Article Contents
Article Contents

# Propagation of chaos: A review of models, methods and applications. Ⅱ. Applications

• *Corresponding author: Antoine Diez

The work of AD is supported by an EPSRC-Roth scholarship cofunded by the Engineering and Physical Sciences Research Council and the Department of Mathematics at Imperial College London

• The notion of propagation of chaos for large systems of interacting particles originates in statistical physics and has recently become a central notion in many areas of applied mathematics. The present review describes old and new methods as well as several important results in the field. The models considered include the McKean-Vlasov diffusion, the mean-field jump models and the Boltzmann models. The first part of this review is an introduction to modelling aspects of stochastic particle systems and to the notion of propagation of chaos. The second part presents concrete applications and a more detailed study of some of the important models in the field.

Mathematics Subject Classification: Primary: 82C22, 82C40, 35Q70; Secondary: 65C35, 92-10.

 Citation:

• Figure 1.  An interaction graph. The vertical axis represents time. Each particle is represented by a vertical line parallel to the time axis. The index of a given particle is written on the horizontal axis. The construction is done backward in time starting from time $t$ where only particle $i$ is present. At each time $t_\ell$, if $i_\ell$ does not already belong to the graph, it is added on the right (with a vertical line which starts at $t_\ell$). The couple $r_\ell = (i_\ell,j_\ell)$ of interacting particles at time $t_\ell$ is depicted by an horizontal line joining two big black dots on the vertical line representing the particles $i_\ell$ and $j_\ell$. for instance, on the depicted graph, $r_2 = (i_2,i)$. Note that at time $t_3$, $r_3 = (i_1,i_2)$ (or indifferently $r_3 = (i_2,i_1)$) where $i_1$ and $i_2$ were already in the system. Index $i_3$ is skipped and at time $t_4$, the route is $r_4 = (i_4,i_1)$. The recollision occurring at time $t_3$ is depicted in red

Figure 2.  A route of size $q$ between $i$ and $j$. The chain of interactions which links $i$ and $j$ are depicted by horizontal lines as explained in Section 2.3.5

Figure 3.  At time $t_1^+$, there is only one particle, at the same position for the BBGKY (in white) and Boltzmann (in red) pseudo-trajectories. At time $t_1^+$, a particle (dotted) is added next to the white particle in a pre-collisional way. At time $t_2^+$, a particle (dashed) is added next to the dotted particle in a post-collisional way. Due to a recollision at time $\tau\in(t_3,t_2)$, the Boltzmann and BBGKY pseudo-trajectories of the white/red particle are no longer close to each other at time $t_3$

•  [1] J. A. Acebrón, L. L. Bonilla, C. J. Pérez Vicente, F. Ritort and R. Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Modern Phys., 77 (2005), 137-185.  doi: 10.1103/RevModPhys.77.137. [2] S. M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys., 51 (2010), 103301.  doi: 10.1063/1.3496895. [3] G. Albi, N. Bellomo, L. Fermo, S.-Y. Ha, J. Kim, L. Pareschi, D. Poyato and J. Soler, Vehicular traffic, crowds, and swarms: From kinetic theory and multiscale methods to applications and research perspectives, Math. Models Methods Appl. Sci., 29 (2019), 1901-2005.  doi: 10.1142/S0218202519500374. [4] D. Aldous, Stopping times and tightness, Ann. Probab., 6 (1978), 335-340.  doi: 10.1214/aop/1176995579. [5] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2nd edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008. [6] L. Andreis, P. Dai Pra and M. Fischer, McKean-Vlasov limit for interacting systems with simultaneous jumps, Stoch. Anal. Appl., 36 (2018), 960-995.  doi: 10.1080/07362994.2018.1486202. [7] N. Ayi, From Newton's law to the linear Boltzmann equation without cut-off, Comm. Math. Phys., 350 (2017), 1219-1274.  doi: 10.1007/s00220-016-2821-6. [8] H. Babovsky, On a simulation scheme for the Boltzmann equation, Math. Methods Appl. Sci., 8 (1986), 223-233.  doi: 10.1002/mma.1670080114. [9] H. Babovsky and R. Illner, A convergence proof for Nanbu's simulation method for the full Boltzmann equation, SIAM J. Numer. Anal., 26 (1989), 45-65.  doi: 10.1137/0726004. [10] M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.  doi: 10.1073/pnas.0711437105. [11] N. Bellomo, P. Degond and E. Tadmor (eds.), Active Particles, Volume 1: Advances in Theory, Models, and Applications, Modeling and Simulation in Science, Engineering and Technology, Springer International Publishing, 2017. doi: 10.1007/978-3-319-49996-3. [12] N. Bellomo, P. Degond and E. Tadmor (eds.), Active Particles, Volume 2: Advances in Theory, Models, and Applications, Modeling and Simulation in Science, Engineering and Technology, Springer International Publishing, 2019. doi: 10.1007/978-3-030-20297-2. [13] S. Benachour, B. Roynette and P. Vallois, Nonlinear self-stabilizing processes - Ⅱ: Convergence to invariant probability, Stochastic Process. Appl., 75 (1998), 203-224.  doi: 10.1016/S0304-4149(98)00019-2. [14] D. Benedetto, E. Caglioti, J. A. Carrillo and M. Pulvirenti, A non-Maxwellian steady distribution for one-dimensional granular media, J. Stat. Phys., 91 (1998), 979-990.  doi: 10.1023/A:1023032000560. [15] D. Benedetto, E. Caglioti and M. Pulvirenti, A kinetic equation for granular media, ESAIM: Mathematical Modelling and Numerical Analysis, 31 (1997), 615-641.  doi: 10.1051/m2an/1997310506151. [16] R. J. Berman and M. Önnheim, Propagation of chaos for a class of first order models with singular mean field interactions, SIAM J. Math. Anal., 51 (2019), 159-196.  doi: 10.1137/18M1196662. [17] E. Bertin, M. Droz and G. Grégoire, Boltzmann and hydrodynamic description for self-propelled particles, Phys. Rev. E, 74 (2006), 022101.  doi: 10.1103/PhysRevE.74.022101. [18] E. Bertin, M. Droz and G. Grégoire, Hydrodynamic equations for self-propelled particles: Microscopic derivation and stability analysis, J. Phys. A: Math. Theor., 42 (2009), 445001.  doi: 10.1088/1751-8113/42/44/445001. [19] L. Bertini, G. Giacomin and K. Pakdaman, Dynamical aspects of mean field plane rotators and the Kuramoto model, J. Stat. Phys., 138 (2009), 270-290.  doi: 10.1007/s10955-009-9908-9. [20] L. Bertini, G. Giacomin and C. Poquet, Synchronization and random long time dynamics for mean-field plane rotators, Probab. Theory Related Fields, 160 (2014), 593-653.  doi: 10.1007/s00440-013-0536-6. [21] P. L. Bhatnagar, E. P. Gross and M. Krook, A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems, Phys. Rev., 94 (1954), 511-525.  doi: 10.1103/PhysRev.94.511. [22] G. A. Bird, Direct simulation and the Boltzmann equation, Phys. Fluids, 13 (1970), 2676.  doi: 10.1063/1.1692849. [23] A. Blanchet and P. Degond, Topological interactions in a Boltzmann-type framework, J. Stat. Phys., 163 (2016), 41-60.  doi: 10.1007/s10955-016-1471-6. [24] A. Blanchet and P. Degond, Kinetic models for topological nearest-neighbor interactions, J. Stat. Phys., 169 (2017), 929-950.  doi: 10.1007/s10955-017-1882-z. [25] T. Bodineau, I. Gallagher and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. math., 203 (2016), 493-553.  doi: 10.1007/s00222-015-0593-9. [26] T. Bodineau, I. Gallagher and L. Saint-Raymond, From hard sphere dynamics to the Stokes-Fourier equations: An analysis of the Boltzmann-Grad limit, Ann. PDE, 3 (2017), 2.  doi: 10.1007/s40818-016-0018-0. [27] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella, One-sided convergence in the Boltzmann-Grad limit, Ann. Fac. Sci. Toulouse Math.(6), 27 (2018), 985-1022.  doi: 10.5802/afst.1589. [28] T. Bodineau, I. Gallagher, L. Saint-Raymond and S. Simonella, Statistical dynamics of a hard sphere gas: Fluctuating Boltzmann equation and large deviations, preprint, arXiv: 2008.10403 [29] N. Boers and P. Pickl, On mean field limits for dynamical systems, Journal of Statistical Physics, 164 (2016), 1-16.  doi: 10.1007/s10955-015-1351-5. [30] F. Bolley, Quantitative concentration inequalities on sample path space for mean field interaction, ESAIM Probab. Stat., 14 (2010), 192-209.  doi: 10.1051/ps:2008033. [31] F. Bolley, J. A. Ca˜nizo and J. A. Carrillo, Stochastic mean-field limit: Non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci., 21 (2011), 2179–2210, Publisher: World Scientific. [32] F. Bolley, J. A. Ca˜nizo and J. A. Carrillo, Mean-field limit for the stochastic Vicsek model, Appl. Math. Lett., 25 (2012), 339–343, Publisher: Elsevier. [33] F. Bolley, I. Gentil and A. Guillin, Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.  doi: 10.1016/j.jfa.2012.07.007. [34] F. Bolley, I. Gentil and A. Guillin, Uniform convergence to equilibrium for granular media, Arch. Ration. Mech. Anal., 208 (2013), 429-445.  doi: 10.1007/s00205-012-0599-z. [35] F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM Math. Model. Numer. Anal., 44 (2010), 867-884.  doi: 10.1051/m2an/2010045. [36] F. Bolley, A. Guillin and C. Villani, Quantitative concentration inequalities for empirical measures on non-compact spaces, Probab. Theory Related Fields, 137 (2006), 541-593.  doi: 10.1007/s00440-006-0004-7. [37] M. Bossy, Some stochastic particle methods for nonlinear parabolic PDEs, ESAIM Proc., 15 (2005), 18-57. [38] M. Bossy, O. Faugeras and D. Talay, Clarification and complement to "Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons", J. Math. Neurosci., 5 (2015), 5-19.  doi: 10.1186/s13408-015-0031-8. [39] M. Bossy and D. Talay, Convergence rate for the approximation of the limit law of weakly interacting particles: Application to the Burgers equation, Ann. Appl. Probab., 6 (1996), 818-861.  doi: 10.1214/aoap/1034968229. [40] M. Bossy and D. Talay, A stochastic particle method for the Mckean-Vlasov and the Burgers equation, Math. Comp., 66 (1997), 157-192.  doi: 10.1090/S0025-5718-97-00776-X. [41] W. Braun and K. Hepp, The Vlasov dynamics and its fluctuation in the 1/N limit of interacting particles, Comm. Math. Phys., 56 (1977), 101-113.  doi: 10.1007/BF01611497. [42] D. Bresch, P.-E. Jabin and Z. Wang, On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak-Keller-Segel model, C. R. Math. Acad. Sci. Paris, 357 (2019), 708-720.  doi: 10.1016/j.crma.2019.09.007. [43] M. Briant, A. Diez and S. Merino-Aceituno, Cauchy theory and mean-field limit for general Vicsek models in collective dynamics, preprint, arXiv: 2004.00883. doi: 10.1137/21M1405885. [44] P. Calderoni and M. Pulvirenti, Propagation of chaos for Burgers' equation, Ann. Inst. Henri Poincaré, Physique théorique, 39 (1983), 85–97. [45] J. A. Cañizo and H. Yolda, Asymptotic behaviour of neuron population models structured by elapsed-time, Nonlinearity, 32 (2018), 464-495.  doi: 10.1088/1361-6544/aaea9c. [46] P. Cardaliaguet, Notes on mean field games (from P.-L. Lions' lectures at Collège de France), in Lecture given at Tor Vergata, 2010, 1–59. [47] P. Cardaliaguet, F. Delarue, J.-M. Lasry and P.-L. Lions, The Master Equation and the Convergence Problem in Mean Field Games, Annals of Mathematics Studies, 201, Princeton University Press, 2019. [48] E. Carlen, M. C. Carvalho, P. Degond and B. Wennberg, A Boltzmann model for rod alignment and schooling fish, Nonlinearity, 28 (2015), 1783-1803.  doi: 10.1088/0951-7715/28/6/1783. [49] E. Carlen, R. Chatelin, P. Degond and B. Wennberg, Kinetic hierarchy and propagation of chaos in biological swarm models, Phys. D, 260 (2013), 90-111.  doi: 10.1016/j.physd.2012.05.013. [50] E. Carlen, P. Degond and B. Wennberg, Kinetic limits for pair-interaction driven master equations and biological swarm models, Math. Models Methods Appl. Sci., 23 (2013), 1339-1376.  doi: 10.1142/S0218202513500115. [51] R. Carmona, Lectures on BSDEs, Stochastic Control, and Stochastic Differential Games with Financial Applications, SIAM, 2016. doi: 10.1137/1.9781611974249. [52] R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications I, Mean Field FBSDEs, Control, and Games, Probability Theory and Stochastic Modelling, 83, Springer International Publishing, 2018. [53] R. Carmona and F. Delarue, Probabilistic Theory of Mean Field Games with Applications Ⅱ, Mean Field Games with Common Noise and Master Equations, Probability Theory and Stochastic Modelling, 84, Springer International Publishing, 2018. [54] K. Carrapatoso, Propagation of chaos for the spatially homogeneous Landau equation for Maxwellian molecules, Kinet. Relat. Models, 9 (2015), 1-49.  doi: 10.3934/krm.2016.9.1. [55] J. A. Carrillo, Y.-P. Choi, C. Totzeck and O. Tse, An analytical framework for consensus-based global optimization method, Math. Models Methods Appl. Sci., 28 (2018), 1037-1066.  doi: 10.1142/S0218202518500276. [56] J. A. Carrillo, M. Fornasier, G. Toscani and F. Vecil, Particle, kinetic, and hydrodynamic models of swarming, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences (eds. G. Naldi, L. Pareschi and G. Toscani), Birkhäuser Boston, 2010,297–336. doi: 10.1007/978-0-8176-4946-3_12. [57] J. A. Carrillo, S. Jin, L. Li and Y. Zhu, A consensus-based global optimization method for high dimensional machine learning problems, ESAIM Control Optim. Calc. Var., 27 (2021), 1-22.  doi: 10.1051/cocv/2020046. [58] J. A. Carrillo, Y.-P. Choi and M. Hauray, The derivation of swarming models: Mean-field limit and Wasserstein distances, in Collective Dynamics from Bacteria to Crowds (eds. A. Muntean and F. Toschi), CISM International Centre for Mechanical Sciences, 553, Springer, Vienna, 2014, 1–46. doi: 10.1007/978-3-7091-1785-9_1. [59] J. A. Carrillo, Y.-P. Choi and S. Salem, Propagation of chaos for the Vlasov-Poisson-Fokker-Planck equation with a polynomial cut-off, Commun. Contemp. Math., 21 (2019), 1850039.  doi: 10.1142/S0219199718500396. [60] J. A. Carrillo, M. Delgadino and G. Pavliotis, A $\lambda$-convexity based proof for the propagation of chaos for weakly interacting stochastic particles, J. Funct. Anal., 279 (2020), 108734.  doi: 10.1016/j.jfa.2020.108734. [61] J. A. Carrillo, M. R. D'Orsogna and V. Panferov, Double milling in self-propelled swarms from kinetic theory, Kinet. Relat. Models, 2 (2009), 363-378.  doi: 10.3934/krm.2009.2.363. [62] J. A. Carrillo, R. J. McCann and C. Villani, Kinetic equilibration rates for granular media and related equations: Entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana, 19 (2003), 971-1018.  doi: 10.4171/RMI/376. [63] J. A. Carrillo, R. J. McCann and C. Villani, Contractions in the 2-Wasserstein length space and thermalization of granular media, Arch. Ration. Mech. Anal., 17 (2006), 217-263.  doi: 10.1007/s00205-005-0386-1. [64] 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), 19-40.  doi: 10.1007/s00440-007-0056-3. [65] P. Cattiaux, F. Delebecque and L. Pédèches, Stochastic Cucker-Smale models: Old and new, Ann. Appl. Probab., 28 (2018), 3239-3286.  doi: 10.1214/18-AAP1400. [66] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Applied Mathematical Sciences, 106, Springer-Verlag New York, 1994. doi: 10.1007/978-1-4419-8524-8. [67] J.-F. Chassagneux, L. Szpruch and A. Tse, Weak quantitative propagation of chaos via differential calculus on the space of measures, preprint, arXiv: 1901.02556. [68] H. Chaté, F. Ginelli, G. Grégoire and F. Raynaud, Collective motion of self-propelled particles interacting without cohesion, Phys. Rev. E, 77 (2008), 046113.  doi: 10.1103/PhysRevE.77.046113. [69] P.-E. Chaudru de Raynal, Strong well posedness of McKean-Vlasov stochastic differential equations with Hölder drift, Stochastic Process. Appl., 130 (2020), 79-107.  doi: 10.1016/j.spa.2019.01.006. [70] P.-E. Chaudru de Raynal and N. Frikha, From the backward Kolmogorov PDE on the Wasserstein space to propagation of chaos for McKean-Vlasov SDE's, preprint, arXiv: 1907.01410. doi: 10.1016/j.matpur.2021.10.010. [71] P.-E. Chaudru de Raynal and N. Frikha, Well-posedness for some non-linear diffusion processes and related PDE on the Wasserstein space, preprint, arXiv: 1811.06904 doi: 10.1016/j.matpur.2021.12.001. [72] J.-Y. Chemin, Fluides Parfaits Incompressibles, Astérisque, 230, Société mathématique de France, 1995. [73] L. Chen, E. S. Daus, A. Holzinger and A. Jüngel, Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems, J. Nonlinear Sci., 31 (2021), 1-38.  doi: 10.1007/s00332-021-09747-9. [74] J. Chevallier, Mean-field limit of generalized Hawkes processes, Stochastic Process. Appl., 127 (2017), 3870-3912.  doi: 10.1016/j.spa.2017.02.012. [75] T.-S. Chiang, McKean-Vlasov equations with discontinuous coefficients, Soochow Journal of Mathematics, 20 (1994), 507-526. [76] L. Chizat and F. Bach, On the Global Convergence of Gradient Descent for Over-parameterized Models using Optimal Transport, in Advances in Neural Information Processing Systems 31 (NeurIPS 2018) (eds. S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi and R. Garnett), Curran Associates, Inc., Montreal, Canada, 2018, 3040–3050. [77] Y.-P. Choi and S. Salem, Propagation of chaos for aggregation equations with no-flux boundary conditions and sharp sensing zones, Math. Models Methods Appl. Sci., 28 (2018), 223-258.  doi: 10.1142/S0218202518500070. [78] Y.-P. Choi and S. Salem, Collective behavior models with vision geometrical constraints: Truncated noises and propagation of chaos, J. Differential Equations, 266 (2019), 6109-6148.  doi: 10.1016/j.jde.2018.10.042. [79] Y.-P. Choi and S. Salem, Cucker-Smale flocking particles with multiplicative noises: Stochastic mean-field limit and phase transition, Kinet. Relat. Models, 12 (2019), 573-592.  doi: 10.3934/krm.2019023. [80] A. J. Chorin, Numerical study of slightly viscous flow, J. Fluid Mech., 57 (1973), 785-796.  doi: 10.1017/S0022112073002016. [81] G. Clarté, A. Diez and J. Feydy, Collective proposal distributions for nonlinear MCMC samplers: Mean-field theory and fast implementation, preprint, arXiv: 1909.08988 [82] M. Coghi and F. Flandoli, Propagation of chaos for interacting particles subject to environmental noise, Ann. Appl. Probab., 26 (2016), 1407-1442.  doi: 10.1214/15-AAP1120. [83] R. Cortez and J. Fontbona, Quantitative propagation of chaos for generalized Kac particle systems, Ann. Appl. Probab., 26 (2016), 892-916.  doi: 10.1214/15-AAP1107. [84] R. Cortez and J. Fontbona, Quantitative uniform propagation of chaos for Maxwell molecules, Commun. Math. Phys., 357 (2018), 913-941.  doi: 10.1007/s00220-018-3101-4. [85] D. Crisan and A. Doucet, A survey of convergence results on particle filtering methods for practitioners, IEEE Trans. Signal Process., 50 (2002), 736-746.  doi: 10.1109/78.984773. [86] I. Csisz{á}r, Sanov property, generalized {I}-projection and a conditional limit theorem, Ann. Probab., 12 (1984), 768–793, https://projecteuclid.org/euclid.aop/1176993227. [87] F. Cucker and S. Smale, On the mathematics of emergence, Jpn. J. Math., 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [88] P. Dai Pra and F. den Hollander, McKean-Vlasov limit for interacting random processes in random media, J. Stat. Phys., 84 (1996), 735-772.  doi: 10.1007/BF02179656. [89] S. Danieri and G. Savaré, Lecture notes on gradient flows and optimal transport, in Optimal Transportation (eds. H. Pajot, Y. Ollivier and C. Villani), Cambridge University Press, Cambridge, 2014,100–144, https://www.cambridge.org/core/product/identifier/CBO9781107297296A015/type/book_part. [90] D. Dawson, Measure-valued Markov processes, in École d'Été de Probabilités de Saint-Flour XXI-1991 (ed. P. Hennequin), Lecture Notes in Mathematics, 1541, Springer Berlin Heidelberg, 1993. doi: 10.1007/BFb0084190. [91] D. Dawson and J. Gärtner, Large deviations from the McKean-Vlasov limit for weakly interacting diffusions, Stochastics, 20 (1987), 247-308.  doi: 10.1080/17442508708833446. [92] D. Dawson and J. Vaillancourt, Stochastic McKean-Vlasov equations, NoDEA Nonlinear Differential Equations Appl., 2 (1995), 199-229.  doi: 10.1007/BF01295311. [93] D. A. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, J. Stat. Phys., 31 (1983), 29-85.  doi: 10.1007/BF01010922. [94] D. A. Dawson and K. J. Hochberg, Wandering random measures in the Fleming-Viot model, Ann. Probab., 10 (1982), 554–580, https://projecteuclid.org/journals/annals-of-probability/volume-10/issue-3/Wandering-Random-Measures-in-the-Fleming-Viot-Model/10.1214/aop/1176993767.full. [95] V. De Bortoli, A. Durmus and X. Fontaine, Quantitative propagation of chaos for SGD in wide neural networks, in Advances in Neural Information Processing Systems 33 (NeurIPS 2020), 2020,278–288, https://proceedings.neurips.cc/paper/2020/file/02e74f10e0327ad868d138f2b4fdd6f0-Paper.pdf. [96] A. De Masi, A. Galves, E. Löcherbach and E. Presutti, Hydrodynamic limit for interacting neurons, J. Stat. Phys., 158 (2015), 866-902.  doi: 10.1007/s10955-014-1145-1. [97] P. Degond and M. Pulvirenti, Propagation of chaos for topological interactions, Ann. Appl. Probab., 29 (2019), 2594-2612.  doi: 10.1214/19-AAP1469. [98] P. Degond, Macroscopic limits of the Boltzmann equation: A review, in Modeling and Computational Methods for Kinetic Equations (eds. N. Bellomo, P. Degond, L. Pareschi and G. Russo), Birkhäuser Boston, Boston, MA, 2004, 3–57, Series Title: Modeling and Simulation in Science, Engineering and Technology. doi: 10.1007/978-0-8176-8200-2_1. [99] P. Degond, Mathematical models of collective dynamics and self-organization, in Proceedings of the International Congress of Mathematicians ICM 2018, 4, Rio de Janeiro, Brazil, 2018, 3943–3964. [100] P. Degond, A. Frouvelle and J.-G. Liu, Phase transitions, hysteresis, and hyperbolicity for self-organized alignment dynamics, Arch. Ration. Mech. Anal., 216 (2015), 63-115.  doi: 10.1007/s00205-014-0800-7. [101] P. Degond, A. Frouvelle, S. Merino-Aceituno and A. Trescases, Alignment of self-propelled rigid bodies: From particle systems to macroscopic equations, in Stochastic Dynamics Out of Equilibrium, Institut Henri Poincaré, Paris, France, 2017 (eds. G. Giacomin, S. Olla, E. Saada, H. Spohn and G. Stoltz), Springer Proceedings in Mathematics & Statistics, 282, Springer, Cham, 2019, 28–66. doi: 10.1007/978-3-030-15096-9_2. [102] P. Degond, J.-G. Liu, S. Merino-Aceituno and T. Tardiveau, Continuum dynamics of the intention field under weakly cohesive social interaction, Math. Models Methods Appl. Sci., 27 (2017), 159-182.  doi: 10.1142/S021820251740005X. [103] P. Degond and S. Merino-Aceituno, Nematic alignment of self-propelled particles: From particle to macroscopic dynamics, Math. Models Methods Appl. Sci., 30 (2020), 1935-1986.  doi: 10.1142/S021820252040014X. [104] P. Degond and S. Motsch, Continuum limit of self-driven particles with orientation interaction, Math. Models Methods Appl. Sci., 18 (2008), 1193-1215.  doi: 10.1142/S0218202508003005. [105] P. Del Moral and J. Tugaut, On the stability and the uniform propagation of chaos properties of Ensemble Kalman-Bucy filters, Ann. Appl. Probab., 28 (2018), 790-850.  doi: 10.1214/17-AAP1317. [106] P. Del Moral, Measure-valued processes and interacting particle systems. Application to nonlinear filtering problems, Ann. Appl. Probab., 8 (1998), 438-495.  doi: 10.1214/aoap/1028903535. [107] P. Del Moral, Feynman-Kac Formulae, Genealogical and Interacting Particle Systems with Applications, Probability and Its Applications, Springer-Verlag New York, 2004. doi: 10.1007/978-1-4684-9393-1. [108] P. Del Moral, Mean Field Simulation for Monte Carlo Integration, Monographs on Statistics and Applied Probability, 126, CRC Press, Taylor & Francis Group, 2013. [109] P. Del Moral, A. Kurtzmann and J. Tugaut, On the stability and the uniform propagation of chaos of a class of extended ensemble Kalman-Bucy filters, SIAM J. Control Optim., 55 (2017), 119-155.  doi: 10.1137/16M1087497. [110] P. Del Moral and J. Tugaut, Uniform propagation of chaos and creation of chaos for a class of nonlinear diffusions, Stoch. Anal. Appl., 37 (2019), 909-935.  doi: 10.1080/07362994.2019.1622426. [111] S. Delattre, N. Fournier and M. Hoffmann, Hawkes processes on large networks, Ann. Appl. Probab., 26 (2016), 216-261.  doi: 10.1214/14-AAP1089. [112] M. G. Delgadino, R. S. Gvalani and G. A. Pavliotis, On the diffusive-mean field limit for weakly interacting diffusions exhibiting phase transitions, Arch. Ration. Mech. Anal., 241 (2021), 91-148.  doi: 10.1007/s00205-021-01648-1. [113] 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, preprint, arXiv: 2112.06304 [114] L. Desvillettes, C. Graham and S. Méléard, Probabilistic interpretation and numerical approximation of a Kac equation without cut-off, Stochastic Process. Appl., 84 (1999), 115-135.  doi: 10.1016/S0304-4149(99)00056-3. [115] A. Diez, Propagation of chaos and moderate interaction for a piecewise deterministic system of geometrically enriched particles, Electron. J. Probab., 25 (2020), 1-38.  doi: 10.1214/20-ejp496. [116] G. Dimarco and S. Motsch, Self-alignment driven by jump processes: Macroscopic limit and numerical investigation, Math. Models Methods Appl. Sci., 26 (2016), 1385-1410.  doi: 10.1142/S0218202516500330. [117] G. Dimarco and L. Pareschi, Numerical methods for kinetic equations, Acta Numerica, 23 (2014), 369-520.  doi: 10.1017/S0962492914000063. [118] Z. Ding and Q. Li, Ensemble Kalman inversion: Mean-field limit and convergence analysis, Stat. Comput., 31 (2021), 9.  doi: 10.1007/s11222-020-09976-0. [119] Z. Ding and Q. Li, Ensemble Kalman sampler: Mean-field limit and convergence analysis, SIAM J. Math. Anal., 53 (2021), 1546-1578.  doi: 10.1137/20M1339507. [120] R. L. Dobrushin, Vlasov equations, Funct. Anal. Appl., 13 (1979), 115-123.  doi: 10.1007/BF01077243. [121] P. Donnelly and T. G. Kurtz, A countable representation of the Fleming-Viot measure-valued diffusion, Ann. Probab., 24 (1996), 698-742.  doi: 10.1214/aop/1039639359. [122] M. R. D'Orsogna, Y. L. Chuang, A. L. Bertozzi and L. S. Chayes, Self-propelled particles with soft-core interactions: Patterns, stability, and collapse, Phys. Rev. Lett., 96 (2006), 104302.  doi: 10.1016/j.physd.2007.05.007. [123] A. Doucet, N. Freitas and N. Gordon (eds.), Sequential Monte Carlo Methods in Practice, Information Science and Statistics, Springer-Verlag New York, 2001. doi: 10.1007/978-1-4757-3437-9_1. [124] S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, New York, 2003. [125] M. Duerinckx, Mean-field limits for some Riesz interaction gradient flows, SIAM J. Math. Anal., 48 (2016), 2269-2300.  doi: 10.1137/15M1042620. [126] B. Düring, N. Georgiou, S. Merino-Aceituno and E. Scalas, Continuum and thermodynamic limits for a simple random-exchange model, preprint, arXiv: 2003.00930 doi: 10.1016/j.spa.2022.03.015. [127] B. Düring, M. Torregrossa and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling the Elo rating system with learning effects, J. Nonlinear Sci., 29 (2019), 1095-1128.  doi: 10.1007/s00332-018-9512-8. [128] A. Durmus, A. Eberle, A. Guillin and K. Schuh, Sticky nonlinear SDEs and convergence of McKean-Vlasov equations without confinement, preprint, arXiv: 2201.07652 [129] A. Durmus, A. Eberle, A. Guillin and R. Zimmer, An elementary approach to uniform in time propagation of chaos, Proc. Amer. Math. Soc., 148 (2020), 5387-5398.  doi: 10.1090/proc/14612. [130] A. Eberle, Reflection couplings and contraction rates for diffusions, Probab. Theory Related Fields, 166 (2016), 851-886.  doi: 10.1007/s00440-015-0673-1. [131] A. Eberle, A. Guillin and R. Zimmer, Quantitative Harris-type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 371 (2019), 7135-7173.  doi: 10.1090/tran/7576. [132] A. Eberle and R. Zimmer, Sticky couplings of multidimensional diffusions with different drifts, Ann. Inst. H. Poincaré Probab. Statist., 55 (2019), 2370-2394. [133] X. Erny, Well-posedness and propagation of chaos for McKean-Vlasov equations with jumps and locally Lipschitz coefficients, preprint, arXiv: 2102.06472 doi: 10.1016/j.spa.2022.04.012. [134] A. Etheridge, An Introduction to Superprocesses, University Lecture Series, 20, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/ulect/020. [135] A. Etheridge, Some Mathematical Models from Population Genetics. École d'Été de Probabilités de Saint-Flour XXXIX-2009, Lecture Notes in Mathematics, 2012, Springer Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-16632-7. [136] S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley series in probability and mathematical statistics, Wiley, New York, 1986. doi: 10.1002/9780470316658. [137] R. Ferland, X. Fernique and G. Giroux, Compactness of the fluctuations associated with some generalized nonlinear Boltzmann equations, Canad. J. Math., 44 (1992), 1192-1205.  doi: 10.4153/CJM-1992-071-1. [138] B. Fernandez and S. Méléard, A Hilbertian approach for fluctuations on the McKean-Vlasov model, Stochastic Process. Appl., 71 (1997), 33-53.  doi: 10.1016/S0304-4149(97)00067-7. [139] R. C. Fetecau, H. Huang and W. Sun, Propagation of chaos for the Keller-Segel equation over bounded domains, J. Differential Equations, 266 (2019), 2142-2174.  doi: 10.1016/j.jde.2018.08.024. [140] A. Figalli, M.-J. Kang and J. Morales, Global well-posedness of the spatially homogeneous Kolmogorov-Vicsek model as a gradient flow, Arch. Ration. Mech. Anal., 227 (2018), 869-896.  doi: 10.1007/s00205-017-1176-2. [141] W. H. Fleming and M. Viot, Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J., 28 (1979), 817-843.  doi: 10.1512/iumj.1979.28.28058. [142] J. Fontbona, H. Guérin and S. Méléard, Measurability of optimal transportation and convergence rate for Landau type interacting particle systems, Probab. Theory Related Fields, 143 (2009), 329-351.  doi: 10.1007/s00440-007-0128-4. [143] M. Fornasier, H. Huang, L. Pareschi and P. Sünnen, Consensus-based optimization on hypersurfaces: Well-posedness and mean-field limit, Math. Models Methods Appl. Sci., 30 (2020), 2725-2751.  doi: 10.1142/S0218202520500530. [144] N. Fournier, Particle approximation of some Landau equations, Kinet. Relat. Models, 2 (2009), 451-464.  doi: 10.3934/krm.2009.2.451. [145] N. Fournier and A. Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probab. Theory Related Fields, 162 (2015), 707–738, Publisher: Springer. [146] N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. Éc. Norm. Supér., 50 (2017), 157-199. [147] N. Fournier and M. Hauray, Propagation of chaos for the Landau equation with moderately soft potentials, Ann. Probab., 44 (2016), 3581-3660.  doi: 10.1214/15-AOP1056. [148] N. Fournier, M. Hauray and S. Mischler, Propagation of chaos for the 2D viscous vortex model, J. Eur. Math. Soc., 16 (2014), 1423-1466.  doi: 10.4171/JEMS/465. [149] N. Fournier and B. Jourdain, Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes, Ann. Appl. Probab., 27 (2017), 2807-2861.  doi: 10.1214/16-AAP1267. [150] N. Fournier and E. Löcherbach, On a toy model of interacting neurons, Ann. Inst. Henri Poincar{é Probab. Stat.}, 52 (2016), 1844–1876, https://projecteuclid.org/journals/annales-de-linstitut-henri-poincare-probabilites-et-statistiques/volume-52/issue-4/On-a-toy-model-of-interacting-neurons/10.1214/15-AIHP701.full. doi: 10.1214/15-AIHP701. [151] N. Fournier and S. Méléard, A Markov process associated with a Boltzmann equation without cutoff and for non-Maxwell molecules, J. Stat. Phys., 104 (2001), 359-385.  doi: 10.1023/A:1010322130480. [152] N. Fournier and S. Méléard, Monte-Carlo approximations and fluctuations for 2D Boltzmann equations without cutoff, Markov Process. Related Fields, 7 (2001), 159-191. [153] N. Fournier and S. Méléard, Monte-Carlo approximations for 2d homogeneous Boltzmann equations without cutoff and for non Maxwell molecules, Monte Carlo Methods Appl., 7 (2001), 177-192.  doi: 10.1515/mcma.2001.7.1-2.177. [154] N. Fournier and S. Méléard, A stochastic particle numerical method for 3D Boltzmann equations without cutoff, Math. Comp., 71 (2002), 583-604.  doi: 10.1090/S0025-5718-01-01339-4. [155] N. Fournier and S. Mischler, Rate of convergence of the Nanbu particle system for hard potentials and Maxwell molecules, Ann. Probab., 44 (2016), 589-627.  doi: 10.1214/14-AOP983. [156] M. Friesen and O. Kutoviy, Stochastic Cucker-Smale flocking dynamics of jump-type, Kinet. Relat. Models, 13 (2020), 211-247.  doi: 10.3934/krm.2020008. [157] T. Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 67 (1984), 331-348. [158] I. Gallagher, L. Saint-Raymond and B. Texier, From Newton to Boltzmann: Hard Spheres and Short-Range Potentials, Zurich Lectures in Advanced Mathematics, 18, European Mathematical Society, 2014. [159] A. Garbuno-Inigo, F. Hoffmann, W. Li and A. M. Stuart, Interacting Langevin diffusions: Gradient structure and ensemble Kalman sampler, SIAM J. Appl. Dyn. Syst., 19 (2020), 412-441.  doi: 10.1137/19M1251655. [160] J. Gärtner, On the McKean-Vlasov limit for interacting diffusions, Math. Nachr., 137 (1988), 197-248.  doi: 10.1002/mana.19881370116. [161] G. Giacomin, K. Pakdaman and X. Pellegrin, Global attractor and asymptotic dynamics in the Kuramoto model for coupled noisy phase oscillators, Nonlinearity, 25 (2012), 1247-1273.  doi: 10.1088/0951-7715/25/5/1247. [162] C. R. Givens and R. M. Shortt, A class of Wasserstein metrics for probability distributions, Michigan Math. J., 31 (1984), 231–240, https://projecteuclid.org/journals/michigan-mathematical-journal/volume-31/issue-2/A-class-of-Wasserstein-metrics-for-probability-distributions/10.1307/mmj/1029003026.full. doi: 10.1307/mmj/1029003026. [163] D. Godinho and C. Quiñinao, Propagation of chaos for a subcritical Keller-Segel model, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 965-992. [164] F. Golse, De Newton à Boltzmann et Einstein: Validation des modèles cinétiques et de diffusion, d'après T. Bodineau, I. Gallagher, L. Saint-Raymond, B. Texier, in Séminaire Bourbaki, Volume 2013/2014, Exposés 1074-1088, vol. 367–368, Astérisque, Société Mathématique de France, 2015,285–326. [165] F. Golse, On the dynamics of large particle systems in the mean field limit, Lecture notes, arXiv: 1301.5494. doi: 10.1007/978-3-319-26883-5_1. [166] H. Grad, Principles of the kinetic theory of gases, in Thermodynamics of Gases (ed. S. Flügge), Encyclopedia of Physics, 12, Springer-Verlag Berlin Heidelberg, 1958,205–294. [167] H. Grad, Asymptotic theory of Boltzmann equation, Phys. Fluids, 6 (1963), 147-181.  doi: 10.1063/1.1706716. [168] C. Graham, McKean-Vlasov Itō-Skorohod equations, and nonlinear diffusions with discrete jump sets, Stochastic Process. Appl., 40 (1992), 69-82.  doi: 10.1016/0304-4149(92)90138-G. [169] C. Graham and S. Méléard, Stochastic particle approximations for generalized Boltzmann models and convergence estimates, Ann. Probab., 25 (1997), 115-132.  doi: 10.1214/aop/1024404281. [170] S. Grassi and L. Pareschi, From particle swarm optimization to consensus based optimization: Stochastic modeling and mean-field limit, Math. Models Methods Appl. Sci., 31 (2021), 1625-1657.  doi: 10.1142/S0218202521500342. [171] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), 325-348.  doi: 10.1016/0021-9991(87)90140-9. [172] F. A. Grünbaum, Propagation of chaos for the Boltzmann equation, Arch. Ration. Mech. Anal., 42 (1971), 323-345.  doi: 10.1007/BF00250440. [173] H. Guérin and S. Méléard, Convergence from Boltzmann to Landau processes with soft potential and particle approximations, J. Stat. Phys., 111 (2003), 931–966, http://link.springer.com/10.1023/A:1022858517569. doi: 10.1023/A:1022858517569. [174] A. Guillin, P. L. Bris and P. Monmarché, Uniform in time propagation of chaos for the 2D vortex model and other singular stochastic systems, preprint, arXiv: 2108.08675 [175] A. Guillin, P. Le Bris and P. Monmarché, Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases, preprint, arXiv: 2105.09070 [176] A. Guillin, W. Liu, L. Wu and C. Zhang, Uniform Poincaré and logarithmic Sobolev inequalities for mean field particles systems, preprint, arXiv: 1909.07051 [177] A. Guillin and P. Monmarché, Uniform long-time and propagation of chaos estimates for mean field kinetic particles in non-convex landscapes, J. Stat. Phys., 185 (2020), 1-20.  doi: 10.1007/s10955-021-02839-6. [178] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Commun. Math. Sci., 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [179] J. Haskovec, Flocking dynamics and mean-field limit in the Cucker-Smale-type model with topological interactions, Phys. D, 261 (2013), 42-51.  doi: 10.1016/j.physd.2013.06.006. [180] J. Haškovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system, Comm. Partial Differential Equations, 36 (2011), 940-960.  doi: 10.1080/03605302.2010.538783. [181] W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika, 57 (1970), 97-109.  doi: 10.1093/biomet/57.1.97. [182] M. Hauray and P.-E. Jabin, N-particles approximation of the Vlasov equations with singular potential, Arch. Ration. Mech. Anal., 183 (2007), 489-524.  doi: 10.1007/s00205-006-0021-9. [183] M. Hauray and P.-E. Jabin, Particles approximations of Vlasov equations with singular forces: Propagation of chaos, Ann. Sci. Éc. Norm. Supér., 48 (2015), 891-940. [184] M. Hauray and S. Mischler, On Kac's chaos and related problems, J. Funct. Anal., 266 (2014), 6055-6157.  doi: 10.1016/j.jfa.2014.02.030. [185] S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes, Stochastic Process. Appl., 120 (2010), 1215-1246.  doi: 10.1016/j.spa.2010.03.009. [186] D. Heydecker, Pathwise convergence of the hard spheres Kac process, Ann. Appl. Probab., 29 (2019), 3062-3127.  doi: 10.1214/19-AAP1475. [187] D. Heydecker, Kac's process with hard potentials and a moderate angular singularity, arXiv: 2008.12943. doi: 10.1007/s00205-022-01767-3. [188] M. Hitsuda and I. Mitoma, Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions, J. Multivariate Anal., 19 (1986), 311-328.  doi: 10.1016/0047-259X(86)90035-7. [189] T. Holding, Propagation of chaos for Hölder continuous interaction kernels via Glivenko-Cantelli, arXiv: 1608.02877. [190] H. Huang, J.-G. Liu and P. Pickl, On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system, J. Stat. Phys., 181 (2020), 1915-1965.  doi: 10.1007/s10955-020-02648-3. [191] P.-E. Jabin, A review of the mean field limits for Vlasov equations, Kinet. Relat. Models, 7 (2014), 661-711.  doi: 10.3934/krm.2014.7.661. [192] P.-E. Jabin and S. Junca, A continuous model for ratings, SIAM J. Appl. Math., 75 (2015), 420-442.  doi: 10.1137/140969324. [193] P.-E. Jabin and Z. Wang, Mean field limit and propagation of chaos for Vlasov systems with bounded forces, J. Funct. Anal., 271 (2016), 3588-3627.  doi: 10.1016/j.jfa.2016.09.014. [194] P.-E. Jabin and Z. Wang, Quantitative estimates of propagation of chaos for stochastic systems with $W^{ -1, \infty}$ kernels, Invent. Math., 214 (2018), 523-591.  doi: 10.1007/s00222-018-0808-y. [195] J.-F. Jabir, Rate of propagation of chaos for diffusive stochastic particle systems via Girsanov transformation, preprint, arXiv: 1907.09096 [196] J.-F. Jabir, D. Talay and M. Tomašević, Mean-field limit of a particle approximation of the one-dimensional parabolic-parabolic Keller-Segel model without smoothing, Electron. Commun. Probab., 23 (2018), 1-14.  doi: 10.1214/18-ECP183. [197] J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Second edition edition, Grundlehren der mathematischen Wissenschaften, 288, Springer Berlin Heidelberg, 2003. doi: 10.1007/978-3-662-05265-5. [198] A. Jakubowski, On the Skorokhod topology, Ann. Inst. Henri Poincaré Probab. Stat., 22 (1986), 263-285. [199] A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab., 18 (1986), 20-65.  doi: 10.2307/1427238. [200] B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM Probab. Stat., 1 (1997), 339-355.  doi: 10.1051/ps:1997113. [201] B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of Metropolis Hastings algorithms: The longtime behavior, Bernoulli, 20 (2014), 1930-1978.  doi: 10.3150/13-BEJ546. [202] B. Jourdain, T. Lelièvre and B. Miasojedow, Optimal scaling for the transient phase of the random walk metropolis algorithm: The mean-field limit, Ann. Appl. Probab., 25 (2015), 2263-2300.  doi: 10.1214/14-AAP1048. [203] B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, Ann. Inst. Henri Poincaré Probab. Stat., 34 (1998), 727–766, Publisher: Gauthier-Villars. doi: 10.1016/S0246-0203(99)80002-8. [204] M. Kac, Foundations of kinetic theory, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 3, University of California Press Berkeley and Los Angeles, California, 1956,171–197. [205] M. Kac, Some Probabilistic Aspects of the Boltzmann Equation, in The Boltzmann Equation. Acta Physica Austriaca (Supplementum X Proceedings of the International Symposium "100 Years Boltzmann Equation" in Vienna 4th-8th September 1972) (eds. E. G. D. Cohen and W. Thirring), Springer Vienna, 1973,379–400, http://link.springer.com/10.1007/978-3-7091-8336-6_17. [206] M.-J. Kang and J. Morales, Dynamics of a spatially homogeneous Vicsek model for oriented particles on the plane, preprint, arXiv: 1608.00185 [207] N. Kantas, A. Doucet, S. S. Singh and J. M. Maciejowski, An overview of sequential Monte Carlo methods for parameter estimation in general state-space models, IFAC Proceedings Volumes, 42 (2009), 774-785.  doi: 10.3182/20090706-3-FR-2004.00129. [208] J. Kennedy and R. Eberhart, Particle swarm optimization, in Proceedings of ICNN'95 - International Conference on Neural Networks, 4, IEEE, Perth, WA, Australia, 1995, 1942–1948. doi: 10.1109/ICNN.1995.488968. [209] F. G. King, BBGKY Hierarchy for Positive Potentials, Ph.D Thesis, University of California,, Berkeley, 1975. [210] D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), 1-11.  doi: 10.1214/18-ECP150. [211] D. Lacker, Hierarchies, entropy, and quantitative propagation of chaos for mean field diffusions, preprint, arXiv: 2105.02983 [212] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasi-linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23, American Mathematical Soc, 1968. [213] O. E. Lanford, Time evolution of large classical systems, in Dynamical Systems Theory and Application, Battelle Seattle 1974 Rencontres (ed. J. Moser), Springer-Verlag Berlin Heidelberg, 1975. [214] D. Lazarovici and P. Pickl, A mean field limit for the Vlasov-poisson system, Arch. Ration. Mech. Anal., 225 (2017), 1201-1231.  doi: 10.1007/s00205-017-1125-0. [215] C. Léonard, Une loi des grands nombres pour des systèmes de diffusions avec interaction et à coefficients non bornés, Ann. Inst. Henri Poincaré Probab. Stat., 22 (1986), 237-262. [216] J.-G. Liu and R. Yang, Propagation of chaos for large Brownian particle system with Coulomb interaction, Res. Math. Sci., 3 (2016), 40.  doi: 10.1186/s40687-016-0086-5. [217] J.-G. Liu and R. Yang, Propagation of chaos for the Keller-Segel equation with a logarithmic cut-off, Methods Appl. Anal., 26 (2019), 319-348.  doi: 10.4310/MAA.2019.v26.n4.a2. [218] W. Liu, L. Wu and C. Zhang, Long-time behaviors of mean-field interacting particle systems related to McKean-Vlasov equations, Commun. Math. Phys., 387 (2021), 179-214.  doi: 10.1007/s00220-021-04198-5. [219] E. Luçon, Large population asymptotics for interacting diffusions in a quenched random environment, in From Particle Systems to Partial Differential Equations Ⅱ (eds. P. Gonçalves and A. J. Soares), Springer Proceedings in Mathematics & Statistics, 129, Springer, Cham, 2015,231–251. doi: 10.1007/978-3-319-93821-9. [220] F. Malrieu, Logarithmic Sobolev inequalities for some nonlinear PDE's, Stochastic Process. Appl., 95 (2001), 109-132.  doi: 10.1016/S0304-4149(01)00095-3. [221] F. Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab., 13 (2003), 540-560.  doi: 10.1214/aoap/1050689593. [222] C. Marchioro and M. Pulvirenti, Hydrodynamics in two dimensions and vortex theory, Comm. Math. Phys., 84 (1982), 483-503.  doi: 10.1007/BF01209630. [223] D. Matthes and G. Toscani, On steady distributions of kinetic models of conservative economies, J. Stat. Phys., 130 (2008), 1087-1117.  doi: 10.1007/s10955-007-9462-2. [224] H. P. McKean, An exponential formula for solving Boltzmann's equation for a Maxwellian gas, Journal of Combinatorial Theory, 2 (1967), 358-382.  doi: 10.1016/S0021-9800(67)80035-8. [225] H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in Stochastic Differential Equations, Lecture Series in Differential Equations, Session 7, Catholic Univ., Air Force Office of Scientific Research, Office of Aerospace Research, Arlington, Va., 1967, 41–57. [226] H. P. McKean, Propagation of chaos for a class of non-linear parabolic equations, in Lecture Series in Differential Equations, Volume 2 (ed. A. K. Aziz), Van Nostrand Mathematical Studies, 19, Van Nostrand Reinhold Company, 1969,177–194. [227] H. P. McKean, Fluctuations in the kinetic theory of gases, Commun. Pure Appl. Math., 28 (1975), 435-455.  doi: 10.1002/cpa.3160280402. [228] S. Mei, A. Montanari and P.-M. Nguyen, A mean field view of the landscape of two-layer neural networks, Proc. Natl. Acad. Sci. USA, 115 (2018), E7665–E7671, http://www.pnas.org/lookup/doi/10.1073/pnas.1806579115. doi: 10.1073/pnas.1806579115. [229] S. Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, in Probabilistic Models for Nonlinear Partial Differential Equations (eds. D. Talay and L. Tubaro), Lecture Notes in Mathematics, 1627, Springer-Verlag Berlin Heidelberg, 1996. doi: 10.1007/BFb0093177. [230] S. Méléard, Convergence of the fluctuations for interaction diffusions with jumps associated with Boltzmann equations, Stochastics, 63 (1998), 195-225.  doi: 10.1080/17442509808834148. [231] S. Méléard, Stochastic approximations of the solution of a full Boltzmann equation with small initial data, ESAIM Probab. Stat., 2 (1998), 23-40.  doi: 10.1051/ps:1998102. [232] S. Méléard, A trajectorial proof of the vortex method for the two-dimensional Navier-Stokes equation, Ann. Appl. Probab., 10 (2000), 1197-1211.  doi: 10.1214/aoap/1019487613. [233] S. Méléard, Monte-Carlo approximations for 2d Navier-Stokes equations with measure initial data, Probab. Theory Related Fields, 121 (2001), 367-388.  doi: 10.1007/s004400100154. [234] S. Méléard and S. Roelly-Coppoletta, A propagation of chaos result for a system of particles with moderate interaction, Stochastic Processes and their Applications, 26 (1987), 317-332.  doi: 10.1016/0304-4149(87)90184-0. [235] S. Méléard and S. Roelly-Coppoletta, Systèmes de particules et mesures-martingales: Un théorème de propagation du chaos, Séminaire de probabilités (Strasbourg), 22 (1988), 438-448. [236] S. Merino-Aceituno, Isotropic wave turbulence with simplified kernels: Existence, uniqueness, and mean-field limit for a class of instantaneous coagulation-fragmentation processes, J. Math. Phys., 57 (2016), 121501.  doi: 10.1063/1.4968814. [237] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 1087-1092.  doi: 10.2172/4390578. [238] N. Metropolis and S. Ulam, The Monte Carlo method, J. Amer. Statist. Assoc., 44 (1949), 335-341.  doi: 10.1080/01621459.1949.10483310. [239] S. Mischler and C. Mouhot, Kac's program in kinetic theory, Invent. Math., 193 (2013), 1–147, Publisher: Springer. [240] S. Mischler, C. Mouhot and B. Wennberg, A new approach to quantitative propagation of chaos for drift, diffusion and jump processes, Probab. Theory Related Fields, 161 (2015), 1-59.  doi: 10.1007/s00440-013-0542-8. [241] Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, Theor. Probability and Math. Statist., 103 (2020), 59-101.  doi: 10.1090/tpms/1135. [242] P. Monmarché, Long-time behaviour and propagation of chaos for mean field kinetic particles, Stochastic Process. Appl., 127 (2017), 1721-1737.  doi: 10.1016/j.spa.2016.10.003. [243] C. Mouhot and L. Pareschi, Fast algorithms for computing the Boltzmann collision operator, Math. Comp., 75 (2006), 1833-1852.  doi: 10.1090/S0025-5718-06-01874-6. [244] A. Muntean and F. Toschi (eds.), Collective Dynamics from Bacteria to Crowds: An Excursion Through Modeling, Analysis and Simulation, CISM International Centre for Mechanical Sciences, 553, Springer, Vienna, 2014. doi: 10.1007/978-3-7091-1785-9. [245] H. Murata, Propagation of chaos for Boltzmann-like equation of non-cutoff type in the plane, Hiroshima Math. J., 7 (1977), 479–515, https://projecteuclid.org/euclid.hmj/1206135751. [246] G. Naldi, L. Pareschi and G. Toscani (eds.), Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, 2010. doi: 10.1007/978-0-8176-4946-3. [247] K. Nanbu, Direct simulation scheme derived from the Boltzmann equation. I. Monocomponent gases, Journal of the Physical Society of Japan, 49 (1980), 2042-2049.  doi: 10.1143/JPSJ.49.2042. [248] K. Oelschläger, A Martingale approach to the law of large numbers for weakly interacting stochastic processes, Ann. Probab., 12 (1984), 458–479, https://projecteuclid.org/euclid.aop/1176993301. [249] K. Oelschläger, A law of large numbers for moderately interacting diffusion processes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 279–322, Publisher: Springer Nature America, Inc. doi: 10.1007/BF02450284. [250] K. Oelschläger, A fluctuation theorem for moderately interacting diffusion processes, Probab. Theory Related Fields, 74 (1987), 591-616.  doi: 10.1007/BF00363518. [251] K. Oelschläger, On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Theory Related Fields, 82 (1989), 565-586.  doi: 10.1007/BF00341284. [252] H. Osada, Propagation of chaos for the two dimensional Navier-Stokes equation, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 8-11. [253] H. Osada and S. Kotani, Propagation of chaos for the Burgers equation, J. Math. Soc. Japan, 37 (1985), 275-294.  doi: 10.2969/jmsj/03720275. [254] K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), 1-26.  doi: 10.1186/2190-8567-4-14. [255] L. Pareschi and T. Rey, On the stability of equilibrium preserving spectral methods for the homogeneous Boltzmann equation, Appl. Math. Lett., 120 (2021), 107187, arXiv: 2011.05811. doi: 10.1016/j.aml.2021.107187. [256] L. Pareschi and G. Russo, An introduction to Monte Carlo method for the Boltzmann equation, ESAIM Proc., 10 (2001), 35-75.  doi: 10.1051/proc:2001004. [257] G. A. Pavliotis, A. M. Stuart and U. Vaes, Derivative-free Bayesian inversion using multiscale dynamics, SIAM J. Appl. Dyn. Syst., 21 (2022), 284-326.  doi: 10.1137/21M1397416. [258] L. Pédèches, Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete Contin. Dyn. Syst., 38 (2018), 2731-2762.  doi: 10.3934/dcds.2018115. [259] R. Pinnau, C. Totzeck, O. Tse and S. Martin, A consensus-based model for global optimization and its mean-field limit, Math. Models Methods Appl. Sci., 27 (2017), 183-204.  doi: 10.1142/S0218202517400061. [260] M. Pulvirenti, Kinetic limits for stochastic particle systems, in Probabilistic Models for Nonlinear Partial Differential Equations (eds. D. Talay and L. Tubaro), Lecture Notes in Mathematics, 1627, Springer-Verlag Berlin Heidelberg, 1996. doi: 10.1007/BFb0093178. [261] M. Pulvirenti and S. Simonella, The Boltzmann-Grad limit of a hard sphere system: analysis of the correlation error, Invent. Math., 207 (2017), 1135-1237.  doi: 10.1007/s00222-016-0682-4. [262] S. Reich and S. Weissmann, Fokker-Planck particle systems for Bayesian inference: Computational approaches, SIAM/ASA J. Uncertain. Quantif., 9 (2021), 446-482.  doi: 10.1137/19M1303162. [263] C. P. Robert and  G. Casella,  Monte Carlo Statistical Methods, Springer Texts in Statistics, Springer, New York, 2004. [264] S. Roelly-Coppoletta, A criterion of convergence of measure-valued processes: Application to measure branching processes, Stochastics, 17 (1986), 43-65.  doi: 10.1080/17442508608833382. [265] G. M. Rotskoff and E. Vanden-Eijnden, Trainability and accuracy of neural networks: An interacting particle system approach, preprint, arXiv: 1805.00915 [266] M. Rousset, A N-uniform quantitative Tanaka's theorem for the conservative Kac's N-particle system with Maxwell molecules, preprint, arXiv: 1407.1965 [267] C. Saffirio, Derivation of the Boltzmann equation: Hard spheres, short-range potentials and beyond, in From Particle Systems to Partial Differential Equations Ⅲ (eds. P. Gon¸calves and A. J. Soares), Springer Proceedings in Mathematics & Statistics, 162, Springer International Publishing, 2016,301–321, Series Title: Springer Proceedings in Mathematics & Statistics. doi: 10.1007/978-3-319-32144-8_15. [268] S. Salem, A gradient flow approach to propagation of chaos, Discrete Contin. Dyn. Syst., 40 (2020), 5729-5754.  doi: 10.3934/dcds.2020243. [269] S. Serfaty, Systems of points with Coulomb interactions, in Proceedings of the International Congress of Mathematicians (ICM 2018), World Scientific, Rio de Janeiro, Brazil, 2019,935–977, https://www.worldscientific.com/doi/abs/10.1142/9789813272880_0033. [270] S. Serfaty, Mean field limit for Coulomb-type flows, Duke Math. J., 169 (2020), 2887–2935, https://projecteuclid.org/journals/duke-mathematical-journal/volume-169/issue-15/Mean-field-limit-for-Coulomb-type-flows/10.1215/00127094-2020-0019.full. doi: 10.1215/00127094-2020-0019. [271] T. Shiga and H. Tanaka, Central limit theorem for a system of Markovian particles with mean field interactions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 69 (1985), 439-459. [272] J. Sirignano and K. Spiliopoulos, Mean field analysis of neural networks: A law of large numbers, SIAM J. Appl. Math., 80 (2020), 725–752, https://epubs.siam.org/doi/10.1137/18M1192184. [273] A.-S. Sznitman, Équations de type de Boltzmann, spatialement homogènes, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 66 (1984), 559-592. [274] A.-S. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, J. Funct. Anal., 56 (1984), 311-336.  doi: 10.1016/0022-1236(84)90080-6. [275] A.-S. Sznitman, A propagation of chaos result for Burgers' equation, Probab. Theory Related Fields, 71 (1986), 581-613.  doi: 10.1007/BF00699042. [276] A.-S. Sznitman, Topics in propagation of chaos, in Éc. Été Probab. St.-Flour XIX-1989, Springer, 1991,165–251. doi: 10.1007/BFb0085169. [277] D. Talay and M. Tomašević, A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The one-dimensional case, Bernoulli, 26 (2020), 1323–1353, https://doi.org/10.3150/19-BEJ1158. doi: 10.3150/19-BEJ1158. [278] H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 46 (1978), 67-105. [279] H. Tanaka, Fluctuation theory for Kac's one-dimensional model of Maxwellian molecules, Sankhyė: The Indian Journal of Statistics, Series A, 44 (1982), 23-46. [280] H. Tanaka, Limit theorems for certain diffusion processes with interaction, in Stochastic Analysis, Proceedings of the Taniguchi International Symposium on Stochastic Analysis (ed. K. Itō), 1982,469–488. doi: 10.1016/S0924-6509(08)70405-7. [281] H. Tanaka, Some probabilistic problems in the spatially homogeneous Boltzmann equation, in Theory and Application of Random Fields, Proceedings of the IFIP-WG 7/1 Working Conference, Bangalore 1982 (ed. G. Kallianpur), Lecture Notes in Control and Information Sciences, Springer-Verlag Berlin Heidelberg, 1983,258–267. doi: 10.1007/BFb0044698. [282] H. Tanaka and M. Hitsuda, Central limit theorem for a simple diffusion model of interacting particles, Hiroshima Math. J., 11 (1981), 415–423, https://projecteuclid.org/euclid.hmj/1206134109. [283] M. Tomašević, Propagation of chaos for stochastic particle systems with singular mean-field interaction of ${L}^{ p }-{L}^{q}$ type, hal preprint: hal-03086253. [284] M. Tomašević, A new McKean-Vlasov stochastic interpretation of the parabolic-parabolic Keller-Segel model: The two-dimensional case, Ann. Appl. Probab., 31 (2021), 432-459.  doi: 10.1214/20-aap1594. [285] G. Toscani, The grazing collisions asymptotics of the non cut-off Kac equation, ESAIM Math. Model. Numer. Anal., 32 (1998), 763-772.  doi: 10.1051/m2an/1998320607631. [286] G. Toscani, Kinetic models of opinion formation, Commun. Math. Sci., 4 (2006), 481-496.  doi: 10.4310/CMS.2006.v4.n3.a1. [287] G. Toscani, A. Tosin and M. Zanella, Kinetic modelling of multiple interactions in socio-economic systems, Netw. Heterog. Media, 15 (2020), 519-542.  doi: 10.3934/nhm.2020029. [288] C. Totzeck, Trends in Consensus-based optimization, preprint, arXiv: 2104.01383 doi: 10.1515/dmvm-2021-0028. [289] C. Totzeck, R. Pinnau, S. Blauth and S. Schotthöfer, A numerical comparison of consensus-based global optimization to other particle-based global optimization schemes, PAMM. Proc. Appl. Math. Mech., 18 (2018), 1–2, https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.201800291. [290] J. Touboul, Propagation of chaos in neural fields, Ann. Appl. Probab., 24 (2014), 1298-1328.  doi: 10.1214/13-AAP950. [291] J. Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab., 41 (2013), 1427-1460.  doi: 10.1214/12-AOP749. [292] J. Tugaut, Phase transitions of McKean-Vlasov processes in double-wells landscape, Stochastics, 86 (2014), 257-284.  doi: 10.1080/17442508.2013.775287. [293] K. Uchiyama, A fluctuation problem associated with the Boltzmann equation for a gas of molecules with a cutoff potential, Japanese Journal of Mathematics. New Series, 9 (1983), 27-53.  doi: 10.4099/math1924.9.27. [294] K. Uchiyama, Fluctuations of Markovian systems in Kac's caricature of a Maxwellian gas, J. Math. Soc. Japan, 35 (1983), 477-499.  doi: 10.2969/jmsj/03530477. [295] K. Uchiyama, Derivation of the Boltzmann equation from particle dynamics, Hiroshima Math. J., 18 (1988), 245–297, https://projecteuclid.org/euclid.hmj/1206129724. [296] K. Uchiyama, Fluctuations in a Markovian system of pairwise interacting particles, Probab. Theory Related Fields, 79 (1988), 289-302.  doi: 10.1007/BF00320923. [297] A. Y. Veretennikov, On strong solutions and explicit formulas for solutions of stochastic integral equations, Math. USSR Sb., 39 (1981), 387–403, http://stacks.iop.org/0025-5734/39/i=3/a=A05?key=crossref.91586277ed28ea996b4d447d5ac7e93a. [298] A. Y. Veretennikov, On ergodic measures for McKean-Vlasov stochastic equations, in Monte Carlo and Quasi-Monte Carlo Methods 2004 (eds. H. Niederreiter and D. Talay), 2006,471–486. doi: 10.1007/3-540-31186-6_29. [299] T. Vicsek, A. Czirók, E. Ben-Jacob, I. Cohen and O. Shochet, Novel type of phase transition in a system of self-driven particles, Phys. Rev. Lett., 75 (1995), 1226-1229.  doi: 10.1103/PhysRevLett.75.1226. [300] T. Vicsek and A. Zafeiris, Collective motion, Phys. Rep., 517 (2012), 71-140.  doi: 10.1016/j.physrep.2012.03.004. [301] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics (eds. S. Friedlander and D. Serre), 1, Elsevier Science, 2002, 71–74. doi: 10.1016/S1874-5792(02)80004-0. [302] C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), 1-141.  doi: 10.1090/S0065-9266-09-00567-5. [303] C. Villani, Optimal Transport, Old and New, Grundlehren der mathematischen Wissenschaften, 338, Springer-Verlag Berlin Heidelberg, 2009. doi: 10.1007/978-3-540-71050-9. [304] W. Wagner, A convergence proof for Bird's direct simulation Monte Carlo method for the Boltzmann equation, J. Stat. Phys., 66 (1992), 1011-1044.  doi: 10.1007/BF01055714. [305] W. Wagner, A functional law of large numbers for Boltzmann type stochastic particle systems, Stoch. Anal. Appl., 14 (1996), 591-636.  doi: 10.1080/07362999608809458. [306] F.-Y. Wang, Distribution dependent SDEs for Landau type equations, Stochastic Process. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006. [307] S. Watanabe, On stochastic differential equations for multi-dimensional diffusion processes with boundary conditions, Kyoto J. Math., 11 (1971), 169-180.  doi: 10.1215/kjm/1250523692. [308] L. Xu, Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials, Ann. Appl. Probab., 28 (2018), 1136-1189.  doi: 10.1214/17-AAP1327. [309] A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. USSR Sb., 22 (1974), 129–149, http://stacks.iop.org/0025-5734/22/i=1/a=A08?key=crossref.2ad44b5b66ab0196526fac25037d275d.

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