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Numerical study of an anisotropic Vlasov equation arising in plasma physics
Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK |
We study contraction for the kinetic Fokker-Planck operator on the torus. Solving the stochastic differential equation, we show contraction and therefore exponential convergence in the Monge-Kantorovich-Wasserstein $ \mathcal{W}_2$ distance. Finally, we investigate if such a coupling can be obtained by a co-adapted coupling, and show that then the bound must depend on the square root of the initial distance.
References:
| [1] |
F. Bolley, I. Gentil and A. Guillin,
Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.
doi: 10.1016/j.jfa.2012.07.007. |
| [2] |
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. |
| [3] |
K. Burdzy and W. S. Kendall,
Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10 (2000), 362-409.
doi: 10.1214/aoap/1019487348. |
| [4] |
M. Chen, Optimal Markovian couplings and applications, Acta Math. Sinica (N. S.), 10 (1994), 260–275; A Chinese summary appears in Acta Math. Sinica, 38 (1995), p575.
doi: 10.1007/BF02560717. |
| [5] |
S. Gadat and L. Miclo,
Spectral decompositions and $ \mathbb{L}^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6 (2013), 317-372.
doi: 10.3934/krm.2013.6.317. |
| [6] |
F. Hérau,
Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.
|
| [7] |
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. |
| [8] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [9] |
W. S. Kendall,
Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.
doi: 10.3150/14-BEJ596. |
| [10] |
K. Kuwada,
Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.
doi: 10.1214/EJP.v14-634. |
| [11] |
S. Mischler and C. Mouhot,
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.
doi: 10.1007/s00205-016-0972-4. |
| [12] |
P. Mörters and Y. Peres, Brownian Motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2010, URL http://books.google.co.uk/books?id=e-TbA-dSrzYC.
doi: 10.1017/CBO9780511750489. |
| [13] |
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. |
| [14] |
D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields, 8 (2002), 163–198, Inhomogeneous random systems (Cergy-Pontoise, 2001). |
| [15] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
| [16] |
C. Villani,
Optimal Transport: Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
| [17] |
R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC. |
show all references
References:
| [1] |
F. Bolley, I. Gentil and A. Guillin,
Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations, J. Funct. Anal., 263 (2012), 2430-2457.
doi: 10.1016/j.jfa.2012.07.007. |
| [2] |
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. |
| [3] |
K. Burdzy and W. S. Kendall,
Efficient Markovian couplings: Examples and counterexamples, Ann. Appl. Probab., 10 (2000), 362-409.
doi: 10.1214/aoap/1019487348. |
| [4] |
M. Chen, Optimal Markovian couplings and applications, Acta Math. Sinica (N. S.), 10 (1994), 260–275; A Chinese summary appears in Acta Math. Sinica, 38 (1995), p575.
doi: 10.1007/BF02560717. |
| [5] |
S. Gadat and L. Miclo,
Spectral decompositions and $ \mathbb{L}^2$-operator norms of toy hypocoercive semi-groups, Kinet. Relat. Models, 6 (2013), 317-372.
doi: 10.3934/krm.2013.6.317. |
| [6] |
F. Hérau,
Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptot. Anal., 46 (2006), 349-359.
|
| [7] |
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. |
| [8] |
R. Jordan, D. Kinderlehrer and F. Otto,
The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998), 1-17.
doi: 10.1137/S0036141096303359. |
| [9] |
W. S. Kendall,
Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.
doi: 10.3150/14-BEJ596. |
| [10] |
K. Kuwada,
Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.
doi: 10.1214/EJP.v14-634. |
| [11] |
S. Mischler and C. Mouhot,
Exponential stability of slowly decaying solutions to the kinetic Fokker-Planck equation, Arch. Ration. Mech. Anal., 221 (2016), 677-723.
doi: 10.1007/s00205-016-0972-4. |
| [12] |
P. Mörters and Y. Peres, Brownian Motion, Cambridge Series in Statistical and Probabilistic Mathematics, Cambridge University Press, 2010, URL http://books.google.co.uk/books?id=e-TbA-dSrzYC.
doi: 10.1017/CBO9780511750489. |
| [13] |
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. |
| [14] |
D. Talay, Stochastic Hamiltonian systems: Exponential convergence to the invariant measure, and discretization by the implicit Euler scheme, Markov Process. Related Fields, 8 (2002), 163–198, Inhomogeneous random systems (Cergy-Pontoise, 2001). |
| [15] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
| [16] |
C. Villani,
Optimal Transport: Old and New, vol. 338 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 2009.
doi: 10.1007/978-3-540-71050-9. |
| [17] |
R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC. |
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