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Open AccessOpen Access JUSTC Mathematics 10 October 2024

Mean field analysis of interacting network model with jumps

  • School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China

Cite this:
CSTR: 32290.14.JUSTC-2023-0163
https://doi.org/10.52396/JUSTC-2023-0163
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  • Author Bio:

    Zeqian Li is a postgraduate student under the tutelage of Prof. Lijun Bo at the School of Mathematical Sciences, University of Science and Technology of China. His research mainly focuses on mean-field analysis

  • Corresponding author: E-mail: lzq7890@mail.ustc.edu.cn
  • Received Date: 30 November 2023
  • Accepted Date: 25 February 2024
    • Available Online: 10 October 2024
    Abstract

  • Abstract

    This paper considers an $ n $-particle jump-diffusion system with mean filed interaction, where the coefficients are locally Lipschitz continuous. We address the convergence as $ n\to\infty $ of the empirical measure of the jump-diffusions to the solution of a deterministic McKean–Vlasov equation. The strong well-posedness of the associated McKean–Vlasov equation and a corresponding propagation of chaos result are proven. In particular, we also provide precise estimates of the convergence speed with respect to a Wasserstein-like metric.

    Graphical abstract

    An estimate on the speed of the convergence of the empirical measure to the limit measure is given with respect to a Wasserstein-like distance.

    Abstract

    This paper considers an $ n $-particle jump-diffusion system with mean filed interaction, where the coefficients are locally Lipschitz continuous. We address the convergence as $ n\to\infty $ of the empirical measure of the jump-diffusions to the solution of a deterministic McKean–Vlasov equation. The strong well-posedness of the associated McKean–Vlasov equation and a corresponding propagation of chaos result are proven. In particular, we also provide precise estimates of the convergence speed with respect to a Wasserstein-like metric.

    • Public Summary

    • Constructing the n-particle system with mean filed interaction, whose dynamic follows a couple of SDEs with jumps.
    • Studying the existence of a solution to the corresponding McKean-Vlasov limit equation.
    • Establishing the propagation of chaos under a proper metric, giving an estimate of the speed of the convergence and proving the uniqueness of the solution to the McKean-Vlasov limit equation.

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    • References(31)
  • [1]
    Dehnen W, Read J I. N-body simulations of gravitational dynamics. The European Physical Journal Plus, 2011, 126: 55. doi: 10.1140/epjp/i2011-11055-3
    [2]
    Sirignano J, Spiliopoulos K. Mean field analysis of neural networks: A central limit theorem. Stochastic Processes and Their Applications, 2020, 130 (3): 1820–1852. doi: 10.1016/j.spa.2019.06.003
    [3]
    Bolley F, Canizo J A, Carrillo J A. Stochastic mean-field limit: non-Lipschitz forces and swarming. Mathematical Models and Methods in Applied Sciences, 2011, 21 (11): 2179–2210. doi: 10.1142/s0218202511005702
    [4]
    Bender M, Heenen P-H, Reinhard P-G. Selfconsistent mean-field models for nuclear structure. Reviews of Modern Physics, 2003, 75 (1): 121–180. doi: 10.1103/revmodphys.75.121
    [5]
    Touboul J. Propagation of chaos in neural fields. Annals of Applied Probability, 2014, 24 (3): 1298–1327. doi: 10.1214/13-aap950
    [6]
    Delarue F, Inglis J, Rubenthaler S, et al. Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Processes and Their Applications, 2015, 125 (6): 2451–2492. doi: 10.1016/j.spa.2015.01.007
    [7]
    Bo L, Capponi A. Systemic risk in interbanking networks. SIAM Journal on Financial Mathematics, 2015, 6 (1): 386–424. doi: 10.1137/130937664
    [8]
    Liu W, Song Y, Zhai J, et al. Large and moderate deviation principles for McKean–Vlasov SDEs with jumps. Potential Analysis, 2023, 59 (3): 1141–1190. doi: 10.1007/s11118-022-10005-0
    [9]
    Guillin A, Liu W, Wu L, et al. Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems. The Annals of Applied Probability, 2022, 32 (3): 1590–1614. doi: 10.1214/21-aap1707
    [10]
    Kac M. Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics. Berkeley, Los Angeles, USA: University of California Press, 1956: 171–197.
    [11]
    Sznitman A S. Topics in propagation of chaos. In: Ecole d'Eté de Probabilités de Saint-Flour XIX—1989. Berlin, Heidelberg: Springer, 1991: 165–251.
    [12]
    McKean H P. A class of Markov processes associated with nonlinear parabolic equations. Proceedings of the National Academy of Sciences, 1966, 56 (6): 1907–1911. doi: 10.1073/pnas.56.6.1907
    [13]
    Mishura Y, Veretennikov A. Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations. Theory of Probability and Mathematical Statistics, 2020, 103: 59–101. doi: 10.1090/tpms/1135
    [14]
    Lacker D. On a strong form of propagation of chaos for McKean–Vlasov equations. Electronic Communications in Probability, 2018, 23: 1–11. doi: 10.1214/18-ecp150
    [15]
    Liu W, Wu L, Zhang C. Long-time behaviors of mean-field interacting particle systems related to McKean–Vlasov equations. Communications in Mathematical Physics, 2021, 387 (1): 179–214. doi: 10.1007/s00220-021-04198-5
    [16]
    Andreis L, Dai Pra P, Fischer M. McKean–Vlasov limit for interacting systems with simultaneous jumps. Stochastic Analysis and Applications, 2018, 36 (6): 960–995. doi: 10.1080/07362994.2018.1486202
    [17]
    Erny X. Well-posedness and propagation of chaos for McKean–Vlasov equations with jumps and locally Lipschitz coefficients. Stochastic Processes and Their Applications, 2022, 150: 192–214. doi: 10.1016/j.spa.2022.04.012
    [18]
    Mehri S, Scheutzow M, Stannat W, et al. Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions. The Annals of Applied Probability, 2020, 30 (1): 175–207. doi: 10.1214/19-aap1499
    [19]
    Baladron J, Fasoli D, Faugeras O, et al. Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. The Journal of Mathematical Neuroscience, 2012, 2: 10. doi: 10.1186/2190-8567-2-10
    [20]
    Bossy M, Faugeras O, Talay D. Clarification and complement to “Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons”. The Journal of Mathematical Neuroscience, 2015, 5: 19. doi: 10.1186/s13408-015-0031-8
    [21]
    Xu L. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. The Annals of Applied Probability, 2018, 28 (2): 1136–1189. doi: 10.1214/17-aap1327
    [22]
    Bo L, Liao H. Probabilistic analysis of replicator–mutator equations. Advances in Applied Probability, 2022, 54 (1): 167–201. doi: 10.1017/apr.2021.22
    [23]
    Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam: Elsevier, 2014.
    [24]
    Luçon E, Stannat W. Mean field limit for disordered diffusions with singular interactions. The Annals of Applied Probability, 2014, 24 (5): 1946–1993. doi: 10.1214/13-aap968
    [25]
    Bard Ermentrout G, Terman D H. Mathematical Foundations of Neuroscience. New York: Springer, 2010.
    [26]
    Hocquet A, Vogler A. Optimal control of mean field equations with monotone coefficients and applications in neuroscience. Applied Mathematics & Optimization, 2021, 84 (2): 1925–1968. doi: 10.1007/s00245-021-09816-1
    [27]
    Garroni M G, Menaldi J L. Green Functions for Second Order Parabolic Integro-Differential Problems. New York: Chapman & Hall/CRC, 1992.
    [28]
    Mandelkern M. Metrization of the one-point compactification. Proceedings of the American Mathematical Society, 1989, 107 (4): 1111–1115. doi: 10.1090/s0002-9939-1989-0991703-4
    [29]
    Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 2015, 162: 707–738. doi: 10.1007/s00440-014-0583-7
    [30]
    Hardy G H, Littlewood J E, Pólya G. Inequalities. Cambridge, UK: Cambridge University Press, 1952.
    [31]
    Protter P E. Stochastic Integration and Differential Equations. Berlin: Springer, 2013.
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    [1]
    Dehnen W, Read J I. N-body simulations of gravitational dynamics. The European Physical Journal Plus, 2011, 126: 55. doi: 10.1140/epjp/i2011-11055-3
    [2]
    Sirignano J, Spiliopoulos K. Mean field analysis of neural networks: A central limit theorem. Stochastic Processes and Their Applications, 2020, 130 (3): 1820–1852. doi: 10.1016/j.spa.2019.06.003
    [3]
    Bolley F, Canizo J A, Carrillo J A. Stochastic mean-field limit: non-Lipschitz forces and swarming. Mathematical Models and Methods in Applied Sciences, 2011, 21 (11): 2179–2210. doi: 10.1142/s0218202511005702
    [4]
    Bender M, Heenen P-H, Reinhard P-G. Selfconsistent mean-field models for nuclear structure. Reviews of Modern Physics, 2003, 75 (1): 121–180. doi: 10.1103/revmodphys.75.121
    [5]
    Touboul J. Propagation of chaos in neural fields. Annals of Applied Probability, 2014, 24 (3): 1298–1327. doi: 10.1214/13-aap950
    [6]
    Delarue F, Inglis J, Rubenthaler S, et al. Particle systems with a singular mean-field self-excitation. Application to neuronal networks. Stochastic Processes and Their Applications, 2015, 125 (6): 2451–2492. doi: 10.1016/j.spa.2015.01.007
    [7]
    Bo L, Capponi A. Systemic risk in interbanking networks. SIAM Journal on Financial Mathematics, 2015, 6 (1): 386–424. doi: 10.1137/130937664
    [8]
    Liu W, Song Y, Zhai J, et al. Large and moderate deviation principles for McKean–Vlasov SDEs with jumps. Potential Analysis, 2023, 59 (3): 1141–1190. doi: 10.1007/s11118-022-10005-0
    [9]
    Guillin A, Liu W, Wu L, et al. Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems. The Annals of Applied Probability, 2022, 32 (3): 1590–1614. doi: 10.1214/21-aap1707
    [10]
    Kac M. Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 3: Contributions to Astronomy and Physics. Berkeley, Los Angeles, USA: University of California Press, 1956: 171–197.
    [11]
    Sznitman A S. Topics in propagation of chaos. In: Ecole d'Eté de Probabilités de Saint-Flour XIX—1989. Berlin, Heidelberg: Springer, 1991: 165–251.
    [12]
    McKean H P. A class of Markov processes associated with nonlinear parabolic equations. Proceedings of the National Academy of Sciences, 1966, 56 (6): 1907–1911. doi: 10.1073/pnas.56.6.1907
    [13]
    Mishura Y, Veretennikov A. Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations. Theory of Probability and Mathematical Statistics, 2020, 103: 59–101. doi: 10.1090/tpms/1135
    [14]
    Lacker D. On a strong form of propagation of chaos for McKean–Vlasov equations. Electronic Communications in Probability, 2018, 23: 1–11. doi: 10.1214/18-ecp150
    [15]
    Liu W, Wu L, Zhang C. Long-time behaviors of mean-field interacting particle systems related to McKean–Vlasov equations. Communications in Mathematical Physics, 2021, 387 (1): 179–214. doi: 10.1007/s00220-021-04198-5
    [16]
    Andreis L, Dai Pra P, Fischer M. McKean–Vlasov limit for interacting systems with simultaneous jumps. Stochastic Analysis and Applications, 2018, 36 (6): 960–995. doi: 10.1080/07362994.2018.1486202
    [17]
    Erny X. Well-posedness and propagation of chaos for McKean–Vlasov equations with jumps and locally Lipschitz coefficients. Stochastic Processes and Their Applications, 2022, 150: 192–214. doi: 10.1016/j.spa.2022.04.012
    [18]
    Mehri S, Scheutzow M, Stannat W, et al. Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions. The Annals of Applied Probability, 2020, 30 (1): 175–207. doi: 10.1214/19-aap1499
    [19]
    Baladron J, Fasoli D, Faugeras O, et al. Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons. The Journal of Mathematical Neuroscience, 2012, 2: 10. doi: 10.1186/2190-8567-2-10
    [20]
    Bossy M, Faugeras O, Talay D. Clarification and complement to “Mean-field description and propagation of chaos in networks of Hodgkin–Huxley and FitzHugh–Nagumo neurons”. The Journal of Mathematical Neuroscience, 2015, 5: 19. doi: 10.1186/s13408-015-0031-8
    [21]
    Xu L. Uniqueness and propagation of chaos for the Boltzmann equation with moderately soft potentials. The Annals of Applied Probability, 2018, 28 (2): 1136–1189. doi: 10.1214/17-aap1327
    [22]
    Bo L, Liao H. Probabilistic analysis of replicator–mutator equations. Advances in Applied Probability, 2022, 54 (1): 167–201. doi: 10.1017/apr.2021.22
    [23]
    Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam: Elsevier, 2014.
    [24]
    Luçon E, Stannat W. Mean field limit for disordered diffusions with singular interactions. The Annals of Applied Probability, 2014, 24 (5): 1946–1993. doi: 10.1214/13-aap968
    [25]
    Bard Ermentrout G, Terman D H. Mathematical Foundations of Neuroscience. New York: Springer, 2010.
    [26]
    Hocquet A, Vogler A. Optimal control of mean field equations with monotone coefficients and applications in neuroscience. Applied Mathematics & Optimization, 2021, 84 (2): 1925–1968. doi: 10.1007/s00245-021-09816-1
    [27]
    Garroni M G, Menaldi J L. Green Functions for Second Order Parabolic Integro-Differential Problems. New York: Chapman & Hall/CRC, 1992.
    [28]
    Mandelkern M. Metrization of the one-point compactification. Proceedings of the American Mathematical Society, 1989, 107 (4): 1111–1115. doi: 10.1090/s0002-9939-1989-0991703-4
    [29]
    Fournier N, Guillin A. On the rate of convergence in Wasserstein distance of the empirical measure. Probability Theory and Related Fields, 2015, 162: 707–738. doi: 10.1007/s00440-014-0583-7
    [30]
    Hardy G H, Littlewood J E, Pólya G. Inequalities. Cambridge, UK: Cambridge University Press, 1952.
    [31]
    Protter P E. Stochastic Integration and Differential Equations. Berlin: Springer, 2013.
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