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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

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