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Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case
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 |
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 [
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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Dumas and F. Golse,
Homogenization of transport equations, SIAM J. Appl. Math., 60 (2000), 1447-1470.
doi: 10.1137/S0036139997332087. |
[14] |
R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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.
|
[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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[13] |
L. Dumas and F. Golse,
Homogenization of transport equations, SIAM J. Appl. Math., 60 (2000), 1447-1470.
doi: 10.1137/S0036139997332087. |
[14] |
R. Z. Has'minskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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