMS)16, 7 (2014), 1423–1466

    2014

  21. [25]

    Fournier and S

    N. Fournier and S. M´ el´ eard. A stochastic particle numerical method for 3D Boltzmann equations without cutoff. Math. Comp., 71(238):583–604, 2002

    2002

  22. [26]

    Fournier and S

    N. Fournier and S. Mischler. Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules.Ann. Probab., 44(1):589–627, 2016

    2016

  23. [28]

    Fournier and C

    N. Fournier and C. Mouhot. On the well-posedness of the spatially homogeneous Boltzmann equation with a moderate angular singularity.Comm. Math. Phys., 289(3):803–824, 2009

    2009

  24. [29]

    Graham and S

    C. Graham and S. M´ el´ eard. Stochastic particle approximations for generalized Boltzmann models and conver- gence estimates.Ann. Probab., 25(1):115–132, 1997

    1997

  25. [30]

    F. A. Gr¨ unbaum. Propagation of chaos for the Boltzmann equation.Arch. Rational Mech. Anal., 42:323–345, 1971

    1971

  26. [31]

    Guillen and L

    N. Guillen and L. Silvestre. The Landau equation does not blow up,Acta Mathematica 234 (2025), no. 2, 315–375

    2025

  27. [32]

    S. Guo. From Kac particles to the Landau equation with hard potentials: BBGKY hierarchy method, arXiv:2508.10697

    arXiv

  28. [33]

    Hauray and S

    M. Hauray and S. Mischler. On Kac’s chaos and related problems.J. Funct. Anal. 266 (2014), no. 10, 6055–6157

    2014

  29. [34]

    Heydecker

    D. Heydecker. Kac’s process with hard potentials and a moderate angular singularity.Arch. Ration. Mech. Anal., 244(3):699–759, 2022

    2022

  30. [36]

    Jacod, A.N

    J. Jacod, A.N. Shiryaev. Limit theorems for stochastic processes, Second edition. Springer-Verlag, 2003

    2003

  31. [37]

    Jakubowski

    A. Jakubowski. On the Skorokhod topology. Annales de l’Institut Henri Poincar´ e Probabilit´ es et Statistiques 22 (1986), no. 3, 263–285

    1986

  32. [38]

    M. Kac. Foundations of kinetic theory. InProceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, vol. III, pages 171–197. Univ. California Press, Berkeley-Los Angeles, Calif., 1956. 48 C. LIU, L. XU, AND A. ZHANG

    1954

  33. [39]

    O. E. Lanford. Time evolution of large classical systems.Lecture Notes in Physics, pages 1–111, 1975

    1975

  34. [41]

    Rougerie

    N. Rougerie. On Two Properties of the Fisher Information InKinetic and Related Models 14.1 (Nov. 2020), pp. 77–88

    2020

  35. [42]

    Silvestre

    L. Silvestre. A new regularization mechanism for the Boltzmann equation without cut-off,Commun. Math. Phys. 348, 69–100 (2016)

    2016

  36. [43]

    H. P. McKean, Jr. An exponential formula for solving Boltmann’s equation for a Maxwellian gas.J. Combina- torial Theory, 2:358–382, 1967

    1967

  37. [44]

    Mischler and C

    S. Mischler and C. Mouhot. Kac’s program in kinetic theory.Invent. Math., 193(1):1–147, 2013

    2013

  38. [45]

    K. Nanbu. Stochastic solution method of the master equation and the model Boltzmann equation.J. Phys. Soc. Japan, 52, 8 (1983), 2654–2658

    1983

  39. [46]

    S. Salem. Derivation of the boltzmann equation with moderately soft potentials from a perturbed nanbu particles system.arXiv:1910.01883v2

    1910 arXiv

  40. [47]

    Sznitman

    A.-S. Sznitman. Topics in propagation of chaosIn: Ecole d’Et´ e de Probabilit´ es de Saint-Flour XIX — 1989. Ed. by Donald L. Burkholder, Etienne Pardoux, Alain-Sol Sznitman, and Paul-Louis Hennequin. Berlin, Heidelberg: Springer, 1991, pp. 165– 251

    1989

  41. [48]

    Sznitman

    A.-S. Sznitman. ´Equations de type de Boltzmann, spatialement homog` enes.Z. Wahrsch. Verw. Gebiete, 66(4):559–592, 1984

    1984

  42. [49]

    C. Tabary. Propagation of chaos for the Landau equation with very soft and Coulomb potentials. arXiv:2506.15795

  43. [50]

    C. Tabary. Propagation of chaos for the Boltzmann equation with very soft potentials.arxiv: 2604.13855

    arXiv

  44. [51]

    H. Tanaka. On the uniqueness of Markov process associated with the Boltzmann equation of Maxwellian molecules. InProceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Wiley-Intersci. Publ., pages 409–...

    1976

  45. [52]

    C. Villani. A review of mathematical topics in collisional kinetic theory. InHandbook of mathematical fluid dynamics, Vol. I, pages 71–305. North-Holland, Amsterdam, 2002. [53]Villani, C.Topics in Optimal Transportation, vol. 58 ofGraduate Studies in Mathematics series. Americ...

    2002

  46. [54]

    C. Villani. Fisher Information in Kinetic Theory,arXiv:2501.00925

    arXiv

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