American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stationary solutions to the boundary value problem for the relativistic BGK model in a slab
  • KRM Home
  • This Issue
  • Next Article
    Evaluating high order discontinuous Galerkin discretization of the Boltzmann collision integral in $ \mathcal{O}(N^2) $ operations using the discrete fourier transform
August  2019, 12(4): 727-748. doi: 10.3934/krm.2019028

Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case

Patrick Cattiaux 1, , Elissar Nasreddine 2, and  Marjolaine Puel 3,,

1. 

Institut de Mathématiques de Toulouse. Université de Toulouse. CNRS UMR 5219, 118 route de Narbonne, F-31062 Toulouse cedex 09, France
2. 

Mathematics Department, Lebanese University, Faculty of Sciences (I), Hadath, Lebanon
3. 

Laboratoire Dieudonné. Université de Nice Sophia Antipolis, Parc Valrose, F-06108 Nice cedex 2, France

* Corresponding author: Marjolaine Puel

Received  January 2018 Revised  October 2018 Published  May 2019

This paper is devoted to the diffusion and anomalous diffusion limit of the Fokker-Planck equation of plasma physics, in which the equilibrium function decays towards zero at infinity like a negative power function. We use probabilistic methods to recover and extend the results obtained in [22]. We prove in particular, in the critical case where the classical diffusion coefficient is no more defined, that the small mean free path limit gives rise to a diffusion equation, with an anomalous time scaling and with a variance breaking.

Keywords: Diffusion limit, Fokker-Planck, heavy tails, general Cauchy distributions, central limit theorem.
Mathematics Subject Classification: Primary: 35K10, 35K65, 60F05, 60J55.
Citation: Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028
References:
[1]

D. Bakry, P. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré, J. Func. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

[2]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262.  doi: 10.1142/S0218202511005738.  Google Scholar

[3]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900.  doi: 10.3934/krm.2011.4.873.  Google Scholar

[4]

P. Cattiaux, D. Chafaï and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA, Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382.   Google Scholar

[5]

P. Cattiaux, D. Chafaï and S. Motsch, Asymptotic analysis and diffusion limit of the persistent Turning Walker model, Asymptot. Anal., 67 (2010), 17-31.   Google Scholar

[6]

P. Cattiaux, N. Gozlan, A. Guillin and C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry, Electronic J. Prob., 15 (2010), 346-385.  doi: 10.1214/EJP.v15-754.  Google Scholar

[7]

P. Cattiaux and A. Guillin, Deviation bounds for additive functionals of Markov processes, ESAIM Probability and Statistics, 12 (2008), 12–29. doi: 10.1051/ps:2007032.  Google Scholar

[8]

P. Cattiaux, A. Guillin and C. Roberto, Poincaré inequality and the ${\mathbb L}^p$ convergence of semi-groups, Elect. Comm. in Probab., 15 (2010), 270-280.  doi: 10.1214/ECP.v15-1559.  Google Scholar

[9]

P. Cattiaux, A. Guillin and P. A. Zitt, Poincaré inequalities and hitting times, Ann. Inst. Henri Poincaré. Prob. Stat., 49 (2013), 95-118.  doi: 10.1214/11-AIHP447.  Google Scholar

[10]

P. Cattiaux and M. Manou-Abi, Limit theorems for some functionals with heavy tails of a discrete time Markov chain, ESAIM P.S., 18 (2014), 468-482.  doi: 10.1051/ps/2013043.  Google Scholar

[11]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous diffusion in plasma physic, Applied Math Letters, 25 (2012), 2344-2348.  doi: 10.1016/j.aml.2012.06.029.  Google Scholar

[12]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[13]

L. Dumas and F. Golse, Homogenization of transport equations, SIAM J. Appl. Math., 60 (2000), 1447-1470.  doi: 10.1137/S0036139997332087.  Google Scholar

[14]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980. Google Scholar

[15]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. of Applied Probab., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.  Google Scholar

[16]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[17]

T. M. Liggett, ${\mathbb L}^2$ rate of convergence for attractive reversible nearest particle systems: the critical case, Ann. Probab., 19 (1991), 935-959.  doi: 10.1214/aop/1176990330.  Google Scholar

[18]

E. Löcherbach and D. Loukianova, Polynomial deviation bounds for recurrent Harris processes having general state space, ESAIM P.S., 17 (2013), 195-218.  doi: 10.1051/ps/2011156.  Google Scholar

[19]

E. Löcherbach, D. Loukianova and O. Loukianov, Polynomial bounds in the ergodic theorem for one dimensional diffusions and integrability of hitting times, Ann. Inst. Henri Poincaré. Prob. Stat., 47 (2011), 425-449.  doi: 10.1214/10-AIHP359.  Google Scholar

[20]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[21]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[22]

E. Nasreddine and M. Puel, Diffusion limit of Fokker-Planck equation with heavy tail equilibria, ESAIM: M2AN, 49 (2015), 1-17.  doi: 10.1051/m2an/2014020.  Google Scholar

[23]

M. Röckner and F. Y. Wang, Weak Poincaré inequalities and L2-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.  Google Scholar

show all references

References:
[1]

D. Bakry, P. Cattiaux and A. Guillin, Rate of convergence for ergodic continuous Markov processes : Lyapunov versus Poincaré, J. Func. Anal., 254 (2008), 727-759.  doi: 10.1016/j.jfa.2007.11.002.  Google Scholar

[2]

N. Ben Abdallah, A. Mellet and M. Puel, Anomalous diffusion limit for kinetic equations with degenerate collision frequency, Math. Models Methods Appl. Sci., 21 (2011), 2249-2262.  doi: 10.1142/S0218202511005738.  Google Scholar

[3]

N. Ben Abdallah, A. Mellet and M. Puel, Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach, Kinet. Relat. Models, 4 (2011), 873-900.  doi: 10.3934/krm.2011.4.873.  Google Scholar

[4]

P. Cattiaux, D. Chafaï and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA, Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382.   Google Scholar

[5]

P. Cattiaux, D. Chafaï and S. Motsch, Asymptotic analysis and diffusion limit of the persistent Turning Walker model, Asymptot. Anal., 67 (2010), 17-31.   Google Scholar

[6]

P. Cattiaux, N. Gozlan, A. Guillin and C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry, Electronic J. Prob., 15 (2010), 346-385.  doi: 10.1214/EJP.v15-754.  Google Scholar

[7]

P. Cattiaux and A. Guillin, Deviation bounds for additive functionals of Markov processes, ESAIM Probability and Statistics, 12 (2008), 12–29. doi: 10.1051/ps:2007032.  Google Scholar

[8]

P. Cattiaux, A. Guillin and C. Roberto, Poincaré inequality and the ${\mathbb L}^p$ convergence of semi-groups, Elect. Comm. in Probab., 15 (2010), 270-280.  doi: 10.1214/ECP.v15-1559.  Google Scholar

[9]

P. Cattiaux, A. Guillin and P. A. Zitt, Poincaré inequalities and hitting times, Ann. Inst. Henri Poincaré. Prob. Stat., 49 (2013), 95-118.  doi: 10.1214/11-AIHP447.  Google Scholar

[10]

P. Cattiaux and M. Manou-Abi, Limit theorems for some functionals with heavy tails of a discrete time Markov chain, ESAIM P.S., 18 (2014), 468-482.  doi: 10.1051/ps/2013043.  Google Scholar

[11]

L. Cesbron, A. Mellet and K. Trivisa, Anomalous diffusion in plasma physic, Applied Math Letters, 25 (2012), 2344-2348.  doi: 10.1016/j.aml.2012.06.029.  Google Scholar

[12]

P. Degond and S. Motsch, Large scale dynamics of the persistent turning walker model of fish behavior, J. Stat. Phys., 131 (2008), 989-1021.  doi: 10.1007/s10955-008-9529-8.  Google Scholar

[13]

L. Dumas and F. Golse, Homogenization of transport equations, SIAM J. Appl. Math., 60 (2000), 1447-1470.  doi: 10.1137/S0036139997332087.  Google Scholar

[14]

R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980. Google Scholar

[15]

M. Jara, T. Komorowski and S. Olla, Limit theorems for additive functionals of a Markov chain, Ann. of Applied Probab., 19 (2009), 2270-2300.  doi: 10.1214/09-AAP610.  Google Scholar

[16]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusion, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[17]

T. M. Liggett, ${\mathbb L}^2$ rate of convergence for attractive reversible nearest particle systems: the critical case, Ann. Probab., 19 (1991), 935-959.  doi: 10.1214/aop/1176990330.  Google Scholar

[18]

E. Löcherbach and D. Loukianova, Polynomial deviation bounds for recurrent Harris processes having general state space, ESAIM P.S., 17 (2013), 195-218.  doi: 10.1051/ps/2011156.  Google Scholar

[19]

E. Löcherbach, D. Loukianova and O. Loukianov, Polynomial bounds in the ergodic theorem for one dimensional diffusions and integrability of hitting times, Ann. Inst. Henri Poincaré. Prob. Stat., 47 (2011), 425-449.  doi: 10.1214/10-AIHP359.  Google Scholar

[20]

A. Mellet, Fractional diffusion limit for collisional kinetic equations: A moments method, Indiana Univ. Math. J., 59 (2010), 1333-1360.  doi: 10.1512/iumj.2010.59.4128.  Google Scholar

[21]

A. Mellet, S. Mischler and C. Mouhot, Fractional diffusion limit for collisional kinetic equations, Arch. Ration. Mech. Anal., 199 (2011), 493-525.  doi: 10.1007/s00205-010-0354-2.  Google Scholar

[22]

E. Nasreddine and M. Puel, Diffusion limit of Fokker-Planck equation with heavy tail equilibria, ESAIM: M2AN, 49 (2015), 1-17.  doi: 10.1051/m2an/2014020.  Google Scholar

[23]

M. Röckner and F. Y. Wang, Weak Poincaré inequalities and L2-convergence rates of Markov semigroups, J. Funct. Anal., 185 (2001), 564-603.  doi: 10.1006/jfan.2001.3776.  Google Scholar

[1]

Roberta Bosi. Classical limit for linear and nonlinear quantum Fokker-Planck systems. Communications on Pure & Applied Analysis, 2009, 8 (3) : 845-870. doi: 10.3934/cpaa.2009.8.845
[2]

Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028
[3]

Jean-Pierre Conze, Stéphane Le Borgne, Mikaël Roger. Central limit theorem for stationary products of toral automorphisms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1597-1626. doi: 10.3934/dcds.2012.32.1597
[4]

James Nolen. A central limit theorem for pulled fronts in a random medium. Networks & Heterogeneous Media, 2011, 6 (2) : 167-194. doi: 10.3934/nhm.2011.6.167
[5]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579
[6]

Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250
[7]

Oliver Díaz-Espinosa, Rafael de la Llave. Renormalization and central limit theorem for critical dynamical systems with weak external noise. Journal of Modern Dynamics, 2007, 1 (3) : 477-543. doi: 10.3934/jmd.2007.1.477
[8]

Shui-Nee Chow, Wuchen Li, Haomin Zhou. Entropy dissipation of Fokker-Planck equations on graphs. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 4929-4950. doi: 10.3934/dcds.2018215
[9]

Sylvain De Moor, Luis Miguel Rodrigues, Julien Vovelle. Invariant measures for a stochastic Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (2) : 357-395. doi: 10.3934/krm.2018017
[10]

Michael Herty, Christian Jörres, Albert N. Sandjo. Optimization of a model Fokker-Planck equation. Kinetic & Related Models, 2012, 5 (3) : 485-503. doi: 10.3934/krm.2012.5.485
[11]

Marco Torregrossa, Giuseppe Toscani. On a Fokker-Planck equation for wealth distribution. Kinetic & Related Models, 2018, 11 (2) : 337-355. doi: 10.3934/krm.2018016
[12]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401
[13]

Michael Herty, Lorenzo Pareschi. Fokker-Planck asymptotics for traffic flow models. Kinetic & Related Models, 2010, 3 (1) : 165-179. doi: 10.3934/krm.2010.3.165
[14]

Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009
[15]

Yves Derriennic. Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the "central limit theorem''. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 143-158. doi: 10.3934/dcds.2006.15.143
[16]

Michael Björklund, Alexander Gorodnik. Central limit theorems in the geometry of numbers. Electronic Research Announcements, 2017, 24: 110-122. doi: 10.3934/era.2017.24.012
[17]

Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
[18]

Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008
[19]

Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028
[20]

Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (86)
  • HTML views (98)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences