June  2014, 7(2): 341-360. doi: 10.3934/krm.2014.7.341

Hypocoercive relaxation to equilibrium for some kinetic models

1. 

Laboratoire de Statistiques et Probabilités, Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex, France

Received  June 2013 Revised  February 2014 Published  March 2014

This paper deals with the study of some particular kinetic models, where the randomness acts only on the velocity variable level. Usually, the Markovian generator cannot satisfy any Poincaré's inequality. Hence, no Gronwall's lemma can easily lead to the exponential decay of $F_t$ (the $L^2$ norm of a test function along the semi-group). Nevertheless for the kinetic Fokker-Planck dynamics and for a piecewise deterministic evolution we show that $F_t$ satisfies a third order differential inequality which gives an explicit rate of convergence to equilibrium.
Citation: Pierre Monmarché. Hypocoercive relaxation to equilibrium for some kinetic models. Kinetic and Related Models, 2014, 7 (2) : 341-360. doi: 10.3934/krm.2014.7.341
References:
[1]

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les Ingalités de Sobolev Logarithmiques, Société mathématique de France, Paris, 2000.

[2]

A. Arnold, E. Carlen and Q. JU, Large-time behavior of non-symmetric Fokker-Planck type equations, Comm. Stoch. Anal., 2 (2008), 153-175.

[3]

A. Arnold, J. A. Carrillo and C. Manzini, Refined long-time asymptotics for some polymeric fluid flow models, Commun. Math. Sci., 8 (2010), 763-782. doi: 10.4310/CMS.2010.v8.n3.a8.

[4]

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

[5]

J.-B. Bardet, A. Christen, A. Guillin, F. Malrieu and P.-A. Zitt, Total variation estimates for the TCP process, Elec. Journ. Probab, 18 (2013), 1-21. doi: 10.1214/EJP.v18-1720.

[6]

M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems, Elec. Com. Probab., 17 (2012), 1-14.

[7]

F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44 (2010), 867-884. doi: 10.1051/m2an/2010045.

[8]

S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations, Comm. Partial Differential Equations, 37 (2012), 1357-1390. doi: 10.1080/03605302.2011.648039.

[9]

D. Chafai, F. Malrieu and K. Paroux, On the long time behavior of the TCP window size process, Stochastic Process. Appl., 120 (2010), 1518-1534. doi: 10.1016/j.spa.2010.03.019.

[10]

L. Desvillettes, Hypocoercivity: The example of linear transport, Contemp. Math., 409 (2006), 33-53. doi: 10.1090/conm/409/07705.

[11]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[12]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[13]

P. Diaconis, S. Holmes and R. M. Neal, Analysis of a nonreversible Markov chain sampler, Ann. Appl. Probab., 10 (2000), 685-1064. doi: 10.1214/aoap/1019487508.

[14]

P. Diaconis and L. Miclo, On the spectral analysis of second-order Markov chains, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 573-621. doi: 10.5802/afst.1383.

[15]

J. Dieudonné, Calcul Infinitésimal, Hermann, Paris, 1968.

[16]

J. Dolbeault, C. Mouhot and C. Shmeiser, Hypocoercivity and stability for a class of kinetic models with mass conservation and a confining potential, Transactions of the American Mathematical Society, to appear.

[17]

J. Dolbeault, C. Mouhot and C. Shmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516. doi: 10.1016/j.crma.2009.02.025.

[18]

R. Douc, G. Fort and A. Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923. doi: 10.1016/j.spa.2008.03.007.

[19]

J.-P. Eckmann and M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys., 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9.

[20]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216.

[21]

K. Fellner, L. Neumann and C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatsh. Math., 141 (2004), 289-299. doi: 10.1007/s00605-002-0058-2.

[22]

F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in $N\log_2N$, SIAM J. Sci. Comput., 28 (2006), 1029-1053. doi: 10.1137/050625175.

[23]

J. Fontbona, H. Guérin and F. Malrieu, Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process, Adv. App. Probab., 44 (2012), 907-1200. doi: 10.1239/aap/1354716586.

[24]

S. Gadat and L. Miclo, Spectral decompositions and $\mathbbL^2$-operator norms of toy hypocoercive semi-groups, Name of the Journal, 6 (2013), 317-372. doi: 10.3934/krm.2013.6.317.

[25]

S. Gadat and F. Panloup, Long time behaviour and stationary regime of memory gradient diffusions, Annales de l'Institut Henri Poincaré, to appear.

[26]

M. Grothaus and P. Stilgenbauer, Hypocoercivity for Kolmogorov backward evolution equations and applications, preprint, arXiv:1207.5447

[27]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing properties and hypocoercivity of a degenerate diffusion arising in technical textile industry, arXiv:1203.4502

[28]

A. Guillin and F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differential Equations, 253 (2012), 20-40. doi: 10.1016/j.jde.2012.03.014.

[29]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005. x+209 pp.

[30]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.

[31]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218. doi: 10.1007/s00205-003-0276-3.

[32]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[33]

J. Inglis, M. Neklyudov and B. Zegarliński, Ergodicity for infinite particle systems with locally conserved quantities, Infin. Dimens. Anal. Quantum Probab. Relat. Top, 15 (2012), 1250005, 28 pp. doi: 10.1142/S0219025712500051.

[34]

M. Ledoux, L'algèbre de Lie des gradients itérés d'un générateur markovien-développements de moyennes et entropies, Ann. Sci. École Norm. Sup. (4), 28 (1995), 435-460.

[35]

P.-A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1-29.

[36]

L. Miclo and P. Monmarché, Étude spectrale minutieuse de processus moins indécis que les autres, Lecture Notes in Mathematics, 2078 (2013), 459-481. doi: 10.1007/978-3-319-00321-4_18.

[37]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[38]

R.-M. Neal, Improving asymptotic variance of MCMC estimators: Non-reversible chains are better, arXiv:math/0407281.

[39]

A. Scemama, T. Lelièvre, G. Stoltz and M. Caffarel, An efficient sampling algorithm for Variational Monte Carlo, Journal of Chemical Physics, 125 (2006). doi: 10.1063/1.2354490.

[40]

M.-B. Tran, Convergence to equilibrium of some kinetic models, ArXiv e-prints, 255 (2013), 405-440. doi: 10.1016/j.jde.2013.04.013.

[41]

C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians, 3 (2006), 473-498.

[42]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., Vol. 202, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/S0065-9266-09-00567-5.

show all references

References:
[1]

C. Ané, S. Blachère, D. Chafaï, P. Fougères, I. Gentil, F. Malrieu, C. Roberto and G. Scheffer, Sur les Ingalités de Sobolev Logarithmiques, Société mathématique de France, Paris, 2000.

[2]

A. Arnold, E. Carlen and Q. JU, Large-time behavior of non-symmetric Fokker-Planck type equations, Comm. Stoch. Anal., 2 (2008), 153-175.

[3]

A. Arnold, J. A. Carrillo and C. Manzini, Refined long-time asymptotics for some polymeric fluid flow models, Commun. Math. Sci., 8 (2010), 763-782. doi: 10.4310/CMS.2010.v8.n3.a8.

[4]

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

[5]

J.-B. Bardet, A. Christen, A. Guillin, F. Malrieu and P.-A. Zitt, Total variation estimates for the TCP process, Elec. Journ. Probab, 18 (2013), 1-21. doi: 10.1214/EJP.v18-1720.

[6]

M. Benaïm, S. Le Borgne, F. Malrieu and P.-A. Zitt, Quantitative ergodicity for some switched dynamical systems, Elec. Com. Probab., 17 (2012), 1-14.

[7]

F. Bolley, A. Guillin and F. Malrieu, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, M2AN Math. Model. Numer. Anal., 44 (2010), 867-884. doi: 10.1051/m2an/2010045.

[8]

S. Calogero, Exponential convergence to equilibrium for kinetic Fokker-Planck equations, Comm. Partial Differential Equations, 37 (2012), 1357-1390. doi: 10.1080/03605302.2011.648039.

[9]

D. Chafai, F. Malrieu and K. Paroux, On the long time behavior of the TCP window size process, Stochastic Process. Appl., 120 (2010), 1518-1534. doi: 10.1016/j.spa.2010.03.019.

[10]

L. Desvillettes, Hypocoercivity: The example of linear transport, Contemp. Math., 409 (2006), 33-53. doi: 10.1090/conm/409/07705.

[11]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316. doi: 10.1007/s00222-004-0389-9.

[12]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation, Comm. Pure Appl. Math., 54 (2001), 1-42. doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[13]

P. Diaconis, S. Holmes and R. M. Neal, Analysis of a nonreversible Markov chain sampler, Ann. Appl. Probab., 10 (2000), 685-1064. doi: 10.1214/aoap/1019487508.

[14]

P. Diaconis and L. Miclo, On the spectral analysis of second-order Markov chains, Ann. Fac. Sci. Toulouse Math. (6), 22 (2013), 573-621. doi: 10.5802/afst.1383.

[15]

J. Dieudonné, Calcul Infinitésimal, Hermann, Paris, 1968.

[16]

J. Dolbeault, C. Mouhot and C. Shmeiser, Hypocoercivity and stability for a class of kinetic models with mass conservation and a confining potential, Transactions of the American Mathematical Society, to appear.

[17]

J. Dolbeault, C. Mouhot and C. Shmeiser, Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516. doi: 10.1016/j.crma.2009.02.025.

[18]

R. Douc, G. Fort and A. Guillin, Subgeometric rates of convergence of $f$-ergodic strong Markov processes, Stochastic Process. Appl., 119 (2009), 897-923. doi: 10.1016/j.spa.2008.03.007.

[19]

J.-P. Eckmann and M. Hairer, Spectral properties of hypoelliptic operators, Comm. Math. Phys., 235 (2003), 233-253. doi: 10.1007/s00220-003-0805-9.

[20]

J.-P. Eckmann and M. Hairer, Non-equilibrium statistical mechanics of strongly anharmonic chains of oscillators, Comm. Math. Phys., 212 (2000), 105-164. doi: 10.1007/s002200000216.

[21]

K. Fellner, L. Neumann and C. Schmeiser, Convergence to global equilibrium for spatially inhomogeneous kinetic models of non-micro-reversible processes, Monatsh. Math., 141 (2004), 289-299. doi: 10.1007/s00605-002-0058-2.

[22]

F. Filbet, C. Mouhot and L. Pareschi, Solving the Boltzmann equation in $N\log_2N$, SIAM J. Sci. Comput., 28 (2006), 1029-1053. doi: 10.1137/050625175.

[23]

J. Fontbona, H. Guérin and F. Malrieu, Quantitative estimates for the long time behavior of an ergodic variant of the telegraph process, Adv. App. Probab., 44 (2012), 907-1200. doi: 10.1239/aap/1354716586.

[24]

S. Gadat and L. Miclo, Spectral decompositions and $\mathbbL^2$-operator norms of toy hypocoercive semi-groups, Name of the Journal, 6 (2013), 317-372. doi: 10.3934/krm.2013.6.317.

[25]

S. Gadat and F. Panloup, Long time behaviour and stationary regime of memory gradient diffusions, Annales de l'Institut Henri Poincaré, to appear.

[26]

M. Grothaus and P. Stilgenbauer, Hypocoercivity for Kolmogorov backward evolution equations and applications, preprint, arXiv:1207.5447

[27]

M. Grothaus, A. Klar, J. Maringer and P. Stilgenbauer, Geometry, mixing properties and hypocoercivity of a degenerate diffusion arising in technical textile industry, arXiv:1203.4502

[28]

A. Guillin and F.-Y. Wang, Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality, J. Differential Equations, 253 (2012), 20-40. doi: 10.1016/j.jde.2012.03.014.

[29]

B. Helffer and F. Nier, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lecture Notes in Mathematics, 1862. Springer-Verlag, Berlin, 2005. x+209 pp.

[30]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.

[31]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171 (2004), 151-218. doi: 10.1007/s00205-003-0276-3.

[32]

L. Hörmander, Hypoelliptic second order differential equations, Acta Math., 119 (1967), 147-171. doi: 10.1007/BF02392081.

[33]

J. Inglis, M. Neklyudov and B. Zegarliński, Ergodicity for infinite particle systems with locally conserved quantities, Infin. Dimens. Anal. Quantum Probab. Relat. Top, 15 (2012), 1250005, 28 pp. doi: 10.1142/S0219025712500051.

[34]

M. Ledoux, L'algèbre de Lie des gradients itérés d'un générateur markovien-développements de moyennes et entropies, Ann. Sci. École Norm. Sup. (4), 28 (1995), 435-460.

[35]

P.-A. Markowich and C. Villani, On the trend to equilibrium for the Fokker-Planck equation: an interplay between physics and functional analysis, Mat. Contemp., 19 (2000), 1-29.

[36]

L. Miclo and P. Monmarché, Étude spectrale minutieuse de processus moins indécis que les autres, Lecture Notes in Mathematics, 2078 (2013), 459-481. doi: 10.1007/978-3-319-00321-4_18.

[37]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998. doi: 10.1088/0951-7715/19/4/011.

[38]

R.-M. Neal, Improving asymptotic variance of MCMC estimators: Non-reversible chains are better, arXiv:math/0407281.

[39]

A. Scemama, T. Lelièvre, G. Stoltz and M. Caffarel, An efficient sampling algorithm for Variational Monte Carlo, Journal of Chemical Physics, 125 (2006). doi: 10.1063/1.2354490.

[40]

M.-B. Tran, Convergence to equilibrium of some kinetic models, ArXiv e-prints, 255 (2013), 405-440. doi: 10.1016/j.jde.2013.04.013.

[41]

C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians, 3 (2006), 473-498.

[42]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., Vol. 202, American Mathematical Society, Providence, RI, 2009. doi: 10.1090/S0065-9266-09-00567-5.

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