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

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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

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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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