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December  2018, 11(6): 1427-1441. doi: 10.3934/krm.2018056

Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus

Helge Dietert , Josephine Evans and  Thomas Holding

Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK

Received  April 2016 Revised  July 2017 Published  June 2018

Fund Project: The authors were supported by the UK Engineering and Physical Sciences Research Council (EPSRC) grant EP/H023348/1 for the University of Cambridge Centre for Doctoral Training, the Cambridge Centre for Analysis).

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.

Keywords: Convergence to equilibrium, hypocoercivity, Monge-Kantorovich-Wasserstein $ \mathcal{W}_2$ distance, Fokker-Planck equation, torus, co-adapted couplings.
Mathematics Subject Classification: 60J60; 35Q84, 60H30, 82C31.
Citation: Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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W. S. Kendall, Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.  doi: 10.3150/14-BEJ596.  Google Scholar

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K. Kuwada, Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.  doi: 10.1214/EJP.v14-634.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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).  Google Scholar

[15]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

[17]

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[9]

W. S. Kendall, Coupling, local times, immersions, Bernoulli, 21 (2015), 1014-1046.  doi: 10.3150/14-BEJ596.  Google Scholar

[10]

K. Kuwada, Characterization of maximal Markovian couplings for diffusion processes, Electron. J. Probab., 14 (2009), 633-662.  doi: 10.1214/EJP.v14-634.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).  Google Scholar

[15]

C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp. doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  Google Scholar

[17]

R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, USA, 2001, URL https://books.google.co.uk/books?id=4cI5136OdoMC.  Google Scholar

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